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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OXI 3LB, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at www.cambridge.org/mathematics 241 Surveys in combinatorics, 1997, R.A. BAILEY (ed) 242 Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) 243 Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) 244 Model theory of groups and automorphism groups, D. EVANS (ed) 245 Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al 246 p-Automorphisms of finite p-groups, E.I. KHUKHRO 247 Analytic number theory, Y. MOTOHASHI (ed) 248 Tame topology and o-minimal structures, L. VAN DEN DRIES 249 The atlas of finite groups: ten years on, R. CURTIS & R. WILSON (eds) 250 Characters and blocks of finite groups, G. NAVARRO 251 Gröbner bases and applications, B. BUCHBERGER & F. WINKLER (eds) 252 Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO & R. STÖHR (eds) 253 The q-Schur algebra, S. DONKIN 254 Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) 255 Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds) 256 Aspects of Galois theory, H. VÖLKLEIN et al 257 An introduction to noncommutative differential geometry and its physical applications 2ed, J. MADORE 258 Sets and proofs, S.B. COOPER & J. TRUSS (eds) 259 Models and computability, S.B. COOPER & J. TRUSS (eds) 260 Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL et al 261 Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL et al 262 Analysis and logic, C.W. HENSON, J. IOVINO, A.S. KECHRIS & E. ODELL 263 Singularity theory, B. BRUCE & D. MOND (eds) 264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds) 265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART 267 Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds) 268 Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJÖSTRAND 269 Ergodic theory and topological dynamics, M.B. BEKKA & M. MAYER 270 Analysis on Lie groups, N.T. VAROPOULOS & S. MUSTAPHA 271 Singular perturbations of differential operators, S. ALBEVERIO & P. KURASOV 272 Character theory for the odd order theorem, T. PETERFALVI 273 Spectral theory and geometry, E.B. DAVIES & Y. SAFAROV (eds) 274 The Mandlebrot set, theme and variations, TAN LEI (ed) 275 Descriptive set theory and dynamical systems, M. FOREMAN et al 276 Singularities of plane curves, E. CASAS-ALVERO 277 Computational and geometric aspects of modern algebra, M.D. ATKINSON et al 278 Global attractors in abstract parabolic problems, J.W. CHOLEWA & T. DLOTKO 279 Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS & A. NOGUEIRA (eds) 280 Characters and automorphism groups of compact Riemann surfaces, T. BREUER 281 Explicit birational geometry of 3-folds, A. CORTI & M. REID (eds) 282 Auslander-Buchweitz approximations of equivariant modules, M. HASHIMOTO 283 Nonlinear elasticity, Y. FU & R.W. OGDEN (eds) 284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SÜLI (eds) 285 Rational points on curves over finite, fields, H. NIEDERREITER & C. XING 286 Clifford algebras and spinors 2ed, P. LOUNESTO 287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE et al 288 Surveys in combinatorics, 2001, J. HIRSCHFELD (ed) 289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE

290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 321 322 323 324 325 326 328 329 330 331 332 333 334 335 336 337 338 339 340 341

Quantum groups and Lie theory, A. PRESSLEY (ed) Tits buildings and the model theory of groups, K. TENT (ed) A quantum groups primer, S. MAJID Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK Introduction to the theory of operator spaces, G. PISIER Geometry and Integrability, L. MASON & YAVUZ NUTKU (eds) Lectures on invariant theory, I. DOLGACHEV The homotopy category of simply connected 4-manifolds, H.-J. BAUES Higher operads, higher categories, T. LEINSTER Kleinian Groups and Hyperbolic 3-Manifolds Y. KOMORI, V. MARKOVIC & C. SERIES (eds) Introduction to Möbius Differential Geometry, U. HERTRICH-JEROMIN Stable Modules and the D(2)-Problem, F.E.A. JOHNSON Discrete and Continuous Nonlinear Schrödinger Systems, M. J. ABLORWITZ, B. PRINARI & A. D. TRUBATCH Number Theory and Algebraic Geometry, M. REID & A. SKOROBOGATOV (eds) Groups St Andrews 2001 in Oxford Vol. 1, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2001 in Oxford Vol. 2, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Peyresq lectures on geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) Surveys in Combinatorics 2003, C. D. WENSLEY (ed) Topology, geometry and quantum field theory, U. L. TILLMANN (ed) Corings and Comodules, T. BRZEZINSKI & R. WISBAUER Topics in Dynamics and Ergodic Theory, S. BEZUGLYI & S. KOLYADA (eds) Groups: topological, combinatorial and arithmetic aspects, T. W. MÜLLER (ed) Foundations of Computational Mathematics, Minneapolis 2002, FELIPE CUCKER et al (eds) Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC Structured ring spectra, A. BAKER & B. RICHTER (eds) Linear Logic in Computer Science, T. EHRHARD et al (eds) Advances in elliptic curve cryptography, I. F. BLAKE, G. SEROUSSI & N. SMART Perturbation of the boundary in boundary-value problems of Partial Differential Equations, D. HENRY Double Affine Hecke Algebras, I. CHEREDNIK Surveys in Modern Mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N. C. SNAITH (eds) Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) Singularities and Computer Algebra, C. LOSSEN & G. PFISTER (eds) Lectures on the Ricci Flow, P. TOPPING Modular Representations of Finite Groups of Lie Type, J. E. HUMPHREYS Fundamentals of Hyperbolic Manifolds, R. D. CANARY, A. MARDEN & D. B. A. EPSTEIN (eds) Spaces of Kleinian Groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) Noncommutative Localization in Algebra and Topology, A. RANICKI (ed) Foundations of Computational Mathematics, Santander 2005, L. PARDO, A. PINKUS, E. SULI & M. TODD (eds) Handbook of Tilting Theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) Synthetic Differential Geometry 2ed, A. KOCK The Navier-Stokes Equations, P. G. DRAZIN & N. RILEY Lectures on the Combinatorics of Free Probability, A. NICA & R. SPEICHER Integral Closure of Ideals, Rings, and Modules, I. SWANSON & C. HUNEKE Methods in Banach Space Theory, J. M. F. CASTILLO & W. B. JOHNSON (eds) Surveys in Geometry and Number Theory, N. YOUNG (ed) Groups St Andrews 2005 Vol. 1, C.M. CAMPBELL, M. R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2005 Vol. 2, C.M. CAMPBELL, M. R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Ranks of Elliptic Curves and Random Matrix Theory, J. B. CONREY, D. W. FARMER, F. MEZZADRI & N. C. SNAITH (eds)

Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues HAYNES R. MILLER Massachusetts Institute of Technology DOUGLAS C. RAVENEL University of Rochester

CAMBRIDGE

UNIVERSITY

PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521700405 © Cambridge University Press 2007 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2007 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloguing in Publication data ISBN 978-0-521-70040-5 paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface Charles Thomas, 1938–2005

page vii xiii 1

1.

Discrete torsion for the supersingular orbifold sigma genus Matthew Ando and Christopher P. French

2.

Quaternionic elliptic objects and K3-cohomology Jorge A. Devoto

26

3.

The M-theory 3-form and E8 gauge theory Emanuel Diaconescu, Daniel S. Freed and Gregory Moore

44

4.

Algebraic groups and equivariant cohomology theories John P. C. Greenlees

89

5.

Delocalised equivariant elliptic cohomology (with an introduction by Matthew Ando and Haynes Miller) Ian Grojnowski

111

6.

On finite resolutions of K(n)-local spheres Hans-Werner Henn

122

7.

Chromatic phenomena in the algebra of BP∗ BP-comodules Mark Hovey

170

8.

Numerical polynomials and endomorphisms of formal group laws Keith Johnson v

204

vi

Contents

Thom prospectra for loopgroup representations Nitu Kitchloo and Jack Morava

214

10.

Rational vertex operator algebras Geoffrey Mason

239

11.

A possible hierarchy of Morava K-theories Norihiko Minami

255

12.

The motivic Thom isomorphism Jack Morava

265

13.

Toward higher chromatic analogs of elliptic cohomology Douglas C. Ravenel

286

14.

What is an elliptic object? Graeme Segal

306

15.

Spin cobordism, contact structure and the cohomology of p-groups C. B. Thomas

318

Brave New Algebraic Geometry and global derived moduli spaces of ring spectra Bertrand Toen and Gabriele Vezzosi

325

9.

16.

17.

The elliptic genus of a singular variety Burt Totaro

360

Preface

A workshop entitled “Elliptic Cohomology and Chromatic Phenomena” was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, on 9–20 December, 2002. The workshop attracted over 75 participants from thirteen nations. The event, an EU Workshop, was the final one in INI’s program New Contexts for Stable Homotopy Theory held in the fall of that year. During the first week nineteen talks described a wide range of perspectives on elliptic genera and elliptic cohomology, including homotopy theory, vertex operator algebras, 2-vector spaces, and open string theories. The second week featured ten talks with a more specifically homotopy theoretic focus, but encompassing the higher chromatic variants of elliptic cohomology. This was the first conference on elliptic cohomology since the one organized by Peter Landweber at the Institute for Advanced Study in Princeton in 1986. The proceedings of that conference were published in [10]. The breadth of that volume is an indication of the multifaceted nature of the subject. From the start it has provided a meeting point for algebraic topology, number theory, and theoretical physics, playing in the present era a role analogous to the role of K-theory in the second half of the last century. Landweber’s introduction to that volume, together with Serge Ochanine’s contribution [13] to it, provide good introduction to the origins of this subject. The starting point was the study of genera of spin manifolds. A genus is a multiplicative bordism invariant, with values in some commutative ring. A classical result asserts that signature provides the universal example of a rational genus of oriented manifolds with the property that it is multiplicative for oriented fiber bundles with connected compact Lie structure groups. Ochinine showed that if a spin genus has this property then it is “elliptic.” In this description, Ochanine used the observation due to Quillen (understood also by Novikov in the rational case) that a genus for almost complex manifolds with values in a ring R is the same as a formal group law over R. Ochanine called a genus “elliptic” if it factors through the genus representing a formal group law used by Euler in 1756 to describe the addition law for certain elliptic integrals. Such a genus automatically factors uniquely through the map from unitary to spin bordism. The value ring can be regarded as a ring of modular forms. vii

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Ochanine conjectured that the converse is true as well, and Edward Witten provided intuitive physics rationale for this conjecture in an article [23] in the IAS conference proceedings, interpreting the elliptic genus as the charactervalued index of a certain hypothetical signature operator on the free loop space of the manifold. The conjecture was proven by Raoul Bott and Cliff Taubes [3], and later by a different method by Kefeng Liu [12]. The traditional genera occur as indexes of differential operators, and hence extend to families versions. The topological manifestation of this phenomenon is that the genus is the effect on homotopy groups of a map of ring specta. Landweber [9] had established a general criterion guaranteeing such a geometric realization, and Landweber, Bob Stong, and Doug Ravenel [11] showed that after suitable localization the Ochanine genus corresponds to a map of ring spectra MU → Ell. The target spectrum represented the first example of an elliptic cohomology theory. One of the principal attractions of elliptic cohomology is the possibility that it admits a geometric description, in analogy with the description of Ktheory via vector bundles. This dream was beautifully enunciated by Graeme Segal in his early survey [17], and expanded on in the his 1988 Oxford University lecture notes, now published as [18]. The idea, very crudely, is to regard a vector bundle with connection as associating a vector space to each point and an isomorphism to each path; the next step would then be to instead associate a Hilbert space to each loop and an operator to each bordism between loops. This is a field theory, and the hope has been that the physics of field theories would provide the ideas necessary to construct this mathematical object, which would, in turn, provide a framework for further physics. These hopes are reviewed in the paper by Segal in this volume. The torch is currently being carried by Stephan Stolz and Peter Teichner [19]. A representative of the string physics arising from the considerations which originally led Witten to his genera is provided by Emmanuel Diaconescu, Dan Freed, and Greg Moore, in their contribution. Other geometric constructions have been attempted. Po Hu and Igor Kriz [6] pursue the conformal field theory approach, while Nils Baas, Bjorn Dundas, and John Rognes [2] start from the philosophy that elliptic cohomology should be a form of the K-theory of the complex K-theory spectrum, and obtain a functor by considering the theory of 2-vector spaces. (This theory was in fact created by Mikhail Kapranov and Vladimir Voevodsky [8] with this application in mind.) Any geometric source of modular forms provides a hint for geometric sources of elliptic cohomology. One such source is vertex operator algebras. It has been globalized by Vassily Gorbunov, Fyodor

Preface

ix

Malikov, and Vadim Schechtman, [5], who use them identify a geometric object associated with a BU 6 structure. A useful introduction to vertex operator algebras, focusing on their modular properties, is provided by Geoffrey Mason’s article in this volume. Another approach to a geometric interpretation of elliptic cohomology starts from Witten’s interpretation of the elliptic genus and develops the notion that the corresponding cohomology theory should be a form of equivariant K-theory of the free loop space. The article of Nitu Kitchloo and Jack Morava pursues this idea. Further evidence of the geometric depth of elliptic genera is provided by the theorems of Lev Borisov, Anatoly Libgober, and Burt Totaro. Igor Krichever and Gerald Höhn had considered certain two-variable variants of the Ochanine genus, taking values in Jacobi forms and exhibiting rigidity for complex manifolds. It turns out that the same genus enjoys quite a different universal property. It is unchanged when one passes from one resolution of singularities of a given complex projective variety to another. One must either specify the process of passing from one to another (by a certain form of complex surgery, “classical flops”) or restrict to certain types of resolution of singularities. Burt Totaro reviews these theorems in his contribution. After early work of Jens Franke [4], Hopkins, Haynes Miller, Paul Goerss, and Charles Rezk have constructed an étale sheaf of commutative ring spectra over the elliptic modular stack. This construction has not yet been fully documented, though it has been the subject of courses ([16]) and a weeklong workshop in Mainz, Germany, in October, 2003. The paper of Bertrand Toën and Gabriele Vezzosi puts this construction into the general context of derived moduli spaces. The rigidity of this construction allows one to form a homotopy inverse limit, which is the spectrum tmf of “topological modular forms.” One of the hallmarks of a geometrically defined cohomology theory is the presence of equivariant versions of the theory. An analytic circle-equivariant theory associated to an elliptic curve E over C, taking values in sheaves of algebras over E, was constructed in 1994 by Ian Grojnowski. This construction has had a large impact on the subject, and his announcement appears here for the first time along with a historical introduction. Quillen [15] described the connection between even multiplicative cohomology theories and formal groups. It appears that lifting the cohomology theory to a circle equivariant form is related to extending the formal group to an algebraic group. This idea is explored in John Greenlees’s contribution. A variant of the problem of an equivariant extension is the possibility of an orbifold theory. Physicists had studied an orbifold elliptic genus, and this

x

Preface

story is brought into contact with the homotopy theoretic enrichment of the Witten genus due to Ando, Hopkins, and Neil Strickland [1] in the paper of Matthew Ando and Christopher French. Elliptic spectra represent rather computable cohomology theories, which carry deep information about stable homotopy in their coefficients and in their operations. Following the approach set out by Frank Adams, these operations are most conveniently studied by means of the self homology Ell∗ Ell. Relations between this object and numerical polynomials (familiar from Adams’s study of K∗ K) are described in Keith Johnson’s article. The second part of the workshop focused on higher analogues of elliptic cohomology. From the chromatic viewpoint on stable homotopy theory, elliptic cohomology is the third in a sequence of types of theories, starting with rational cohomology and K-theory; and, locally at a prime at least, this sequence can be continued. One still wishes to realize these still higher “height” theories by some geometric data. A futuristic venture, representing joint work with the late Charles Thomas, is described by Jorge Devoto in this volume. Here the objective is to construct a variant of elliptic cohomology using moduli of K3-surfaces, and geometrically realize this theory by means of a quaternionic field theory. A step towards a different extension of the notion of elliptic cohomology is proposed in Ravenel’s article, in which he observes that the Jacobians of certain curves over Fp have one-dimensional formal factors of high height. The theory of operations of these higher height theories is the topic of the review article here by Mark Hovey. More detailed computations and homotopy theoretic constructions carried out by Goerss, Hans-Werner Henn, Mark Mahowald, and Rezk in the case of elliptic cohomology are reviewed and in part extended to higher height by Henn in his paper. Nori Minami explores a variety of analogies and connections between the various localizations of stable homotopy theory associated with these higher height theories. We dedicate this volume to the memory of our friend Professor Charles Thomas of the University of Cambridge, who died suddenly on December 16, 2005. Charles was constant in his belief in the geometric promise of elliptic cohomology. His book Elliptic Cohomology [20] and his review article [21] were important contributions to this development. His generosity and his warmth are missed by the whole community. We are pleased to publish here a paper derived from the lecture he gave at the workshop. Cambridge, MA, [email protected] Rochester, NY, [email protected] October 25, 2006

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References [1] Matthew Ando, Michael Hopkins, and Neil Strickland, Elliptic spectra, the Witten genus, and the theorem of the cube, Invent. math. 146 (2001) 595–687. [2] Nils A. Baas, Bjørn Ian Dundas, and John Rognes, Two-vector bundles and forms of elliptic cohomology. In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages 18–45. Cambridge Univ. Press, Cambridge, 2004. [3] Raoul Bott and Clifford Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989) 137–186. [4] Jens Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992) 43–65. [5] Vassily Gorbunov, Fyodor Malikov, and Vadim Schechtman, Gerbes of chiral differential operators, Math. Res. Lett. 7 (2000) 55–66. [6] P. Hu and I. Kriz, Conformal field theory and elliptic cohomology. Adv. Math., 189(2):325–412, 2004. [7] M. J. Hopkins, Algebraic topology and modular forms, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press. [8] M. Kapranov and V. A. Voevodsky, 2-categories and Zamolodchikov tetrahedra equations, in Algebraic groups and their generalizations: quantum and infinitedimensional methods (University Park, PA, 1991), Proc. Symp. Pure Math. 56 (1994) 177–259. [9] P. S. Landweber, Homological properties of comodules over MU∗ MU and BP∗ BP, Amer. J. Math 98 (1976) 591–610. [10] P. S. Landweber, editor, Elliptic curves and modular forms in algebraic topology, volume 1326 of Lecture Notes in Mathematics, Berlin, 1988. Springer-Verlag. [11] P. S. Landweber, D. C. Ravenel, and R. E. Stong, Periodic cohomology theories defined by elliptic curves. In Mila Cenkl and Haynes Miller, editors, The ˇ Cech Centennial, volume 181 of Contemporary Mathematics, pages 317–338, Providence, Rhode Island, 1995. American Mathematical Society. [12] Kefeng Liu, On SL2 Z and topology, Math. Res. Lett. 1 (1994) 53–64. [13] Serge Ochanine, Genres elliptiques equivariants, in P. S. Landweber, editor, Elliptic curves and modular forms in algebraic topology, Lecture Notes in Mathematics 1326, 1988, Springer-Verlag. [14] Serge Ochanine, Sur les genres multiplicatifs définis par des intégrales elliptiques. Topology, 26(2):143–151, 1987. [15] D. G. Quillen, On the formal group laws of oriented and unoriented cobordism theory. Bulletin of the American Mathematical Society, 75:1293–1298, 1969. [16] Charles Rezk, Supplementary notes for Math 512, Course Notes, Northwestern University, 2002. [17] G. Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others), Séminaire Bourbaki, Vol. 1987/88, Astérisque 161–162 (1989) 187–201. [18] Graeme Segal, The definition of conformal field theory. In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages 247–343. Cambridge Univ. Press, Cambridge, 2004.

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[19] Stephan Stolz and Peter Teichner, What is an elliptic object? In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages. Cambridge Univ. Press, Cambridge, 2004. [20] Charles B. Thomas, Elliptic cohomology. The University Series in Mathematics, Kluwer Academic/Plenum Publishers, New York, 1999. [21] Charles B. Thomas, Elliptic cohomology. In Survesy on surgery theory, Vol. 1 pages 409–439 Ann. of Math. Studies 145, Princeton University Press, Princeton, 2000. [22] Burt Totaro, Chern numbers for singular varieties and elliptic homology. Ann. of Math. (2), 151(2):757–791, 2000. [23] Edward Witten, The index of the Dirac operator in loop space. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math. 1326 (1988) 161–181.

Charles Thomas, 1938–2005

Charles B. Thomas, Professor of Algebraic Topology in the University of Cambridge died on 16 December 2005. Although some ill health had come his way, his death came as a shock, particularly to those who wished him well at his retirement dinner but nine days earlier. He was born on 17 August 1938 and educated at the Benedictines’ Douai School near Reading. After two years’ service in the Royal Air Force he entered Trinity College, Cambridge, in 1958. An initial intention to read physics was soon converted into a life-long commitment to the study of mathematics. He worked for his PhD initially at Trinity College under D.B.A. Epstein and then under A. Dold at Heidelberg (where he met his wife, Maria). He was a Research Associate at Cornell University (1965–1967), a Lecturer at the University of Hull (1967–1969) and a Lecturer at University College London (1969–1979). He returned to Cambridge as a University Lecturer and Fellow of Robinson College in 1979. Charles Thomas’ research concerned the interplay between algebra (in particular the intricacies of finite group theory), algebraic topology, number theory, the topology of manifolds and various structures in differential geometry. He studied finite group actions on spheres and homotopy spheres and worked on many aspects of the spherical space form problem. His study of 3-manifold groups and Poincaré complexes culminated in his book Elliptic structures on 3-manifolds. Work on characteristic classes, classifying spaces and the cohomology of finite groups led to another book; problems on the cohomology of a wide range of types of group were a continuing interest to him. In two papers in the 1970s Charles was one of the first to note the significance of contact structures on smooth manifolds. He returned to this topic with renewed vigour some twenty years later when contact and symplectic structures became fashionable. Aspects of his research led to his writing books on elliptic cohomology, on differential manifolds (written with D. Barden) and on representation theory. He edited the volumes of the proceedings of xiii

xiv

Charles Thomas

several meetings he had organised and was particularly proud to be co-editor of a selection of the works of J.F. Adams, whom he particularly admired. Several of Charles’ former PhD students now propagate his high standards to students of their own. Charles valued academic excellence. A list of his mathematical achievements gives only a limited impression of his intellect. He spoke something of most of the main European languages and was fluent in French and German. He translated into English four mathematics books from German and one from French. He could converse on any academic subject but had a passion for medieval European and Byzantine history. Whilst giving first priority to research, Charles did his share of university administration. Recently, he was Chairman of the Faculty Board and editor of the Mathematical Proceedings of the Cambridge Philosophical Society. His careful lecturing was illuminated by his immaculate handwriting on the blackboard and on duplicated notes; an occasional stutter allowed the student a moment to catch up. He planned in his retirement to continue his work partly in Cambridge, partly in California at Santa Cruz. His wife, two sons, a daughter and a very new grandson survive him. Raymond Lickorish Slightly edited, first appearing in the Newsletter of the London Mathematical Society (No. 345 February 2006). Used by permission.

Charles Thomas, Hong Kong, 2002

DISCRETE TORSION FOR THE SUPERSINGULAR ORBIFOLD SIGMA GENUS MATTHEW ANDO AND CHRISTOPHER P. FRENCH Abstract. The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of [HKR00] to the Borel-equivariant genus associated to the sigma orientation of [AHS01] to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the same sort of formula as the orbifold “two-variable” genus of [DMVV97] and [BL02]. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable genus in the formulae of [And03] to define an analytic equivariant genus in Grojnowski’s equivariant elliptic cohomology, and we show that this gives precisely the orbifold two-variable genus. The second purpose of this paper is to study the effect of varying the BU h6i-structure in the Borelequivariant sigma orientation. We show that varying the BU h6i structure by a class in H 3 (BG; Z), where G is the orbifold group, produces discrete torsion in the sense of [Vaf85]. This result was first obtained by Sharpe [Sha], for a different orbifold genus and using different methods.

1. Introduction Let E be an even periodic, homotopy commutative ring spectrum, let C be an elliptic curve over SE = spec π0 E, and let t be an isomorphism of formal groups b∼ t:C = spf E 0 (CP ∞ ), so that C = (E, C, t) is an elliptic spectrum in the sense of [Hop95, AHS01]. In [AHS01], Hopkins, Strickland, and the first author construct a map of homotopy commutative ring spectra σ(C) : MU h6i → − E called the sigma orientation; it is conjectured in [Hop95] that this map is the restriction to MU h6i of a similar map MOh8i → E. (MU h6i is the bordism Date: Version 1.7, July 2005. We thank the participants in the BCDE seminar at UIUC, particularly David Berenstein and Eric Sharpe, for teaching us the physics which led to this paper. Ando was supported by NSF grants DMS—0071482 and DMS—0306429. Part of this work was carried out while Ando was a visitor at the Isaac Newton Institute for Mathematical Sciences. He thanks the Newton Institute for its hospitality.

1

2

Ando and French

theory of SU -manifolds with a trivialization of the second Chern class, and MOh8i is the bordism theory of spin manifolds with a trivialization of the characteristic class p21 .) The sigma orientation is natural in the elliptic spectrum, and, if KTate = (K[[q]], Tate, t) is the elliptic spectrum associated to the Tate elliptic curve, then the map of homotopy rings π∗ MU h6i → π∗ KTate

(1.1)

is the restriction from π∗ MSpin of the Witten genus. Explicitly, let M be a Riemannian spin manifold, and let D be its Dirac operator. Let T denote the tangent bundle of M. If V is a (real or complex) vector bundle over M, let V C be the complex vector bundle V C = V ⊗ C. R

If V is a complex vector bundle, let rV be the reduced bundle rV = V − rank V , and let St V =

X

tk S k V

k≥0

be the indicated formal power series in the symmetric powers of V . The operation V 7→ St V extends to an exponential operation K(X) → K(X)[[t]] because of the formula St (V ⊕ W ) = (St V )(St W ). The Witten genus of M is given by the formula w(M) = ind D ⊗

O

Sqk (rT C )

k≥1

and the diagram

!

∈ Z[[q]],

(1.2)

π∗ MU h6i

/ π∗ MSpin NNN NNN w NN π∗ σ(KTate ) NNN&

Z[[q]]

commutes [AHS01]. The Witten genus first arose in [Wit87], where Witten showed that various elliptic genera of a manifold M are essentially one-loop amplitudes of quantum field theories of closed strings moving in M. Locally on a spin manifold M, the quantum field theory associated to w(M) is a conformal field theory, and the obstruction to assembling a conformal field theory globally on M is c2 M.

Discrete torsion and the sigma genus

3

1

This gives a physical proof that, if c2 M = 0, then w(M) is the q-expansion of a modular form. Suppose that M is an SU -manifold. The formula (1.2) shows that the Witten genus is an invariant of the spin structure of M. On the other hand the sigma orientation depends on a choice of BU h6i structure, that is, a lift in the diagram BU h6i x

x

x

x

x<

c2 / BSU / K(Z, 4). M It is an interesting problem to understand how the orientation depends on this choice. The fibration sequence ι

c

2 K(Z, 3) → − BU h6i → BSU − → K(Z, 4)

shows that a lift exists precisely when c2 (M) = 0, and that the set of lifts is a quotient of H 3 (M; Z). The dependence on the choice of lift appears to have an explanation in string theory. The action for the QFT described by the Witten genus is a function on the space of maps X:Σ→M of 2-dimensional surfaces Σ to M. If the theory is anomaly-free, that is, if c2 M = 0, then one is free to add to the action a term of the form Z X ∗ B, Σ

2

where B ∈ Ω M is a differential 2-form on M (called the “B-field”), provided that H = dB is an integral three-form. It seems clear that the physics of the B-field should account for the variation in the sigma orientation from at least torsion classes in H 3 (M, Z).

This situation has already received a good deal of attention, particularly in the case of orbifolds. Eric Sharpe [Sha] showed that Vafa’s discrete torsion [Vaf85] arises from, as he put it us once, “the action of the orbifold group on the B-field.” Lupercio and Uribe [LU] have explained how discrete torsion arises from a gerbe on a total quotient orbifold M/G with finite orbifold group (and so gives rise to a class in the Borel cohomology group H 3 (MG ; Z)). The techniques used to date in the study of the sigma orientation have quite a different flavor from those used in the study of gerbes on orbifolds. 1There

And03]).

are various ways to understand this obstruction ([Wit87, BM94, GMS00,

4

Ando and French

One goal of the research which led to this paper was to find using the sigma orientation some of the phenomena associated to the B-field. In this direction, we show that varying the BU h6i structure in the orbifold sigma genus of a supersingular elliptic curve indeed produces a variant of discrete torsion. More precisely, suppose that M is a complex manifold with an action by a group G, and suppose that V is a complex G-vector bundle over M. Let T denote the tangent bundle of M. If X is a space, let XG denote the Borel construction EG ×G X. If c1 (TG ) = c1 (VG ) c2 (TG ) = c2 (VG ), then there is a lift in the diagram BU h6i `

w

MG

w

w

(1.3)

w;

w

/ VG −TG

BSU,

and a choice of lift gives a Thom class U (M, `, C)G ∈ E 0 (VG − TG ). The relative zero section together with the Pontrjagin-Thom construction provide a map τ (V )G : E 0 (VG − TG ) → E 0 (−TG ) → E 0 (BG), and τ (V )G (U (M, `, C)G ) ∈ E 0 (BG) is the (Borel) equivariant sigma genus of M twisted by V (see §4). To get from it an “orbifold” genus taking its values in E 0 , we use the character theory of Hopkins, Kuhn, and Ravenel ([HKR00]; see also §5). It associates to a pair (g, h) of commuting elements of G a ring homomorphism Ξg,h : E 0 (BG) → D, where D is a complete local E-algebra which depends on the formal group of the spectrum E. It turns out that the quantity X σorb (M, `, C)G = Ξg,h τ (V )G (U (M, `, C)G ) (1.4) gh=hg

takes its values in E 0 ; we call it the orbifold sigma genus of M twisted by V (see §6).

There is already an extensive literature on the subject of “orbifold elliptic genera”, particularly the “two-variable” elliptic genus of [Kri90, EOTY89]; see for example [DMVV97, BL02]. In §6, we show that the formula (1.4) is formally analogous to the formula for the orbifold two-variable genus. More

Discrete torsion and the sigma genus

5

precisely, by applying the topological Riemann-Roch formula to the expression (1.4) and using the Thom class associated to the the two-variable elliptic genus in place of U , we obtain a formula nearly identical to that of [BL02]. Unfortunately our use of the topological Riemann-Roch formula in this context, while illuminating, is not useful for calculation, because we work with the Borel-equivariant elliptic cohomology associated to a supersingular elliptic curve, which is a highly completed situation. In order to locate the orbifold two-variable genus more precisely in the setting of equivariant elliptic cohomology, we consider in §7 the case of a finite cyclic group G = T[n] ⊂ T. We use the principle suggested by Shapiro’s Lemma to define def

EG (X) = ET (T ×G X), where ET is the uncompleted analytic equivariant elliptic cohomology of Grojnowski. We adapt the formulae in [And03], which descends from [Ros01, AB02], to write down an euler class in EG (X). The associated genus Ellan (M, G) takes its value in Γ(EG (∗)) ∼ = Γ(OC[n] ) ∼ = CG×G , and we prove the following. Theorem 1.5. Suppose that G is a finite cyclic group. Summing the analytic equivariant two-variable genus over the torsion points of the elliptic curve gives the orbifold two-variable genus: more precisely, we have 1 X Ellorb (X, G) = Ellan (M, G, g, h). |G| gh=hg We were pleased to be able to confirm that orbifold elliptic genera are (in the cyclic case) so simply obtained from equivariant elliptic genera. It would be interesting to use this observation to investigate more subtle properties of orbifold genera, such as, for example, the “McKay correspondence” of Borisov and Libgober. The rest of the paper is devoted to the study of the dependence of the orbifold sigma genus on the choice ` of BU h6i structure in (1.3). Suppose that we have chosen an element u ∈ H 3 (BG; Z), represented as a map u : BG → K(Z, 3). If π : MG → BG denotes the projection in the Borel construction, then we obtain an element π ∗ u = uπ ∈ H 3 (MG ; Z), and ` + ιuπ is another BU h6i-structure on VG − TG . In §8, we use the character theory and the sigma orientation to associate to u an alternating bilinear map δ = δ(u, C) : G2 → D × , where G2 denotes the set of pairs of commuting elements of G of p-power order. In §9 we obtain the

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Ando and French

Theorem 1.6. The orbifold sigma genus associated to the BU h6i structure `+ιuπ is related to the equivariant sigma genus associated to ` by the formula X σorb (M, ` + ιuπ, C)G = δ(g, h)Ξg,h τ (V )G (U (M, `, C)G ). gh=hg

In [Vaf85], Vafa observed that if φ=

X

φg,h

gh=hg

is an orbifold elliptic genus associated to a theory of strings on M, and if c = c(g, h) is a 2-cocycle with values in U (1), then X c(g, h)φg,h (1.7) gh=hg

is again modular; he called this phenomenon “discrete torsion”. Eric Sharpe [Sha] showed that the genus (1.7) arises from adding a B-field on the orbifold M/G. As expressed by Lupercio and Uribe, this is a U (1)-gerbe B on the orbifold M/G, whose associated cohomology class [B] ∈ H 3 (MG ; Z) satisfies [B] = [c] ∈ H 3 (MG ; Z), where [c] is the cohomology class in MG obtained from c by pulling back along MG → BG. Our result shows that varying the BU h6i-structure of MG by an element u ∈ H 3 (BG) has a similar effect on the orbifold sigma genus. When G is an abelian group of order dividing n = ps , the map δ may be viewed as a two-cocycle on G with values in D × [n] ∼ = Z/n, and as such it represents a 3 H (BG). It is not quite the cohomology cohomology class in H 2 (BG; Z/n) ∼ = class u: instead, as we shall see in §10, if c is a 2-cocycle representing u ∈ H 3 (BG; Z) ∼ = H 2 (BG; Z/n), then δ(g, h) = c(g, h) − c(h, g). 2. The sigma orientation and the sigma genus In this section we recall some results from [AHS01]. Definition 2.1. An elliptic spectrum consists of (1) an even, periodic, homotopy commutative ring spectrum E with formal group PE = spf E 0 CP ∞ over π0 E; (2) a generalized elliptic curve C over π0 E; b of PE with the formal completion of C. (3) an isomorphism t : PE → − C

Discrete torsion and the sigma genus

7

A map (f, s) of elliptic spectra E1 = (E1 , C1 , t1 ) → − E2 = (E2 , C2 , t2 ) consists of a map f : E1 → − E2 of multiplicative cohomology theories, together with an isomorphism of elliptic curves s

C2 → − (π0 f )∗ C1 , extending the induced isomorphism of formal groups. Theorem 2.2. An elliptic spectrum C = (E, C, t) determines a map σ(C) : MU h6i → E of homotopy-commutative ring spectra. The association C 7→ σ(C) is modular, in the sense that if (f, s) : C1 → C2 is a map of elliptic spectra, then the diagram σ(C )

1 /E 1 GG GG GG f G σ(C2 ) GG#

MU h6i

E2

commutes up to homotopy. If KTate = (K[[q]], Tate, t) is the elliptic spectrum associated to the Tate curve, then the diagram MU h6i

/ MSpin KK KK KK w K σ(KTate ) KK%

K[[q]] commutes, where w is the orientation associated to the Witten genus. 3. The sigma genus Definition 3.1. Let W be a virtual complex vector bundle on a space M. A BU h6i-structure on W is a map `:M → − BU h6i such that the composition `

M→ − BU h6i → − BU classifies rW . Now let M be a connected compact closed manifold with complex tangent bundle T , and let V be another complex vector bundle on M. Let d = 2 rankC T − V Let τ (V ) : S 0 −−−→ M −T → − M V −T P −T

ζ

8

Ando and French

be the composition of the Pontrjagin-Thom map with the relative zero section. If ` : M → BU h6i is a BU h6i-structure on V − T , and if C = (E, C, t) is an elliptic spectrum, let U (M, `, C) ∈ E −d (M V −T ) be the class given by the map σ(C)

`

U (M, `, C) : Σd M V −T → − MU h6i −−−→ E. Definition 3.2. The sigma genus of ` in C is the element def

σ(M, `, C) = τ (V )∗ (U (M, `, C)) ∈ E −d (S 0 ) = πd E. Example 3.3. Suppose that c1 T = 0 = c2 T , so that T itself admits a BU h6istructure, say ` : M → BU h6i, and d = 2 dim M. Then we have a Thom isomorphism E 0 (M) ∼ = E −d (M −T ) 1 7→ U (M, `, C). and the usual Umkehr map π!M associated to the projection πM : M → ∗ is the composition ∼ =

π!M : E 0 (M) − → E −d (M −T ) −−−→ E −d (S 0 ) ∼ = πd E. P −T

Thus σ(M, `, C) = π!M (1) = πd (σ(C))([M]) ∈ πd E is just the genus of M with BU h6i-structure `, associated to the sigma orientation σ(C) MU h6i −−−→ E. 4. The Borel-equivariant sigma genus Now suppose that G is a compact Lie group, and, if X is a space, let XG denote the Borel construction def

XG = EG ×G X. Suppose that G acts on the compact connected manifold M, that V is an equivariant complex vector bundle, and that ` : MG → BU h6i is a BU h6i-structure on the bundle VG −TG . Since TG is the bundle of tangents along the fiber of MG → BG, we have a Pontrjagin-Thom map P −T

BG+ −−−→ (MG )−TG ,

Discrete torsion and the sigma genus

9

and so a map ζ

P −T

τ (V )G : BG+ −−−→ (MG )−TG → − (MG )VG −TG . Let U (M, `, C)G ∈ E −d (MGVG −TG ) be given by the map `

σ(C)

U (M, `, C)G : Σd (MG )VG −TG → − MU h6i −−−→ E. Definition 4.1. The (Borel) equivariant sigma genus of ` in C is the element def

σ(M, `, C)G = τ (V )G (U (M, `, C)G ) ∈ E −d (BG). 5. Character theory The equivariant sigma genus described in §4 is not so familiar, because E (BG) is not. In this section we review the character theory of [HKR00], which gives a sensible way to understand E ∗ (BG). In the next section, we apply the character theory to produce the orbifold sigma genus from the equivariant sigma genus; as we shall see, it is given by the same sort of formula as those for “orbifold elliptic genera” in for example [DMVV97, BL02] ∗

We suppose that E is an even periodic ring spectrum, and that π0 E is a complete local ring of residue characteristic p > 0. We write P for CP ∞ , so PE = spf E 0 P is the formal group of E. We assume that PE has finite height h. Let Λ∞ = (Zp )h , and for n ≥ 1, let Λn = Λ∞ /pn Λ∞ . If A is an abelian def group, let A∗ = hom(A, C× ) denote its group of complex characters, so for example Λ∗∞ ∼ = (Z[ 1p ]/Z)h . = (Qp /Zp )h ∼ Each λ ∈ Λ∗n defines a map Bλ

BΛn −→ P. Choose a coordinate x ∈ E 0 P. For each λ ∈ Λ∗n , let x(λ) = (Bλ)∗ x ∈ E 0 BΛn . Let S ⊂ E 0 BΛn be the multiplicative subset generated by {x(λ)|λ 6= 0}. Let Ln = S −1 E 0 BΛn , and let Dn be the image of E 0 BΛn in Ln . In other words, Dn is the quotient of E 0 BΛn by the ideal generated by annihilators of euler classes of non-zero characters of Λn . It is clear that Ln and Dn are independent of the choice of coordinate x. Now suppose that G is a finite group. Let α : Λn → G be a homomorphism: specifying such α is equivalent to specifying an h-tuple of commuting elements of G of order dividing pn .

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Ando and French

Definition 5.1. The character map associated to α is the ring homomorphism E 0 Bα

Ξα : E 0 BG −−−→ E 0 BΛn → − Dn . One may check directly from the definition that the map Λn+1 Λn induces maps Dn → Dn+1 Ln → Ln+1 . Let D = colim Dn n

L = colim Ln .

(5.2)

n

Since G is finite, any homomorphism α:Λ→G factors as α

n α:Λ→ − Λn −→ G for sufficiently large n, and we may unambiguously attach a character homomorphism Ξα : E 0 BG → − D such that, for sufficiently large n, the diagram

E 0 BGH

Ξαn

/D n HH HH HH Ξα HH#

D

commutes. A great deal is known about the ring Dn , because it turns out [AHS03] that spf Dn is the scheme of level Λ∗n -structures on the PE . For example, it is easy to check that the action of Aut(Λn ) on E 0 BΛn induces an action of Aut(Λn ) on Dn . Using the description of Dn in terms of level structures, one may prove the following. Proposition 5.3. The ring Dn is finite and faithfully flat over E. If PE is the universal deformation of a formal group of height h (i.e. if E is a Morava E-theory), then Dn is a complete Noetherian local domain, and in that case, and in general if p is regular in π0 E, then Ln = 1p Dn . The structural map E 0 → Dn identifies E 0 with the Aut(Λn )-invariants in Dn . Proof. [Dri74] or [Str97].

Discrete torsion and the sigma genus

11

If w ∈ Aut(Λn ) and α : Λn → G, then we have two homomorphisms from E (BG) to Dn , namely wΞα and Ξαw . 0

Lemma 5.4. The diagram E 0 BGG

Ξα

/ Dn GG GG GG w Ξαw GG#

Dn

commutes.

Corollary 5.5. The expression X

Ξα

α:Λn →G

defines an additive map E 0 BG → π0 E. In the case that the height of PE is two, the sum is over all pairs of commuting elements of G of p-power order. If G is a p-group, then we write X Ξg,h : E 0 BG → − π0 E gh=hg

for the map in the Corollary. 6. The orbifold sigma genus There has been much study of the orbifold version of the two-variable elliptic genus of [EOTY89]; see for example [DMVV97, BL02]. In this section we introduce an orbifold version of the sigma genus, in the case of a supersingular elliptic curve. Our definition is intentionally as simple as possible: we consider only total quotient orbifolds, and then extract the orbifold sigma genus from the Borel genus using the map of Corollary 5.5. Explicitly, suppose that G is a finite group acting on a manifold M with complex tangent bundle T , that V is an equivariant complex vector bundle, and that ` : MG → BU h6i is a BU h6i-structure on the bundle VG − TG . Let C be the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p > 0, and let C = (E, C, t) be the associated elliptic spectrum.

12

Ando and French

Definition 6.1. The orbifold sigma genus of ` in C is the element X def σorb (M, `, C)G = Ξg,h σ(M, `, C)G ∈ π−d E.

(6.2)

gh=hg

The rest of this section is devoted to showing that the formula (6.2) is formally analogous to the formula for the orbifold two-variable genus. In section 7, we show that, in the case of a finite cyclic group, the orbifold twovariable genus is precisely the genus in Grojnowski’s circle-equivariant elliptic cohomology obtained from the characteristic series defining the two-variable genus by following the construction of [Ros01, AB02, And03]. These sections are logically independent of the discussion of discrete torsion and the proof of Theorem 1.6, and readers interested primarily in that formula may prefer to skip to section 8. Our comparison in this section is based on the analogue of the formula (6.2) in the case of a genus given by a complex orientation t : MU → − E, so that E has Thom classes and Umkehr maps for complex vector bundles. If M is a compact manifold of real dimension d, then we write π M for the projection M → ∗, M and πt for the Umkehr map πtM : E ∗ (M) → πd−∗ E. This Umkehr map is often denoted π!M ; our notation emphasizes the dependence on the orientation t. In any case, the genus associated to t is the map Φt : π∗ MU → π∗ E given by the formula Φt (M) = πtM (1). If a compact Lie group G acts on M, then we write π M,G for the projection π M,G : MG → BG, and πtM,G for the associated Umkehr map πtM,G : E 0 (MG ) → − E −d (BG). If G is a finite p-group, then the analogue of our formula (6.2) is the quantity X Φtorb (M) = Ξg,h πtM,G (1). gh=hg

If g and h are commuting elements of G, let def

M (g,h) = M g ∩ M h

13

Discrete torsion and the sigma genus

be the subset of M fixed by both g and h. Let V(g, h) be a normal bundle of M (g,h) in M. We view (g, h) as a homomorphism (g,h)

Λn −−→ G : this makes Λn act trivially on M (g,h) , and we let et (V(g, h)) ∈ E ∗ (BΛn ) ⊗ E ∗ (M (g,h) ) ∼ = E ∗ (EΛn ×Λn M (g,h) ) be the Λn -equivariant euler class of V(g, h) in the orientation t. Recall from (5.2) that L = colim Ln is the colimit of the rings Ln obtained from E ∗ BΛn by inverting the euler classes of non-trivial characters of Λ. Proposition 6.3. Suppose that (g, h) 6= (0, 0). (1) The euler class et (V(g, h)) is a unit of L ⊗E ∗ E ∗ (M (g,h) ). (2) The quantity 1⊗

(g,h) πtM

lies in the subring D ⊂ L. (3) As elements of D we have Ξg,h (πtM,G (1))

=1⊗

1 et (V(g, h))

(g,h) πtM

1 . et (V(g, h))

Thus the orbifold genus of M associated to t is given by the formula X 1 t t M (g,h) Φorb (M) = Φ (M) + 1 ⊗ πt . et (V(g, h)) gh=hg

(6.4)

(g,h)6=(0,0)

Proof. Keeping in mind that M (g,h) is a compact manifold, the first assertion follows by the argument originally due to [AS68]. Now examine the diagram j

BΛ × M (g,h) −−−→ EΛ ×(g,h) M −−−→ EG ×G M M,G (g,h) M,Λ M π y y yπ 1×π i

BΛ The right square is a pull-back, so

BΛ

B(g,h)

−−−−→

B(g, h)∗ πtM,G = πtM,Λ j ∗ . It follows that Ξ(g,h) π!M,G (1) = πtM,Λ (1),

BG

14

Ando and French

considered as an element of D. The fixed-point formula asserts that π!M,Λ (1)

=1⊗

(g,h) π!M

1 et (V(g, h))

in L: but in fact we know that the left-hand side is an element of D ⊂ L. It follows that the right hand side is too, and 1 M,G M (g,h) . Ξ(g,h) π! (1) = 1 ⊗ π! et (V(g, h)) The rest is easy.

The formula (6.4) is the analogue for the t-genus of the orbifold elliptic genera of [DMVV97, BL02]. To see this, let A be the (abelian) subgroup of G generated by g and h, and suppose that V(g, h) decomposes as a sum V(g, h) ∼ = L1 ⊕ · · · ⊕ L r of complex line bundles, with A acting on Li by the character χi . Let e(χi ) ∈ E(BA) be the euler class of the character χi , using the orientation t, and let yi ∈ E(M (g,h) ) be the (non-equivariant) euler class of the line bundle Li . Then Y et (V(g, h)) = yi +F e(χi ). i

We can be even more explicit. Let F (x, y) ∈ E[[x, y]] be the formal group law over E induced by the orientation t. If R is a complete local E-algebra, let us write F (R) for the maximal ideal of R, considered as an abelian group using the power series F to perform addition. Associating to a character λ ∈ Λ∗n its first chern class in E-theory using the orientation t defines a group homomorphism Λ∗n → − F (E(BΛn )), which gives rise to a homomorphism Λ∗n → − F (Dn ); in fact, this is the “level structure” referred to in §5. The dual of the epimorphism Λ → Λn → A

Discrete torsion and the sigma genus

15

is a monomorphism A∗ → Λ∗n which composes with the level structure to give a homomorphism φ : A∗ → Λ∗n → − F (Dn ). By construction, φ(χi ) = e(χi ). If e1 , e2 are a basis for Λ∗n , then we can write χi = a i e 1 + b i e 2 in Λ∗n , where ai , bi ∈ Z/n. If vi = `(ei ) for i = 1, 2, then φ(χi ) = [ai ](v1 ) +F [bi ](v2 ). Our typical summand in the formula (6.4) for the orbifold genus becomes ! Y 1 (g,h) 1 ⊗ π!M . (6.5) y + [a ](v ) + [b ](v ) i F i 1 F i 2 i It is customary to calculate expressions like (6.5) by using the topological Riemann-Roch formula to pass to ordinary cohomology. In fact this approach is not available in our situation. To do so, one introduces the exponential ba → F exp : G

b a (L) such that of the group law F , and finds xi ∈ L ⊗ E(M (g,h) ) and wi ∈ G yi = f (xi )

v1 = f (w1 )

v2 = f (w2 ). However, if v1 = f (w1 ) then 0 = [n]F (v1 ) = f (nw1 ), which implies that nw1 = 0, and, as L is torsion free, we conclude that w1 must be zero! Nevertheless, we shall proceed formally in order to compare our formula with those of [DMVV97, BL02]. We have Y et (V(g, h)) = f (xi + ai w1 + bi w2 ). i

Let uj , j = 1, . . . , r be the roots of the total Chern class of the tangent bundle of M (g,h) : Y c(M (g,h) ) = (1 − uj ). j

Then the Riemann-Roch formula gives ! Z Y uj Y 1 1 M (g,h) πt = . et (V(g, h)) f (uj ) f (xi + ai w1 + bi w2 ) M (g,h) j i

(6.6)

16

Ando and French

In [BL02], Borisov and Libgober use the two-variable elliptic whose exponential is θ(x, τ ) f (x) = , (6.7) θ(x − z, τ ) where 1

x

x

θ(x, τ ) = −iq 8 (e 2 − e− 2 )

Y

(1 − q n )(1 − q n ex )(1 − q n e−x ),

(6.8)

n≥1

τ is a complex number with positive imaginary part, and q = e2πiτ . (We have adopted slightly different conventions regarding factors of 2π. The simplest way to compare is to say that we work with the elliptic curve C/(2πiZ + 2πiτ Z), while they work with the elliptic curve C/(Z + τ Z)) Their expression for the orbifold two-variable genus is Ellorb (M, G) =

1 X Φg,h , |G| gh=hg

where, with our conventions, ! Z Y uj Y 1 Φg,h = ezBi /n . (6.9) f (uj ) f (xi + Ai (1/n) − Bi (τ /n)) M (g,h) j i and the Ai and Bi are integer representatives of ai and bi . This differs from the formal expression (6.6) by only the factors ezbi /n . These factors are familiar from the study of equivariant genera; for example 1/k they are analogous to the factors νs in (11.26) of [BT89], S(c1 (V 1/n )T ) in k ˆ ¯ in (5.16) of [And03]. Their role is to make the (6.17) of [AB02], or u n I(m) expression (6.9) independent of the choice of representatives Ai and Bi . It is necessary to introduce these factors because f is not doubly periodic; instead we have f (x + 2πi` + 2πikτ ) = y −k f (x),

(6.10)

where y = ez , as one checks easily using (6.8). So the expression doesn’t depend on the choice of Ai . If Bi0 = Bi + nδi , then Y Y P P 0 f (xi + Ai /n − Bi0 τ /n)ezBi /n = f (xi + Ai /n − Bi τ /n)y − δi ezBi /n y δi i

i

=

Y

f (xi + Ai /n − Bi τ /n)ezBi /n .

i

This is not an issue in our expression (6.5), and so the factors ezBi /n have no role in our genus.

Discrete torsion and the sigma genus

17

7. Comparison with the analytic equivariant genus In fact, we can use the expression (6.7) for the two-variable elliptic genus in terms of theta functions to construct a Thom class in Grojnowski’s equivariant cohomology, following [Ros01, AB02, And03]. When G = T[n] is a cyclic group of order n acting on a compact manifold M, we can write down a formula for a T-equivariant genus on T ×G M, which by Shapiro’s lemma is a sensible notion of G-equivariant genus on M. When we do so, we obtain the formula of [DMVV97, BL02]. Let Λ be the lattice (2πiZ + 2πiτ Z), let C be the elliptic curve C/Λ, and let ET be Grojnowski’s equivariant elliptic cohomology associated to C. For convenience we identify T∼ = R/Z so that 1 G∼ = Z[ ]/Z. n We identify T×T∼ =C by the formula (r + Z, s + Z) 7→ 2πir + 2πisτ + Λ. (7.1) Let M be a G-manifold with an equivariant complex structure on its tangent bundle T . We define def

EG (M) = ET (T ×G M). For a ∈ C, (T ×G M)a = 0 unless a ∈ C[n], so ET (T ×G M) is a metropolitan sheaf (collection of skyscraper sheaves) supported at C[n]. The stalk at a point a of order dividing n is H(EG ×G X a ). Let Tˇ be the lattice of cocharacters in Spin(2d). In [And03], the first author constructed orientations for theta functions Θ = Θ( − , τ ) : Tˇ ⊗ C → C satisfying Θ(x + 2πi` + 2πikτ, τ ) = exp(−2πiI(k, x)) exp(−φ(k))Θ(x, τ ) for x ∈ Tˇ ⊗ C and k, ` ∈ Tˇ , where φ :Tˇ → Z I :Tˇ × Tˇ → Z

(7.2)

18

Ando and French

are respectively quadratic and bilinear functions related by φ(` + `0 ) = φ(`) + I(`, `0 ) + φ(`0 ). The building block of the orientation is a family of functions F which we now describe. For simplicity we have supposed that T M is a complex vector bundle, and so our structure group is U (d) instead of Spin(2d). We let T be the maximal torus of diagonal matrices; our choices so far identify the lattice Tˇ = hom(T, T ) of cocharacters with Zd in the usual way. Let (g, h) ∈ G2 ; let a be the corresponding point of C[n], and let A ⊆ G be the subgroup generated by g and h. The action of A on T M|M A is described by characters m = (m1 , . . . , md ) ∈ hom(A, T ) ∼ = Tˇ /|A|Tˇ ∼ = (Z/|A|)d . Choose integer lifts

m ¯ = (m ¯ 1, . . . , m ¯ d ) ∈ Tˇ ∼ = Zd .

Choose a¯ = 2πi

` k + 2πi τ n n

so that a=a ¯ + Λ. In terms of these choices, the stalk of the orientation at (g, h) is built from the function k k ¯ exp(2πi φ(m)τ ¯ )Θ(x + m ¯ ⊗a ¯) F (x) = exp(2πi I(x, m)) n n The essential feature of F is that the functional equation (7.2) satisfied by Θ implies that F is Weyl invariant and independent of the choice of preimage m, ¯ and its dependence on a ¯ is under control. Now consider the exponential f (6.7) associated to the two-variable elliptic genus. Comparison of its functional equation (6.10) with the functional equation (7.2) for Θ suggests that we should build the orientation for the two-variable genus from the simpler function Y k X F (x) = exp(2πi z m ¯ j) f (xj + 2πim ¯ j `/n + 2πiτ m ¯ j k/n). (7.3) n j

The argument at the end of §6 shows that indeed, this F is independent of the choice of lift m. ¯ In fact, the simple transformation rule (6.10) for f implies that F is also independent of the choice of representative a¯ for a.

Now use this F to write down a class Ellan (M, G) ∈ EG (M) following the instructions in [And03]. (In general one gets a section of the cohomology of the Thom space, but the Thom isomorphism in ordinary cohomology defined by f identifies this with EG (M)). More precisely, the formula for the value in the stalk at a ∈ C[n] is the one for γa before Lemma 8.12, taking V 0 to be

Discrete torsion and the sigma genus

19

trivial and θ 0 to be 1. That formula refers to an expression R which is defined in terms of F in Lemma 5.28. (The expression for R also includes a product of σ functions, which should be replaced with the corresponding product of f ’s). Theorem 7.4. With these substitutions, the value of Ell an (M, G) in the stalk at (g, h) is the class in H(EG ×G M (g,h) ) whose restriction to H(BA × M (g,h) ) is the integrand in the summand Φg,−h of the orbifold two-variable elliptic genus (see (6.9)). With the remarks so far, the proof is straightforward. We omit the details, except note that if (g, h) = (`/n + Z, k/n + Z), then in (6.9), (Ai , Bi ) can be taken to run over the set (`m ¯ i , km ¯ i ). Thus the typical factor in (6.9) can easily be seen to identify with the typical factor in (7.3). The genus associated to Ellan (M, G) is the global section of OC[n] whose value at (g, h) is Z Ellan (M, G)g,h . M (g,h)

Summing over (g, h) ∈ G, we get exactly |G| times the orbifold two-variable elliptic genus of [DMVV97, BL02]. 8. The cocycle We now return to the orbifold sigma genus, and study the effect of varying the BU h6i structure. The fibration of infinite loop spaces K(Z, 3) → BU h6i → BSU gives a map of E∞ ring spectra i

Σ∞ K(Z, 3)+ → − MU h6i. If C = (E, C, t) is an elliptic spectrum, then the sigma orientation σ(C) : MU h6i → E gives rise to a map of ring spectra def

w(C) = σ(C)i : Σ∞ K(Z, 3)+ → − E. In particular, w(C) is a Thom class for the trivial bundle over K(Z, 3), and so it is a unit of E 0 K(Z, 3). If α = (g, h) : Λ → G, then we define def

δn (α) = δn (u, C, α) = Ξg,h (u∗ w(C)) ∈ D × .

(8.1)

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Ando and French

Lemma 8.2. If α

α0 = Λn+1 → Λn − → G, then δn+1 (α0 ) = δn (α),

(8.3)

and so we have a well-defined unit δ(g, h) ∈ D × . It satisfies δ(g, h)n = 1 δ(h, g) = δ(g, h)−1 δ(g + g 0 , h) = δ(g, h)δ(g 0, h) δ(g, h + h0 ) = δ(g, h)δ(g 0, h) for any g, g 0 , h, and h0 in G for which these equations make sense. Proof. The arguments for the various claims are similar to each other; as an illustration we show that δ is exponential in the first variable, and for simplicity we suppose that G is abelian. The proof in the general case will be given in §10. Let C be the cyclic group of order n. The universal example of an abelian group with three elements (g1 , g2 , h) of order n is C 3 . Let (g1 ,h)

C 2 −−−→ C 3 (g2 ,h)

C 2 −−−→ C 3 (g1 +g2 ,h)

C 2 −−−−−→ C 3 be the maps which represent the selection of the indicated pairs elements formed from the triple (g1 , g2 , h). It suffices to show that for every homotopy class u : BC 3 → − K(Z, 3), the outside rectangle of the diagram BC 2 ∆y

(g1 +g2 ,h)

−−−−−→

(g1 ,h)×(g2 ,h)

BC 3

u

−−−→

u×u

K(Z, 3) x

σ(C)

−−−→

σ(C)×σ(C)

E x

BC 2 × BC 2 −−−−−−−→ BC 3 × BC 3 −−−→ K(Z, 3) × K(Z, 3) −−−−−−→ E × E (8.4)

21

Discrete torsion and the sigma genus

commutes. The right-side rectangle commutes, because σ(C) is a map of ring spectra. To show that the left-side rectangle commutes, consider the diagram BC 2 ∆y

(g1 +g2 ,h)

−−−−−→

BC 3

(g1 ,h)×(g2 ,h)

v

−−−→

v×v

K(C, 2) x

β

−−−→

β×β

K(Z, 3) x

BC 2 × BC 2 −−−−−−−→ BC 3 × BC 3 −−−→ K(C, 2) × K(C, 2) −−−→ K(Z, 3) × K(Z, 3). (8.5) Once again, the right square commutes, this time because the Bockstein is an additive group homomorphism. An easy calculation shows that the Bockstein β : H 2 (BC 3 ; C) → H 3 (BC 3 ; Z) is surjective, so there is a v such that βv = u. Moreover H 2 (BC 3 ; C) ∼ = C{µ12 , µ13 , µ23 }, where µij : BC 3 ∼ = K(C, 1)3 → K(C, 2) is the map which represents the natural transformation µij : H 1 (X; C)3 → H 2 (C; C) given by µij (x1 , x2 , x3 ) = xi ∪ xj . If v = aµ12 + bµ13 + cµ23 , then the top row of the diagram represents the natural transformation H 1 (X; C)2 → − H 3 (X; Z) given by (x, y) 7→ β(ax ∪ x + bx ∪ y + cx ∪ y), while the other (i.e. counterclockwise) composition represents the natural transformation (x, y) 7→ β(bx ∪ y + cx ∪ y). These coincide since β(x ∪ x) = 0. In other words, the diagram (8.5) commutes for any v, and so the diagram (8.4) does too, as required.

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Ando and French

8.1. The Weil pairing. As an example, let’s consider the case that G = Λ, and u : BG → K(Z/N, 2) is the map representing the cup product: indeed u is a generator of E 0 BG. A homomorphism α:Λ→G gives rise to a homomorphism G∗ → − Λ∗ → C[N ]. Thus we may view the homomorphism α = (g, h) as a pair of N -torsion points of C[N ]. The argument of [AS01] then shows Lemma 8.6. δ(u) is the Weil pairing of the elliptic curve C.

9. Discrete torsion We are now ready to state our basic formula. Theorem 9.1. If α = (g, h) : Λn → G, then Ξα σ(M, ` + ιuπ, C)G = δ(u, C, α)Ξα σ(M, `, C)G , and so abbreviating δ(g, h) = δ(u, C, α), we have X σorb (M, ` + ιuπ, C)G = δ(g, h)Ξg,h σ(M, `, C)G ∈ π−d E. gh=hg

Proof. Since σ(C) : MU h6i → E is a map of ring spectra, and since the multiplication on MU h6i arises from the addition on BU h6i, we have U (M, ` + ιuπ, C)G = U (M, `, C)G U (M, ιuπ, C)G = U (M, `, C)G π ∗ u∗ w(C). Recall that E 0 (VG − TG ) is an E 0 (MG )-module, and so an E 0 (BG)-module via E 0 (π). As such the Pontrjagin-Thom map τ (V )G : E 0 (VG − TG ) → E 0 (BG) is a homomorphism of E 0 (BG)-modules. It follows that σ(M, ` + ιuπ, C)G = τ (V )G (U (M, ` + ιuπ, C)G ) = τ (V )G (U (M, `, C)G π ∗ u∗ w(C)) = u∗ w(C)σ(M, `, C)G in E ∗ (BG). If α : Λn → G,

Discrete torsion and the sigma genus

23

then applying Ξα gives Ξα σ(M, ` + ιuπ, C)G = Ξα u∗ w(C)Ξα σ(M, `, C)G = δn (u, C, α)Ξα σ(M, `, C)G . as required.

10. The non-abelian Case

In this section, we prove Lemma 8.2 in the case that G is non-abelian. We fix an n sufficiently large that |G| divides n. We first construct an isomorphism H 3 (BΛn ) ∼ = Z/n. Consider the following commutative diagram, where the columns are universal coefficient exact sequences. Ext1 (H1 (BΛn ); Z) /

Ext1 (H1 (BΛn ); Z/n)

H 2 (BΛn ) /

H 2 (BΛn ; Z/n)

Hom(H2 (BΛn ); Z) /

/ H 3 (BΛ ) n l l l l l∼ l = ul

Hom(H2 (BΛn ); Z/n).

Here, the middle row is part of Bockstein long exact sequence. Note that this is a short exact sequence since multiplication by n kills group cohomology of Λn . Since H2 (BΛn ) is torsion, Hom(H2 (BΛn ); Z) = 0. We therefore obtain the dotted arrow. Since Ext1 (H1 (BΛn ); −) is right exact,the top map is a surjection. It follows that the dotted arrow is an isomorphism. Now it is not hard to check that e1 ⊗ e2 − e2 ⊗ e1 is a generator for H2 (BΛn ) ∼ = Z/n. Composing the dotted arrow above with evaluation on this generator yields an isomorphism H 3 (BΛn ) ∼ = Z/n. Definition 10.1. If u is an element in H 3 (BG) and α = (g, h) : Λn → G is any map, then we obtain a class α∗ u ∈ H 3 (BΛn ). Let (g, h) = nu (g, h) be the image of this class in Z/n under the isomorphism above. Remark 10.2. It is easy to check that (g, h) = u˜(g, h) − u ˜(h, g) where u ˜ : G × G → Z/n is a 2-cocycle whose cohomology class maps to u under the Bockstein. Lemma 10.3. Whenever the expressions are defined, the following properties hold. (g, h) = −(h, g)

24

Ando and French (h, j) − (gh, j) + (g, hj) − (g, h) = 0 (gg 0 , h) = (g, h) + (g 0 , h) (g, hh0 ) = (g, h) + (g, h0 )

Proof. The first and second properties follow easily from the remark. For the third property, it suffices to show that u ˜(gg 0 , h) − u ˜(h, gg 0) − u ˜(g, h) + u ˜(h, g) − u ˜(g 0 , h) + u ˜(h, g 0 ) is zero. Since u ˜ is a cocycle, we may rewrite the expression using the following equations: u ˜(gg 0 , h) − u˜(g 0 , h) = u ˜(g, g 0 h) − u ˜(g, g 0), −˜ u(h, gg 0 ) + u ˜(h, g) = u ˜(g, g 0 ) − u ˜(hg, g 0 ), −˜ u(g, h) + u ˜(h, g 0 ) = u ˜(gh, g 0 ) − u˜(g, hg 0 ). Then, canceling terms, we get u ˜(g, g 0 h) + u ˜(gh, g 0) − u ˜(hg, g 0) − u ˜(g, hg 0 ). Since gh = hg and g 0 h = hg 0 , this is zero as needed. The last property follows similarly, or directly from the first and third. Proof of Lemma 8.2. Let F : BΛn → K(Z, 3) represent the element 1 ∈ Z/n ∼ = H 3 (BΛn ) under the isomorphism above. Let x ∈ D be the image of F ∗ w(C) ∈ E 0 BΛn under the map tautological map E 0 BΛn → − Dn → − D. It is easy to check from the definitions (8.1) and (10.1) of δ and that δ(g, h) = xg,h . The result now follows from Lemma 10.3.

References [AB02] [AHS01]

[AHS03] [And03]

Matthew Ando and Maria Basterra. The Witten genus and equivariant elliptic cohomology. Mathematische Zeitschrift, 240(4):787–822, 2002. Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. Elliptic spectra, the Witten genus, and the theorem of the cube. Inventiones Mathematicae, 146:595–687, 2001. DOI 10.1007/s002220100175. Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. The sigma orientation is an H∞ map. Amer. J. Math., To appear. Matthew Ando. The sigma orientation for analytic circle-equivariant elliptic cohomology. Geometry and Topology, 7:91–153, 2003. math.AT/0201092.

Discrete torsion and the sigma genus

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[AS68]

M. F. Atiyah and G. B. Segal. The index of elliptic operators. II. Ann. of Math. (2), 87:531–545, 1968. [AS01] M. Ando and N. P. Strickland. Weil pairings and Morava K-theory. Topology, 40(1):127–156, 2001. [BL02] Lev A. Borisov and Anatoly Libgober. Elliptic genera of singular varieties, orbifold elliptic genus and chiral de Rham complex. In Mirror symmetry, IV (Montreal, QC, 2000), volume 33 of AMS/IP Stud. Adv. Math., pages 325–342. Amer. Math. Soc., Providence, RI, 2002. [BM94] J.-L. Brylinski and D. A. McLaughlin. The geometry of degree-four characteristic classes and of line bundles on loop spaces. I. Duke Math. J., 75(3):603–638, 1994. [BT89] Raoul Bott and Clifford Taubes. On the rigidity theorems of Witten. J. of the Amer. Math. Soc., 2, 1989. [DMVV97] Robbert Dijkgraaf, Gregory Moore, Erik Verlinde, and Herman Verlinde. Elliptic genera of symmetric products and second quantized strings. Comm. Math. Phys., 185(1):197–209, 1997. [Dri74] V. G. Drinfeld. Elliptic modules. Math. USSR-Sb., 23(4):561–592, 1974. [EOTY89] H. Eguchi, H. Ooguri, A. Taormina, and S.-K. Yang. Superconformal algebras and string compactification on manifolds with SU (N ) holonomy. Nucl. Phys. B, 315, 1989. [GMS00] Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman. Gerbes of chiral differential operators. Math. Res. Lett., 7(1):55–66, 2000. math.AG/9906117. [HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. Generalized group characters and complex oriented cohomology theories. J. Amer. Math. Soc., 13(3):553–594 (electronic), 2000. [Hop95] Michael J. Hopkins. Topological modular forms, the Witten genus, and the theorem of the cube. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), pages 554–565, Basel, 1995. Birkh¨ auser. [Kri90] Igor Moiseevich Krichever. Generalized elliptic genera and Baker-Akhiezer functions. Mat. Zametki, 47(2):34–45, 158, 1990. [LU] Ernesto Lupercio and Bernardo Uribe. Deligne cohomology for orbifolds, discrete torsion, and B-fields. hep-th/0201184. [Ros01] Ioanid Rosu. Equivariant elliptic cohomology and rigidity. Amer. J. Math., 123(4):647–677, 2001. [Sha] Eric Sharpe. Recent developments in discrete torsion. hep-th/0008191. [Str97] Neil P. Strickland. Finite subgroups of formal groups. J. Pure and Applied Algebra, 121:161–208, 1997. [Vaf85] C. Vafa. Modular invariance and discrete torsion. Nuclear Physics B, 261:678– 686, 1985. [Wit87] Edward Witten. Elliptic genera and quantum field theory. Comm. Math. Phys., 109, 1987. E-mail address: [email protected] E-mail address: [email protected] Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana IL 61801, USA

QUATERNIONIC ELLIPTIC OBJECTS AND K3-COHOMOLOGY JORGE A. DEVOTO Abstract. We shall outline a research program which aims to provide a geometric approach to K3-cohomology. We define for this a quaternionic analogue of Segal’s elliptic objects.

1. Introduction The relation between one dimensional formal group laws and complex oriented generalised cohomology theories, which is a consequence of Quillen’s theorem [19, 20] which relates Lazard’s ring – the ring over which is defined the universal formal group law – and the coefficient ring of complex cobordism, has been a very fruitful source of research in algebraic topology. Among one dimensional formal group laws three have a direct geometric interpretation. They are obtained from the additive group, the multiplicative group and elliptic curves respectively by taking completion at the origin. The corresponding cohomology theories, Ordinary cohomology, K-theory and Elliptic cohomology also have geometric interpretations 1. These interpretations have extended their use, and helped to find applications of them, to other fields of mathematics and physics besides algebraic topology. No other one dimensional commutative formal group law is associated to a group in this direct way. However M. Artin and B. Mazur [2] showed that one can associate formal groups to certain algebraic varieties – among them K3 surfaces – in a way which generalises the case of elliptic curves. M. Hopkins used their work to define an associated generalised cohomology theory: K3cohomology. The geometric origin of the formal group laws associated to K3 surfaces gives a reason to hope that there exists a geometric definition of K3 cohomology. There is some evidence that the K3-cohomology of a manifold M should be related to some form of K-theory on the double loop space L (2) M = C ∞ (S 1 × S 1 , M) of M. This relation should be analogous to the relation between the elliptic cohomology of M and K-theory of the loop space of M. If this belief is true, then it is possible that one can find some interesting Date: 1st September 2003. 1991 Mathematics Subject Classification. 55N35, 14J81. Key words and phrases. K3 cohomology, K3 surfaces, Elliptic objects, Quaternionic Geometry. 1 Conjectural in the case of elliptic cohomology

26

Quaternionic elliptic objects and K3-cohomology

27

applications of K3 cohomology to other areas of mathematics and physics, as in the case of K-theory, Ordinary cohomology and Elliptic cohomology. The aim of the present article is to outline a research program whose purpose is to give a geometric interpretation of K3 cohomology. This interpretation is based on an appropriate modification of G. Segal’s elliptic objects. The main difference is that in the case of K3-cohomology the relevant geometry is defined over the quaternions rather than the complex numbers, as in the case of elliptic objects. We expect that the geometric interpretation will be related to: • Dirac operators on double loop spaces. • Representation theory of double loop groups. • Representation theory of DiffΩ (S 1 ×S 1 ) where Ω is a fixed volume form and DiffΩ (S 1 × S 1 ) is the group of diffeomorphisms which preserve Ω. • Siegel modular forms. • Classical and quantum gravity. • D-Branes. There are few published references for K3-cohomology. We refer the reader to [27, 28] for an exposition of different aspects of K3 cohomology. This project is a joint project with Charles Thomas and is a pleasure to express my gratitude to him for sharing with me his insight and for his continuous support over the years. I want also to express my gratitude to the funders of the Newton Institute for the invitation to be in the conference. Finally I want to thank Haynes Miller and the referee for their suggestions which greatly helped to improve the present manuscript.

2. Definitions of K3-cohomology 2.1. Formal groups and algebraic varieties. In this section we will describe the work of Artin and Mazur [2] which is at the basis of the definition of K3 cohomology. Let Gm denote the multiplicative group scheme. If X is an algebraic curve defined over a field k, then applying Gm to the sheaf O of regular functions on X and taking identity component of the first cohomology group H 1 (X, Gm ) one obtains Pic(X), the Picard group of X. If X is an elliptic curve and j = 1 the identity component of Pic(X) is isomorphic to X. One can ask if it is possible to obtain in this way the formal completion of the Picard group, which in the case of elliptic curves is the relevant formal group for elliptic cohomology. A positive answer, and a generalisation of this construction, was obtained by Artin and Mazur [2]. Let X be a variety defined over a field k. If A is a local Artin k-algebra with spectrum Spec A, then XA = X ×Spec k Spec A can be thought as an infinitesimal thickening of X which can be used to study deformation theory. The inclusion Spec k → Spec A induces a morphism X → XA .

28

Jorge A. Devoto

Consider now, for each natural number j, the functor from the category of Artin local algebras over k to the category of groups defined by: (2.1)

A → ker{H j (XA , Gm ) → H j (X, Gm )}.

Cohomology in this case means etale cohomology. One can ask if this functor is prorepresentable by some formal object. If X is a X and j = 1, then the answer is yes and the representing object is the formal completion of Pic(X). Artin and Mazur showed that the answer is also yes in other cases. 2.2. K3-Surfaces. Let us consider Artin and Mazur’s construction for the case j = 2. They showed that if H 1,0 (X) = H 0,1 (X) = 0, then the functor (2.1) is prorepresented by a smooth formal group, defined over Spec k, of dimension dim H 2,0 (X). The representing object is called the formal Brauer group. This group has a natural coordinatisation – see Theorem 4.9 and Corollary 4.10 of [25] – and hence it has an associated formal group law, which is the real object of interest in K3-cohomology. We shall call this formal group c law the formal Brauer group law and we shall denote it by Br(X).

Definition 2.1. A K3-surface X is a compact surface with h1,0 (X) = 0 and trivial canonical bundle. If X is a K3-surface, then the Hodge diamond has the following form: 1 0 0 1 20 1 0 0 1. From this it follows that the formal Brauer group has dimension one. 2.2.1. Examples of K3 surfaces. Example 2.2. The Fermat quartic S = {[z0 , z1 , z2 , z3 ] ∈ CP3 : z04 + z14 + z24 + z34 = 0} Brauer’s formal group law is defined over Z and has as logarithm X (4m)! x4m+1 log(x) = . 4 4m + 1 (m!) m≥0 Example 2.3. Any nonsingular quartic in CP3

Example 2.4. A complete intersection of a cubic and a quadric in CP4 Example 2.5. Let Λ be a lattice in C2 . Then the quotient T = C2 /Λ is a 4-torus. Define a map σ : T → T by σ : (z1 , z2 )+Λ → (−z1 , −z2 )+Λ. Then σ defines an action of Z2 which fixes the 16 points {(z1 , z2 ) + Λ | (z1 , z2 ) ∈ 12 Λ}. The quotient space by this action T/σ is a singular complex orbifold with 16 singular points. Let K be the blow-up of T/σ at the singular points. Then

Quaternionic elliptic objects and K3-cohomology

29

K is a K3 surface called a Kummer surface. The map T → K is called the Kummer map. If one is not working with smooth surfaces the surface T/σ is also called a Kummer surface. For more references about Kummer surfaces see [12]. 2.3. K3 cohomology. Let p be a prime. Then in characteristic p the formal c group law Br(X) is either the additive group law, of height h = ∞, or a formal group law of height h less or equal than 10.

Definition 2.6. Let X be a K3-surface defined over a field of characteristic p 6= 0. Then if h = ∞ we will say that X is supersingular.

Example 2.7. The formal Brauer group associated to the Fermat quartic is supersingular for the primes p ≡ 3 mod 4. The filtration by height induces in characteristic p a stratification M1 ⊃ M2 ⊃ · · · ⊃ M10 ⊃ M11 = M∞ of the moduli space of K3 surfaces. In this case M∞ corresponds to supersingular surfaces. The strata M(i) corresponding to K3 surfaces of height h = i are, for i < 11 regular and locally complete intersection of codimension h − 1 [29, Thm 15.1]. From Theorem 8.21A and Proposition 8.23 of [11] it follows that if X is a K3 surface which is not supersingular, then the associated formal group law satisfies Landweber exactness criterion reduced modulo p. One could study different theories associated to this construction. The first one is defined in the spirit of elliptic cohomology, trying to consider c a universal example.The construction X → Br(X) induces a map b from the moduli space of K3 surfaces to the space F of formal group laws. If MU (X) denotes the complex cobordism of a space X, then one has a sheaf over F associated to the MU -module structure of MU (X). Here MU is identified with Lazard’s ring. One can use b to pull-back this sheaf of modules associated to the moduli space of K3 surfaces. This corresponds to consider a cohomology theory associated to a generic K3 surface. A second approach is to study theories associated to certain families of K3 surfaces, like Kummer surfaces. We hope that this approach will produce a theory with a geometric model. There is a second approach to K3 cohomology – see [28] – inspired by the h8i work of M. Kreck and S. Stoltz in elliptic cohomology [16]. Let Ω∗ denote the bordism ring of manifolds with w1 (M) = w2 (M) = 12 p1 (M) = 0. Let h8i I ⊂ Ω∗ be the ideal generated by elements of the form [E] − [OP2 ][M], where E → M is a fibration. The fibre OP2 , is the Cayley projective plane, defined for example in [5, 28]. The structural group is the exceptional Lie group F4 which is the isometry group of OP2 . Define then the functor M → h8i h8i Ω∗ (M)/IΩ∗ (M). Is this functor, may be after inverting some primes, a generalised homology theory? How is it related to the previously mentioned construction?

30

Jorge A. Devoto 3. Elliptic Objects

The physically motivated heuristic relation between elliptic cohomology and Dirac operators on loop spaces – see for example [31, 32] – provided evidence that there exists a geometric definition of elliptic cohomology related to some form of Diff+ (S 1 ) equivariant K-theory on free loop spaces. The group Diff+ (S 1 ) is the group of orientation preserving diffeomorphisms of the circle and the action on the loop spaces is by reparametrisation of loops. G Segal proposed a definition of elliptic objects and conjectured in [22] that they might be used to give the sought for geometric construction of elliptic cohomology. Elliptic objects are closely related to Segal’s axiomatic formulation of conformal field theory [23]. Conformal field theories are defined as functors from a cobordism category C to suitable topological vector spaces. We shall describe now Segal’s definitions. Then in the next section we shall generalise the category C and later we shall consider some generalisations of the functors involved in conformal field theory. Let C be the category which has an object set {C0 , C1 , . . .}. The set C0 = {∅} and for each n ≥ 0 the set {Cn } contains all the 1-dimensional manifolds formed by the disjoint union of n parametrised circles. A morphism Σ : Cm → Cn is a Riemann surface Σ with boundary ∂Σ together with an orientation preserving identification Cn t Cm → ∂Σ. Here Cm means Cm with the orientation reversed. The circles in Cm are called incoming and the e are equivalent if there is circles in Cn are called outgoing. Two surfaces Σ, Σ e which is analytic in the interiors of the surfaces a diffeomorphism f : Σ → Σ and respects the identifications of the boundary. Composition of morphisms e is defined by sewing surfaces together along the boundary. The set Σ◦Σ Cm,n of morphisms Σ : Cn → Cm is a topological space with one connected component for each topological type of surface Σ. Segal showed that one can define a structure of holomorphic manifold on Cm,n and that composition Ck,m × Cm,n → Ck,n is a holomorphic map [23, Section 4]. There are a number of natural operations which can be performed on C . (1) The symmetric groups Sm and Sn act on Cm,n by permuting the numbering of the circles. (2) If Σ ∈ Cm,n , then the complex conjugated surface Σ ∈ Cn,m and Σ → Σ is an antiholomorphic map. (3) By reversing the orientation of various boundary circles one can obtain several holomorphic crossing maps Cm,n → Cp,q , with m + n = p + q. (4) By sewing k incoming circles to k outgoing one obtains a holomorphic map Cm,n → Cm−k.n−k which is referred to as collapsing. Definition 3.1. Let H be a topological vector space with a hermitian form and a given real structure induced by an anti-involution H → H . A conformal field theory based on H is a continuous functor U from C to the category of topological vector spaces with the following properties:

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(1) For U (∅) = C and U (Cn ) = H ⊗n for n ≥ 1. ⊗m ⊗ H ⊗n . The image (2) For each Σ ∈ Cm,n the element U (Σ) ∈ H of the natural homomorphism H ⊗ H → End(H ) consists of trace class operators [24]. Therefore the element U (Σ) can be identified with a trace class operator. (3) The map Cm,n × H m → H n is compatible with the action of the symmetric groups Sm and Sn . (4) U (Σ) = U (Σ)∗ for all morphisms Σ. This property is termed reflection positivity. ⊗m ⊗ (5) The crossing maps Cm,n → Cp,q are compatible with the maps H ⊗p ⊗n ⊗q H →H ⊗H induced by the real structure of H . (6) The collapsing maps Cm,n → Cm−k.n−k are compatible with the maps ⊗m ⊗(m−k) H ⊗H ⊗n → H ⊗H ⊗(n−k) induced by the hermitian structure. Remark 3.2. The set A of equivalence classes of morphisms Σ : C1 → C1 which are topologically equivalent to an annulus form a semigroup. Segal shows that A plays the role of a complexification of Diff+ (S 1 ). As a consequence of the definition of a conformal field theory it follows that the space H carries a representation of Diff+ (S 1 ) which can be extended to a representation of A . A first step in the construction of a conformal field theory is to understand which representations of Diff+ (S 1 ) can be extended to representations of A . These representations are called positive energy representations. ∂ be the tangent vector field on S 1 which generates the rigid rotations. Let ∂θ ∂ Then ∂θ defines an operator on H . A representation of Diff+ (S 1 ) is a positive ∂ energy representation if H = −i ∂θ is a self-adjoint operator with spectrum bounded from below. Proposition 3.3. [23, Proposition 3.1] There is a one to one correspondence between positive energy representations of Diff+ (S 1 ) and holomorphic (contraction) representations of A . Remark 3.4. let Σ ∈ A and let U be a conformal field theory based on the space H . Let Σ be the complex torus obtained by sewing the incoming and outgoing circles. This torus can be seen as a morphism Σ : ∅ → ∅. In physical terms a vacuum to vacuum transition. The last condition in the definition of conformal field theory implies that the number U (Σ) is the trace of the operator U (Σ). The torus Σ has a distinguished point given by the image of 1 ⊂ S 1 which induces the structure of elliptic curve on it. The assignment Σ → U (Σ) for all Σ ∈ A defines a modular function. In this way the theory of elliptic objects gives a geometric explanation of the appearances of elliptic curves and modular forms in elliptic cohomology. Let M be a manifold. An elliptic object with target M is a modification of the definition of a conformal field theory which reflects some of the properties

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of a sigma model with target M. One considers the category C (M) with objects pairs {Cn , f }, where Cn is an object of C and f : Cn → M is a smooth map. A morphism (Σ, H) : (Cm , f ) → (Cn , g) is a pair formed by a morphism Σ : Cm → Cn and a smooth function H : Σ → M which restricts on the boundary, via the corresponding identifications, to f and g respectively. An elliptic object is a functor U from C (M) to topological vector spaces satisfying some conditions similar to Definition 3.1. One can relate elliptic objects to vector bundles on L M in the following way: The vector space U (S 1 , f ) can be seen as the fibre of a vector bundle E → L M over the element f ∈ L M. The linear transformations U (Σ, H) associated to morphisms (Σ, H), where Σ ∈ A , can be seen as the holonomy of a connection on E over the path H(Σ) in L M. 4. The categories T

2

and T 3

If one tries to mimic Segal’s definition of elliptic objects in order to define objects related to vector bundles over double loop spaces, then the natural thing to do is to consider a cobordism category T 2 whose objects are a finite disjoint union of oriented manifolds diffeomorphic to the standard two torus T = S 1 × S 1 . The next logical step would be to understand which are the morphisms in T 2 . They should be 3-dimensional manifolds Σ with some extra structure. If this construction is going to be related to K3 cohomology, then the geometry of the morphisms should give natural explanations of the role played by K3-surfaces in a way analogous to Remark 3.4. There is a clear problem with this approach. Morphisms in T 2 are manifolds of real dimension 3 instead of 4, which is the real dimension of K3-surfaces. We shall consider for this reason two different categories at the same time. One will be T 2 . A notion of elliptic objects based on this category will be related to bundles on the double loop spaces. The other, called T 3 will be explain the appearance of K3 surfaces. Both categories are related by a functor which we call the change of dimensions functor. 4.1. The category T 3 . Let T3 = S 1 × S 1 × S 1 be the standard torus equipped with the volume form Ω = dθ1 ∧ dθ2 ∧ dθ3 . A parametrised torus is a pair (M, ΩM ) formed by a manifold M with a fixed diffeomorphism f : T → M and a volume form ΩM such that f ∗ ΩM = Ω. For each n ≥ 0 the category T 3 has a space of objects Tn3 . For n > 0 the objects are pairs (M, Ω), where M is the disjoint union of n copies of parametrised tori. In the case n = 0 it has only one object which is the empty set. The morphisms Σ : (M1 , Ω1 ) → (M2 , Ω2 ) are equivalence classes of 4-manifolds with a hyperk¨ ahler structure smooth up to the boundary together with a volume preserving diffeomorphism ∂Σ ' M 1 t M2 . Two hyperk¨ahler manifolds Σ1 and Σ2 representing morphisms (M1 , Ω1 ) → (M2 , Ω2 ) are equivalent if there exists a diffeomorphism f : Σ1 → Σ2 analytic in the interior of the manifolds ,

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which preserves the hyperk¨ahler structure. Composition of morphisms is obtained by gluing manifolds along the boundary. The reasons for this choices are explained in the next sections. 4.2. Hyperk¨ ahler Geometry. Let X be a 4n dimensional manifold. An almost hyperk¨ ahler structure on X is a quadruple (J1 , J2 , J3 , g) where Ji are anticommuting almost complex structures on X with J1 J2 = J3 and g is a Riemannian metric on X which is hermitian with respect to J1 , J2 and J3 . The family of complex structures Ji defines a H-module structure in each of the tangent spaces Tx X. The form g is called a hyperhermitian form. An almost hyperk¨ahler structure on X is called hyperk¨ ahler if in addition ∇Ji = 0, i = 1, 2, 3, where ∇ is the Levi-Civita connection of g. Hyperk¨ahler structures are a quaternionic analogue of K¨ahler structures. The basic examples of hyperk¨ahler manifolds are the quaternionic n-dimensional spaces Hn . Identify Hn = R4n using coordinates (q1 , . . . qn ) with qj = x0j + x1j J1 + x2j J2 + x3j J3 , where J1 = i, J2 = j, J3 = k are the usual quaternionic units. Introduce complex coordinates (z1 , . . . , z2n ) by z2j−1 = x0j + ix1j and P z2j = x2j + ix3j then the form qj q j defines a metric g and three symplectic forms ωi , i = 1, 2, 3 by g=

2n X j=1

2n

2

|dzj | ,

iX ω1 = dzj ∧ dz j , 2 j=1

ω2 + iω3 =

n X

dz2j−1 ∧ dz2j .

j=1

The forms g and ωi , i = 1, 2, 3 can be defined in any almost hyperk¨ahler manifold. The next result gives different characterisations of hyperk¨ahler manifolds. Proposition 4.1. [13, Proposition 7.1.2] Suppose that X is a 4n manifold and that (J1 , J2 , J3 , g) is an almost hyperk¨ ahler structure on X. Let ω1 , ω2 , ω3 be the hermitian forms of J1 , J2 , J3 . Then the following conditions are equivalent (1) (J1 , J2 , J3 , g) is a hyperk¨ ahler structure, (2) dω1 = dω2 = dω3 = 0, (3) ∇ω1 = ∇ω2 = ∇ω3 = 0, where ∇ is the Levi-Civita connection of g. (4) The holonomy of ∇ is contained in Sp(n) and J1 , J2 , J3 are the complex structures induced by the holonomy. In the category T 3 closed hyperk¨ahler manifolds X appear as a vacuum to vacuum transition X : ∅ → ∅ and one can expect that they will be related to the partition function in any quantum field theory defined on T 3 . In this way, due to the next result, one can see that smooth K3-surfaces play a natural and essential part of the theory. Theorem 4.2. [13, Thm 7.3.13] Let (X, J) be a K3 surface. Then each K¨ ahler class on X has a unique metric with holonomy SU (2) = Sp(1). Conversely any compact Riemannian 4 manifold (X, g) with holonomy SU (2)

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admits a covariantly constant complex structure J such that (X, J) is a K3 surface. 4.3. The semigroup of tubes. Let M be the set of morphisms Σ : (T3 , Ω) → (T3 , Ω) which are topologically tubes T3 × [a, b]. This set forms a semigroup which is an analogue of the semigroup A considered by Segal. We shall presents some arguments that show that M could be considered as a quaternionification of the volume preserving diffeomorphisms of T3 . In conformal field theory the diffeomorphisms of a circle arise in relation to Lorentz invariance in a two dimensional Mikowskian space S 1 × R where the space is the circle [23, Section 1]. One of the reasons for the choice of the volume preserving diffeomorphisms of a three dimensional manifold T3 as the relevant group of symmetry is that we want to make contact with the work of Ashtekar [3] which relates this group to general relativity, in particular to self dual solutions of Einstein equations on T3 × R 2. The arguments are based on a work of S. Donaldson which we will briefly describe now. Let M be a three dimensional compact closed manifold with a fixed volume form Ω and let DiffΩ (M) be the Lie group of diffeomorphisms of M preserving the form Ω. Donaldson defined in [8] a set of manifolds equipped with extra geometric structures which could be used as a the elements of a semigroup SM . He suggested that the semigroup SM plays the role of a complexification of DiffΩ (M). In particular he showed that the natural adjoint representation of DiffΩ (M) on its Lie algebra VectΩ (M), consisting of divergence free vector fields, has a natural extension to an action of SM on VectΩ (M)⊗C. The basis of his construction is the following definition. Let G be a real Lie group, with Lie algebra g, which acts on a real vector space V . We shall say that two elements U + , U − of V ⊗ C are GC -related if there are: (1) An element g ∈ G. (2) Smooth one parameter families U1 (t), U2 (t) of elements of V parametrised by [−a, a] such that U1 (−a) + iU2 (−a) = U − , U1 (a) + iU2 (a) = gU + . (3) A smooth one parameter family α(t) of elements of g such that d (4.1) (U1 + iU2 ) = iα(t)(U1 + iU2 ). dt Remark 4.3. When G has a complexification GC , then two elements U + and U − of a complex representation V of G are GC related if and only if they are in the same GC orbit of the natural extension of the action of G to GC . Remark 4.4. If the case G = Diff+ (S 1 ) then two elements U + and U − are GC related if and only if there is an element A in Segal’s semigroup of annuli 2There

are also some other contexts in physics where this group appears as the natural generalisation of Diff+ (S 1 )

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35

such that AU − = U + . What makes Segal’s construction special is that in this case there is an intrinsic representation of this construction in terms of complex structures on the annuli. 4.3.1. Nahm’s equations. Nahm’s equations are a system of ordinary differential equations for three smooth one parameter families Ti (t) with i = 1, 2, 3 of elements of a real Lie algebra g. The equations are: dT1 (4.2) = [T2 , T3 ], dt dT2 = [T3 , T1 ], dt dT3 = [T1 , T2 ], dt where t ∈ [−a, a] for some a ∈ R. One can combine the first two equations into a single equation d (4.3) (T1 + iT2 ) = [iT3 , T1 + iT2 ], dt defined for elements of the complexified Lie algebra g ⊗ C. Definition 4.5. Two elements U + , U − of elements of g ⊗ C are related by a Nahm’s solution if U + = T1+ + iT2+ , U − = T1− + iT2− , where Ti± = Ti (±a) and Ti , i = 1, 2, 3 is a solution of Nahm’s equations. A solution of Nahm’s equation defines a pair of GC -related elements in the adjoint representation. Take U1 = T1 , U2 = T2 and α = T3 . When G is compact the converse holds. Two elements of g ⊗ C are GC related if and only if they are related by a Nahm’s solution, see for example [7] for the case of the unitary groups. The result has been extended to other groups by Saskida [21]. 4.3.2. The geometry behind Nahm’s equations. The essential idea in Donaldson’s complexification of DiffΩ (M) is to find an intrinsic geometric characterisation of solutions of Nahm equations. Recall that a symplectic form in a 2n dimensional manifold M is a nondegenerate closed two form ω. That ω is non-degenerate means that ω n is a volume form. Definition 4.6. A complex symplectic surface Σ is a smooth four dimensional manifold with a complex structure and a holomorphic symplectic form w ∈ Ω2,0 (Σ) . In the complex symplectic context the non-degeneracy condition means that w ∧ w is a complex valued volume form. If we write w = ω1 + iω2 , then the algebraic constraints imposed by the non-degeneracy condition are (4.4)

ω1 ∧ ω1 6= 0,

ω2 ∧ ω2 6= 0,

at each point of Σ. Both forms are closed.

and ω1 ∧ ω2 = 0,

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Remark 4.7. Donaldson showed that if one have two real closed 2-forms ω1 and ω2 on a 4 manifold Σ which satisfy conditions (4.4) at each point, then they define a complex symplectic structure on Σ. Example 4.8. If X is a hyperk¨ahler manifold, then the form ω2 + iω3 defines a complex symplectic structure on X. Let M be a three dimensional manifold with a fixed volume form Ω and suppose that vi (t), i = 1, 2, 3 are three families of vector fields parametrised by t ∈ [−a, a] such that for each t the fields are divergence free and linearly independent. Suppose that also d (4.5) (v1 + iv2 ) = [iv3 , v1 + iv2 ]. dt We shall now define a complex symplectic structure on Σ = M × [−a, a]. For each t let 1 , 2 , 3 be the one forms dual to the vector fields v1 (t), v2 (t), v3 (t). Then there exists a smooth function f : Σ → R such that Ω = f 1 ∧ 2 ∧ 3 . Define then (4.6)

ω1 = f (dt ∧ 1 + 2 ∧ 3 ), ω2 = f (dt ∧ 2 + 3 ∧ 1 ).

Let w = ω1 + iω2 . An elementary algebraic computation shows that the forms ω1 and ω2 satisfy the algebraic conditions (4.4). The forms ω1 and ω2 are closed if and only if the vector fields vi (t) satisfy equation (4.5). Thus by Donaldson’s construction quoted in Remark 4.7 w defines a complex symplectic structure on M. Remark 4.9. It is an open problem to know if any complex symplectic structure on a 4 manifold Σ ' M × [−a, a] comes from a solution of Nahm’s equations. See the discussion in Donaldson’s paper. 4.4. The semigroup M as a quaternionification of DiffΩ (T3 ). We shall use some work of Atiyah and Bielawski [4] to argue that Nahm’s equations really define a quaternionification of a Lie Algebra. Then we shall show that the set of manifolds that appears in Donaldson’s work are elements of the semigroup M . 4.4.1. Algebraic Point of View. Let g be a real Lie algebra. Then in Lie theory one considers the usual complexified Lie algebra gC = g ⊗ C ' g ⊗ R2 . The elements of g ⊗ C can be identified to pairs g1 , g2 of elements of g. In the case of Segal’s theory the elements v1 + iv2 , where vi are vector fields on the circle can be seen as infinitesimal annuli so, quoting Segal, A has the right size to be a complexification of Diff+ (S 1 ). In the case of Nahm’s equations one has to work with a family of three elements T1 , T2 , T3 of g, which can be naturally identified to an element of g ⊗ R3 . Atiyah and Bielawski argued in [4] that the one should identify R3 with the imaginary quaternions I and use time t as the real coordinate so

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37

H = {t + ix1 + jx2 + kx3 }. Introduce a fourth function T0 : [−a, a] → g and consider the one form (4.7)

ω = T0 dt + T1 dx1 + T2 dx2 + T3 dx3 .

as an R3 invariant connection on H ⊗ g → H. Nahm’s equations have a natural interpretation in terms of the curvature of the connection. When g is the Lie algebra of a compact Lie group G, the natural gauge group to consider consists of G valued functions of t. Then one can set T0 = 0 using a convenient gauge transformation. In this way we can argue that Nahm’s equations are related to the quaternionification g ⊗ H of g. 4.4.2. Geometric Point of View. In Section 4.3.2 we used the vector fields v1 and v2 to define an complex symplectic form w1 , but the cyclic symmetry of Nahm’s equations implies that we can define another two complex symplectic forms w2 and w3 . Donaldson showed that these forms are compatible with a Riemannian metric in the sense that g + iw1 + jw2 + kw3 define a hyperher∂ mitian metric which is a hyperk¨ahler structure. In this case ∂t becomes the gradient flow for a harmonic time function. From this observation if follows that the manifolds defined by solutions of Nahm’s equations define elements of M . 4.5. The Category T 2 . The category T 3 seems to have many of the desired properties for a geometric model of K3 cohomology, however it has no clear relation with double loop spaces. This relation will reappear if we consider the category T 2 . Let us now define the morphisms in T 2 . Definition 4.10. A contact structure φ on a three dimensional manifold M is a one form φ with the property that at each point x ∈ M the form φ ∧ dφ defines a volume form. Definition 4.11. A 3 dimensional manifold M has a taut contact circle if there is a pair φ1 , φ2 of contact structures with the property that for each pair of real numbers λ1 , λ2 such that λ21 + λ22 = 1 the contact one form λ1 φ1 + λ2 φ2 defines the same volume form Ω = (λ1 φ1 + λ2 φ2 ) ∧ d(λ1 φ1 + λ2 φ2 ). The category T 2 has objects T which are two dimensional manifolds with fixed volume forms. Each of the connected components Tc of T is parametrised by an area preserving diffeomorphism φ : S 1 × S 1 → Tc . The relevant group is the group of area preserving diffeomorphisms of S 1 × S 1 . This group is the natural generalisation of Diff+ (S 1 ) in many physical situations, particularly 2d-gravity, W-gravity, matrix models, see [15] and references therein. A morphism Σ : T1 → T2 is a equivalence class of oriented 3-manifolds with a taut contact circle, together with an identification ∂Σ = T1 t T2 . Two manifolds Σ1 and Σ2 are equivalent if there is a diffeomorphism φ : Σ1 → Σ2 which preserves the taut contact circles and the identifications on the boundary.

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4.6. Change of Dimensions. Manifolds with a taut contact circle have been classified by Gonzalo and Geiges in [10]. They used the following result and the classification of complex surfaces. Theorem 4.12. [10] There is a one to one correspondence between 3-manifolds with a taut contact circle and four dimensional manifolds with a complex symplectic structure and a distinguished copy of S 1 . The correspondence is defined by M → M × S 1 . This results suggests that there is a functor, given by multiplication by S 1 between T 2 and T 3 . We can use this functor to extend and relate constructions in T 2 and T 3 . 5. Positive energy representations of DiffΩ (M) In the definition of quaternionic elliptic objects one would need to consider functors from T 3 to an appropriate category of topological vector spaces satisfying axioms similar to the axioms of conformal field theory. The first of these axioms implies that these spaces should carry a representation of DiffΩ (T 3 ) and the representation should extend to a representation of M . The notion of positive energy representations can be extended to DiffΩ (T), where T = S 1 × S 1 × S 1 in the following way. Begin with R3 , then the space of spinors can be identified with the quaternions H using the identification Spin(3) ' Sp(1) and the Dirac operator on R3 is (5.1)

D=i

∂ ∂ ∂ +j +k ∂x1 ∂x2 ∂x3

acting on the space of H-valued Schwartz functions S (R3 , H). If one takes the standard L2 metric it is easy to show that D is a self-adjoint operator. This results can be easily extended to spinors on T; just take periodic functions f : R3 → H. Then again D is a self adjoint operator with a discrete spectrum which acts on the Hilbert space H = L2 (T, H). Using this operator we can define a polarisation of H = H+ ⊕ H− into the spaces generated by eigenvectors corresponding to the positive and negative eigenvalues of D and e−tD restricted to H+ is a contraction operator for positive t. Definition 5.1. A representation H of DiffΩ (T) will be termed admissible if (1) H is a H-module. (2) the metric on H is hyperhermitian. (3) VectΩ (T) acts by skew-adjoint operators Let H be an admissible representation of DiffΩ (T). Then identify the partial derivatives ∂/∂θi with the vector fields which generate rigid rotations in each of the copies of S 1 in T. In this way we obtain three elements of the Lie algebra of DiffΩ (T). If we think of these elements as operators on H , then there is a natural operator D on H defined by the formula (5.1).

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Definition 5.2. An admissible representation H of DiffΩ (T) is a positive energy representation if D has a discrete spectrum bounded from bellow. If H is a positive energy representation we can associate to any manifold T × [0, t] equipped with the natural hyperhermitian metric a contraction operator e−tD . If Σ ∈ M is a morphism defined by a solution of Nahm’s equations, then let t : Σ → R be the time function and let v1 (t), v2 (t), v3 (t) be the vector fields on Σ which are the solutions of Nahm’s equations. Assume that t has no critical points, then t induces a foliation on Σ by manifolds Tt diffeomorphic to T. For each t we have an operator Dt = iv1 (t) + jv2 (t) + kv3 (t) defined on H . Then we can use the one parameter family e−tDt to define an operator associated to Σ. So in principle any positive energy representation of DiffΩ (T3 ) can be extended to a representation of M . The next step is to construct positive energy representations. In the case of conformal field theory the two basic representations, free bosons and free fermions, are related to sections of holomorphic bundles over Riemann surfaces. We shall argue that something similar should be true in the present case. If v ∈ H is a vector and vt = e−tD v, where D is the operator defined in (5.1), then the “evolution equation” for vt is ∂vt ∂vt ∂vt ∂vt = −Dvt = −i −j −k ∂t ∂θ1 ∂θ2 ∂θ3 which is the Cauchy-Riemann-Fueter equation. This equation is a quaternionic analogue of the Cauchy-Riemann equations. The use of the CauchyRiemann-Fueter equation allowed Fueter to extend the theory of holomorphic functions of one variable to the case of functions of a quaternionic variable, descriptions of Fueter’s work can be found in [26]. So the final idea is that the holomorphic structure in Segal’s elliptic objects has to be replaced by some form of quaternionic manifolds and quaternionic holomorphic bundles over them. (5.2)

6. Quaternionic algebra In this section we shall summarise some of the ideas of D. Joyce, see [14, 30], about quaternionic algebra and quaternionic geometry and see how they lead to natural definitions of quaternionic Fock spaces. The basic construction in Joyce’s theory is the quaternionic tensor product. This product is also necessary in principle to extend condition one of the definition of conformal field theory. An alternative approach to quaternionic quantum mechanics and quaternionic quantum field theory is given in [1]. Let I = {ai + bj + ck ∈ H} denote the imaginary quaternions. If U is a (left) H-module U , we shall write U ∗ for the set (6.1)

U ∗ = {φ : U → H, | φ(qu) = qφ(u), ∀ q ∈ H, u ∈ U }.

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Let U be a (left) H-module U with a distinguished (real) real subspace U 0 and define U † = {φ ∈ U ∗ such that φ(u) ∈ I, ∀u ∈ U 0 } An AH-module is a generalisation of the pair (H, I); it consists of a (left) H-module U with a distinguished real subspace U 0 such that if u ∈ U 0 and φ(u) = 0, ∀φ ∈ U † , then u = 0. Usually we shall not write U 0 which is ∗ understood. If U is a H-module, then define a map iU : U → H ⊗R (U † ) by iU (u).α = α(U ). It is not difficult to show that iU is a quaternionic linear ∗ monomorphism, where H acts on H ⊗R (U † ) in the obvious way. Definition 6.1. Let U and V be two AH-modules and consider the H-module ∗ ∗ ∗ H ⊗R (U † ) ⊗R (V † ) . Exchanging the factors we may regard (U † ) ⊗R iV (V ) ∗ ∗ ∗ and iU (U ) ⊗R (V † ) as submodules of H ⊗R (U † ) ⊗R (V † ) . The quaternionic tensor product U ⊗H V is the H-module n o n o ∗ ∗ (6.2) U ⊗H V = iU (U ) ⊗R (V † ) ∩ (U † ) ⊗R iV (V ) . The module U ⊗H V has a distinguished real subspace ∗

∗

(U ⊗H V )0 = (U ⊗H V ) ∩ (I ⊗ (U † ) ⊗ (V † ) ), and the pair (U ⊗H V, (U ⊗H V )0 ) defines a AH-module. Lemma 6.2. Let U, V, W be AH-modules. Then there are canonical AHhomomorphisms H ⊗H U ∼ = (U ⊗H V ) ⊗H W. = V ⊗H U and U ⊗H (V ⊗H W ) ∼ = U, U ⊗H V ∼ Two important consequences of this lemma are that firstly one can form symmetric and antisymmetric powers of an AH-module U denoted by S k U and ∧k U and secondly one can define AH-algebras, examples of the last construction are the symmetric algebra S ∗ U and the exterior algebra ∧∗ U . An appropriate choice of U gives a candidate for a quaternionic bosonic Fock space of a particle 3. Recall that using holomorphic quantisation – see for example the first section of Fadeev Lectures in [6] or chapter two of [9] one can identify the space of quantum states of a harmonic oscillator in terms of the holomorphic functions on the plane. In this representation the symmetric power S n (C) appears both as the space of homogeneous polynomials of degree n (in the power series expansion of holomorphic functions) or as the n-particle space. Quoting Fadeev [9, Section 2.2]: This representation is very useful for the field theory because in this case the free Hamiltonian can be represented by the sum of Hamiltonians of an infinite set of oscillators. Let U be the set of linear functions f : H → H of the form f (x0 + ix1 + jx2 + kx3 ) = q0 x0 + q1 x1 + q2 x2 + q3 x3 3We

ignore if there is a real physical content in these definitions.

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for certain quaternions qi with i = 0, . . . , q3 . The function f is q-holomorphic if and only if q0 + q1 j + q2 j + q3 k = 0. Thus the set U of q-holomorphic linear functions is isomorphic to H3 . Let U 0 = {(q1 , q2 , q3 ) ∈ H3 : qj ∈ I for j = 1, 2, 3 and q1 i + q2 j + q3 k ∈ I}. Then (U, U 0 ) defines an AH module. The k-th symmetric power SHk U has a natural identification with the q-holomorphic functions which are homogeneous polynomials of degree k. So the sum ⊕k SHk U could serve as a quaternionic analogue of the state space of a harmonic oscillator. One might hope that this Fock space could be the basis for the definition of a Fock space for free bosonic fields 4. 7. Other Aspects of the Program It is our aim in this section to briefly discuss some other aspects which need to be developed. Everything is highly speculative. 7.1. The coefficient ring of K3 cohomology. In this section we shall conjecture about the possible nature of the coefficient ring of K3 cohomology. This ring should be related to sections of some bundle over the moduli space of K3-surfaces. One question if is possible to relate this ring to modular forms. We will follow closely the exposition of [28] where we refer the reader for details. 7.1.1. Siegel Modular Forms. One of the interesting thing about elliptic cohomology is the relation with modular forms. Can we find a relation between K3 cohomology and modular forms? Definition 7.1. The Siegel upper half plane h2 is the space of 2 × 2 complex matrices τ which are symmetric and have a positive definite imaginary part. The symplectic group Sp4 (R) acts on h2 by A B (7.1) × τ → (Aτ + B)(Cτ + D)−1 C D Definition 7.2. A Siegel modular from of degree 2 and weight k is a holoA B morphic function f : h2 → C such that for each matrix C D ∈ Sp4 (R) and τ ∈ h2 (7.2)

f (τ ) = det (Cτ + D)−k f ((Aτ + B)(Cτ + D)−1 )

Siegel modular forms can be seen as sections over the moduli space of principally polarised Abelian surfaces. They are related to the moduli space of K3-surfaces via the Kummer map. One might speculate that if one considers only Kummer surfaces one can show the coefficient ring of K3 cohomology is some subset of the set of Siegel Modular forms. 4This

definition has be extended to consider free boson fields. This is work in progress

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Jorge A. Devoto

7.2. Singular K3-surfaces. Besides the smooth K3-surfaces there are also important examples of singular ones. Which role they play in the theory? Can one extend the definition of morphisms in M 3 to include them? 7.3. Projective representations of DiffΩ (M). How can one build other positive energy representations of DiffΩ (M)? Are they related to the representations of double or triple loop groups? One aspect to consider is that representations coming from quantum physics are projective and therefore they correspond to genuine representations of central extensions of groups. Pressley and Segal showed in [18] that in a certain sense there are no interesting central extensions of double and triple loop groups. Since we are working with the quaternions it is reasonable to conjecture that one might need to consider some non-central extensions. Projective quaternionic representations are studied in [1]. 7.4. Index theory on L (2) X. The most important question if there is some formal operator on the double loop space related to the genus associated to a K3 surface. References 1. S. Adler, Quaternionic quantum mechanics and quantum fields, Oxford University Press, 1995. ´ 2. M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. Ecole Norm. Sup. 10 (1977), 87–131. 3. A. Ashtekar, T. Jacobsen and L. Smolin, A new characterization of half-flat solutions to Einstein’s equations, Commun. Math. Phys. 115 (1988), 631-648. 4. M. Atiyah and R. Bielawski, Nahm’s equations, configuration spaces and flag manifolds, xxx.lanl.gov – math.RT/0110112v1, 2001. 5. J. Baez, The octonions, Bull. of the A.M.S. 39 (2002), no. 2, 145–206. 6. P [et all] Deligne (ed.), Quantum fields and strings: A course for mathematicians, vol. I, A.M.S., 1999. 7. S. Donaldson, Nahm’s equations and the classification of monopoles, Commun. Math. Phys. 96 (1984), 387–407. , Complex cobordisms, astekar’s equations and diffeomorphisms, London Math. 8. Soc. Lecture Notes (1993), no. 192, 45–55. 9. L. Faddeev and A. Slavnov, Gauge fields: An introduction to quantum theory, second edition ed., Frontiers in Physics, Perseus Books, 1991. 10. H Geiges and J. Gonzalo, Contact geometry and complex surfaces, Inv. Mathematicae 121 (1995), no. 1, 147–211. 11. R. Hartshorne, Algebraic geometry, Springer-Verlag (1983). 12. R. W. H. T. Hudson, Kummer’s quartic surfaces, Cambridge Univ. Press, 1990. 13. D. Joyce, Compact manifolds with special holonomy, Oxford University Press, 2000. 14. , A theory of quaternionic algebra with applications to hypercomplex geometry, Available from arXiv.math.DG/0010079, October 2001. 15. D. Karachanyan , R. Mavelyan and R. Mkrtchyan, Area preserving structure of 2d gravity, Available from arXiv.hep-th/9401031. 16. M. Kreck and S. Stolz, HP2 bundles and elliptic cohomology, Acta Math. 171 (1993), 232–261.

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17. L. J. Mason and E. T. Newman, A connection between the Einstein equations and Yang-Mill equations, Commun. Math. Phys. 121 (1989), 659–668. 18. A. Pressley and G. B. Segal, Loop groups and their representations, Oxford University Press, 1986. 19. D. Quillen, On the formal group laws of unoriented and complex cobordism, Bull. of the A. M. S. 75 (1969), 1293–1298. 20. , Elementary proof of some results of cobordism theory using Steenrod operations, Adv. in Math. 7 (1971), 29–56. 21. P Saskida, Unpublished dissertation. 22. G. B. Segal, Elliptic cohomology, S´eminaire Bourbaki 1987/88, Ast´erisque, vol. 161-162, Soc. Mathematique de France, 1988, pp. 187–201. , The definition of conformal field theory, Unpublished Notes Oxford, 1989. 23. 24. Simon, B, Trace ideals and their applications London Mathematical Society. Lecture Notes vol. 35, (1979). 25. J. Stienstra, Formal group laws arising from algebraic varieties, Am. J. of Math. 109 (1987), no. 5, 907–926. 26. A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Phyl. Soc. 85 (1979), 199–225. 27. C. Thomas, Elliptic cohomology, Surveys on Surgery Theory (S Capell, A Ranicki, and Rosenberg J., eds.), Annals of Mathematics Studies, Princeton University Press, 1999, pp. 409–439. 28. C. Thomas, Elliptic cohomology, Plenum, 1999. 29. G. van der Geer and T. Katsura, On a stratification of the moduli of K3 surfaces, Preprint (1999), Available from math.AG/9910061. 30. D. Widdows, Quaternion algebraic geometry, Ph.D. thesis, Univ. of Oxford, 2000. 31. E. Witten, Elliptic genera and quantum field theory, Communications in Mathematical Physics 109 (1987), 525–536. 32. , The index of the Dirac operator on the loop space, Elliptic curves and Modular Forms in Algebraic Topology (P. Landweber, ed.), Lecture Notes in Math, vol. 1326, Springer-Verlag, 1988, pp. 161–181.

Dept. of Math. ITBA Av. Madero 399, Buenos Aires, Argentina Dept. of Math. FCECN, University of Buenos Aires, Ciudad Universitaria, Buenos Aires, Argentina [email protected]

THE M -THEORY 3-FORM AND E8 GAUGE THEORY EMANUEL DIACONESCU, GREGORY MOORE, AND DANIEL S. FREED

Abstract. We give a precise formulation of the M -theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8 gauge field. This leads to a precise version of the Chern-Simons interaction of 11-dimensional supergravity on manifolds with and without boundary. As an application of the formalism we give a formula for the electric C-field charge, as an integral cohomology class, induced by self-interactions of the 3-form and by gravity. As further applications, we identify the M theory Chern-Simons term as a cubic refinement of a trilinear form, we clarify the physical nature of Witten’s global anomaly for 5-brane partition functions, we clarify the relation of M -theory flux quantization to K-theoretic quantization of RR charge, and we indicate how the formalism can be applied to heterotic M -theory.

1. Introduction This paper summarizes a talk given at the conference on Elliptic Cohomology at the Isaac Newton Institute, in December, 2002 [1] In this paper we will discuss the relation of M-theory to E8 gauge theory in 10, 11, and 12 dimensions. Our basic philosophy is that formulating Mtheory in a mathematically precise way, in the presence of nontrivial topology, challenges our understanding of the fundamental formulation of the theory, and therefore might lead to a deeper understanding of how one should express the unified theory of which 11-dimensional supergravity and the five 10-dimensional string theories are distinct limits. To be more specific, let us formulate three motivating problems for the formalism we will develop. The first problem concerns 11-dimensional supergravity. We will be considering physics on an 11-dimensional, oriented, spin manifold, Y . When it has a boundary we will denote ∂Y = X. The basic fields of 11-dimensional supergravity are a metric g, (Lorentzian or Euclidean) a “C-field,” and a gravitino ψ ∈ Γ(S ⊗ T ∗ Y ) where S is the spin bundle on Y . Our main concern in this paper is with the mathematical nature of the C-field. In the standard formulation of supergravity the C-field is regarded as a 3-form gauge potential (1.1)

C ∈ Ω3 (Y ).

Date: Nov. 30, 2003; Revised March 22, 2004.

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The M-theory 3-form and E8 gauge theory

45

This generalizes the Maxwell potential A ∈ Ω1 (M) of electromagnetism on a manifold M. The fieldstrength is defined to be (1.2)

G = dC ∈ Ω4 (Y ).

Whereas in classical electromagnetism the gauge potential A is a global 1form, in the quantum theory Dirac’s law of charge quantization changes the geometric nature of A: it is now a connection on a U (1) line bundle over M. (See [2] (sec. 3) and [3] (sec.2) for expository accounts of how charge quantization leads to a U (1) connection and its generalizations in differential cohomology theory.) One of our goals is to give a similar geometric description of the C-field. In standard supergravity the path integral measure is a canonical formal measure on the space of fields, defined by the metric g and the Faddeev-Popov procedure, weighted by an action. Schematically, the exponentiated action is given by " # Z 1 1 / Φ(C) (1.3) exp −2π vol(g)R(g) + 3 G ∧ ∗G + ψ¯Dψ 9 2` Y ` plus 4-fermion terms, where ` is the 11-dimensional Planck length and, roughly speaking, Z 1 2 (1.4) Φ(C) ∼ exp 2πi CG − CI8 (g) Y 6 Here I8 (g) is a quartic polynomial in the curvature tensor. We will give precise normalizations for the C-field and I8 in sections 3.2 and 4.1 below. Now let us suppose we wish to formulate the action in the presence of nontrivial topology. This means, among other things, that we wish to allow the fieldstrength G in (3.37) to define a nontrivial class in the DeRham cohomology of Y . Evidentally, we cannot use (1.4). One might be tempted to introduce a 12-manifold bounding Y and use Stokes formula. This procedure works (after accounting for several subtleties) when Y is closed but fails when ∂Y 6= ∅. Thus our first problem is: Find a mathematically precise definition of Φ(C) when G is cohomologically nontrivial, and ∂Y = X is nonempty. We will give a complete answer to this question in section 5 below. Having formulated the measure we can next turn to applications. When ∂Y = X is nonempty, the path integral for the C-field on a manifold with boundary Y defines a wavefunction of the boundary values CX of C. We may denote this wavefunction as Ψ(CX ). Now, there is a group of gauge transformations G of the C-field and the wavefunction must be suitably gauge invariant. Our second problem is If Ψ(CX ) is a nonvanishing gauge invariant wavefunction, what conditions are imposed on the values of CX ? Put more simply: What is the Gauss law for the C-field? We will find nontrivial conditions on CX in section 7.1 and will interpret them as the condition that

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

the induced electric C-field charge associated to a C-field configuration and gravity must vanish. An analogy might be useful at this point. Because of the Chern-Simons phase Φ(C), the path integral is a generalization of the familiar 3-dimensional massive gauge theory: Z Z Z 1 k 2 3 (1.5) Ψ(AX ) = [dAY ] exp − TrF ∧ ∗F + i Tr AdA + A 2e2 4π Y 3

where AY is a connection on a bundle PY over the three-manifold Y and AX is the fixed boundary value. The “Gauss law” is the statement that this “function” is suitably gauge invariant. The Gauss law implies, among other things, that one can only define a nonvanishing gauge invariant wavefunction when c1 (PX ) = 0, where PX is the restriction of PY . In the case e2 = ∞ we have pure Chern-Simons theory. The Gauss law then implies F (AX ) = 0 and leads directly to the mathematical interpretation of Ψ as a section of a complex line bundle over the moduli space of flat connections on a Riemann surface. The analogy to Chern-Simons gauge theory cannot be pushed too far. In (1.5) the integer parameter k appears. In (1.4) the analogous parameter is, roughly speaking, k = 1/6. It is precisely this fractional value which makes the proper definition of Φ(C) subtle. Finally, and most importantly, let us turn to the third motivating problem, namely an understanding of the emergence of E8 gauge symmetry in Mtheory. There are several hints that there is a fundamental role for the group E8 in M-theory. First, there are the famous U -duality global symmetries that arise when M-theory is compactified on tori. These symmetries involve exceptional groups. Next the duality relating M theory on K3 to heterotic string theory on T 3 implies that there are enhanced gauge symmetries when the K3-surface develops A − D − E singularities. These gauge symmetries certainly include E8 × E8 . Next, a construction of Horava-Witten shows that a quotient of M-theory on X × S 1 by an orientation reversing isometry of S 1 leads to the E8 × E8 heterotic string, with E8 gauge fields propagating on the boundary [4, 5]. Furthermore, in [6] Witten gave a definition of Φ(C) that used E8 gauge theory in 12 dimensions. This definition was then used in [7] to establish a connection to the K-theoretic classification of RR fluxes in the limit that M theory reduces to type II string theory. All this suggests a hidden E8 structure in M theory which might point the way to a useful reformulation of the theory. Thus we have our third problem: What is the precise relation of the C-field of 11-dimensional supergravity to an E 8 gauge field? We will propose an answer to this question in section 3. The remainder of the paper endeavors to demonstrate that this answer can be useful. Finally, we would like to close this introduction with a general remark. In the physics literature gauge fields are traditionally modeled as belonging to a space with a group action, or simply as an element of the quotient space. For

The M-theory 3-form and E8 gauge theory

47

example, a 1-form gauge field is typically taken to be a connection on a fixed principal bundle, and the group of gauge transformations acts on the space of all such connections. It is technically sounder to use a different model. Namely, we consider instead the groupoid of all connections on all principal bundles; morphisms are maps of principal bundles which preserve the connections. There is still a quotient space of equivalence classes of connections, and this is the space over which one writes the functional integral. This model is local, whereas fixing a particular principal bundle is not. Also, we can replace the groupoid by an equivalent groupoid without changing the physics. This allows us the convenience of using different models adapted to different purposes. In section 3 we introduce a few different models for the C field. One (described in secs. 3.1-3.3) is closer to the physics tradition, while the other two (described in secs. 3.4-3.5) involves a groupoid model. In this paper we mostly rely on the first model, which emphasizes the connection to E8 . There is an alternative approach to some of the issues in this paper [8] which uses yet other models, following the ideas developed in [9]. That point of view leans more heavily on the (differential) algebraic topology to construct the cubic form which appears in the M-theory action. In particular, the model for the C-field does not involve E8 gauge fields. Two very recent papers on subjects closely related to the present work are [10, 11]. 2. The gauge equivalence class of a C-field Our first task is to give a precise answer to the question: “What is a Cfield?” In this section we will give a partial answer, by describing the gauge equivalence class (gec) of a C-field. The answer will be that the gec of a C-field is a (shifted) differential character. To motivate this description let us consider the description of the gauge equivalence class of a U (1) gauge field on a manifold M. The key to answering this problem turns out to be to consider the holonomy around 1-cycles γ. This is certainly gauge-invariant information, and it turns out to be all the gauge invariant information. More precisely, given a connection A on a line bundle over M we may regard the holonomy as a map on closed 1-cycles χ ˇA : Z1 (M) → U (1) given by I (2.1) χˇA (γ) = exp i A γ

It is then natural to ask how χ ˇA differs from an arbitrary such map. The answer is that there exists a closed 2-form, the fieldstrength of χ, ˇ such that if γ = ∂B2 is a boundary then Z (2.2) χ ˇA (γ) = exp i F B2

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

Note that it follows that F has 2π ZZ-periods, denoted F ∈ Ω22πZZ (M), and that F is closed. Note also that it follows from (2.2) that χ ˇ is a homomorphism of abelian groups. It is smooth in an appropriate sense. Such maps χ ˇ such that a fieldstrength exist are in 1-1 correspondence with the gec of U (1) bundles with connection. In supergravity theories one often encounters p-form gauge potentials. One natural mathematically precise formulation of such gauge potentials is in terms of differential characters, which generalize the above description of U (1) gauge connections. By definition a Cheeger-Simons character, or differential ˇ p+1 (M) is a homomorphism character χ ˇ∈H (2.3)

χˇ : Zp (M) → U (1)

where Zp (M) is the group of p-cycles on M, such that there is a fieldstrength ω(χ) ˇ ∈ Ωp+1 ˇ ZZ (M). That is, there exists a closed differential (p + 1)-form ω(χ) with ZZ-periods such that Z (2.4) Σp = ∂Bp+1 ⇒ χ(Σ ˇ p ) = exp 2πi ω(χ) ˇ . Bp+1

We will identify the space of gec’s of a p-form gaugefield with the space of ˇ p+1 (M). In order to establish some notation let differential characters χˇ ∈ H us recall the basic facts about differential characters. (See [9] and references therein for further details about differential characters and cohomology). The gauge invariant information in a Cheeger-Simons character can be expressed in two distinct ways, each of which is summarized by an exact sequence. The first sequence is related to the space of flat characters, that is, characters with ω(χ) ˇ = 0: (2.5)

ˇ p+1 (M) → Ωp+1 (M) → 0 0 → H p (M, U (1)) → H ZZ

This is clear because a flat character defines a homomorphism of abelian groups Hp (M, ZZ) → U (1), which, by Poincar´e duality is H p (M, U (1)). In order to define the second sequence we begin with the topologically trivial characters. If A ∈ Ωp (M) is globally defined, then we may define a differential character: Z (2.6) χˇA (Σp ) := exp[2πi A]. Σp

Note that, first of all, χˇA only depends on A modulo ΩpZZ , and secondly, the fieldstrength ω(χˇA ) = dA is trivial in cohomology. Thus, the cohomology class of the fieldstrength is an obstruction to writing a character as a trivial character (2.6). In fact we have the second sequence: (2.7)

ˇ p+1 (M) → H p+1 (M, ZZ) → 0 0 → Ωp /ΩpZZ → H

The projection map in this sequence defines the characteristic class of χˇ which we will denote as a(χ) ˇ ∈ H p+1 (M, ZZ). Note that this is a class in integral

The M-theory 3-form and E8 gauge theory

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cohomology, not DeRham cohomology. The compatibility relation between the two sequences states that (2.8)

a(χ) ˇ IR = [ω(χ)] ˇ DR

where a(χ) ˇ IR denotes the image of a(χ) ˇ in DeRham cohomology. It turns out that the gec of a C-field is not quite a differential character, but is rather a “shifted” differential character [6]. The reason for this is best explained in terms of the coupling to the “membrane.” In the formulation of M-theory - as it is presently understood - one posits the existence of fundamental “electrically charged” membranes with 3-dimensional worldvolumes. In Maxwell theory, a charged particle, of charge e, moving along a worldline γ, couples to the background gauge potential A via the holonomy: Z (2.9) exp[i eA]. γ

In M theory the membrane couples to the C-field in an analogous way. The standard coupling of supergravity fields to the membrane wrapping a 3-cycle Σ is usually written: Z (2.10) ∼ exp[2πi C]. Σ

More accurately, because of the worldvolume fermions on the membrane, the topologically interesting part of the membrane amplitude is Z q / S(N ) exp 2πi C DetD (2.11) Σ

where N is the rank 8 normal bundle of the embedding ι : Σ ,→ Y and S(N ) is an associated chiral spinor bundle. The squareroot has an ambiguous sign when one considers (2.11) on the space of all 3-cycles Z3 (Y ) and metrics, and the holonomy (2.10) has a compensating sign such that the product is welldefined [6]. The net effect is that it is the gec of the difference [C1 − C2 ] of ˇ 4 (Y ). We can be slightly C-fields which is an honest differential character in H more precise about the nature of the shift. As shown in [6] the requirement that (2.11) is well-defined implies quantization condition on the fieldstrength: 1 [G]DR = aIR − λIR 2 4 where a ∈ H (Y, ZZ) is an integral class and λ is the canonical integral class of the spin bundle of Y . 1 In conclusion the we have a partial answer to the question “What is a C-field?” The gauge equivalence class of a C-field is a shifted differential character. We have not yet defined precisely what is meant by “shifted.”

(2.12)

1Put

differently, only twice the characteristic class of the twisted differential character is well-defined, and it is constrained to be equal to w4 modulo two.

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

This will be explained in the section 3.4 (equation 3.25). The space of shifted ˇ 41 (Y ). characters, with shift 12 λ will be denoted H λ 2

3. Models for the C-field In the previous section we have explained what the gauge equivalence class of a C-field is, but have not answered the question: “ What is a C-field?” An analogous situation would be to have in hand a formulation of nonabelian gauge theory in terms of gauge invariant quantities without having introduced connections on principal bundles. In physical theories the requirement of locality, that we be able to formulate the theory in terms of local fields, forces us to introduce redundant variables, such as gauge potentials. Similarly, in the standard formulation of supergravity one takes the C field to be an ordinary 3-form C ∈ Ω3 (Y ) subject to a gauge invariance C → C + ω where ω is a closed 3-form. When ω is exact such gauge transformations are referred to as small gauge transformations. When ω is closed but not exact the gauge transformation is a large gauge transformation. In M-theory, quantization of membrane charge requires that ω have integral periods. In this view, the gauge equivalence class of a C-field is (3.1)

[C] ∈ Ω3 (Y )/Ω3ZZ (Y ).

According to (2.7) such fields define topologically trivial differential characters. However, many interesting nontrivial phenomena in string/M-theory involve topologically nontrivial characters and hence we must modify the geometric description of the C-field. In this paper we will focus on the “E8 model for the C-field,” which seems well-suited to describing the M-theory action. As we will discuss below, there are other models of the C-field which can be considered to be equivalent to the E8 model. 3.1. The E8 model for the C-field. The E8 model is motivated by Witten’s definition [6] of the M-theory action as an integral in 12-dimensions. (We will review Witten’s definition in section 4.1 below.) This model is based on the topological fact that there is a homotopy equivalence (3.2)

E8 ∼ K(ZZ, 3)

up to the 14-skeleton. Equivalently, the homotopy groups πi (E8 ) of E8 vanish for 4 ≤ i ≤ 14. It follows from (3.2) that BE8 ∼ K(ZZ, 4) and therefore, for dim M ≤ 15, there is a one-one correspondence between integral classes (3.3)

a ∈ H 4 (M, ZZ)

The M-theory 3-form and E8 gauge theory

51

and isomorphism classes of principal E8 bundles over M. 2 For each a we pick a specific bundle P (a) → M. 3 We now come to a central definition; we will say that a “C-field on Y with characteristic class a ” is an element of (3.4)

EP (Y ) := A(P (a)) × Ω3 (Y )

where A(P ) is the space of smooth connections on the principal bundle P . Thus, our “gauge potentials,” or “C-fields,” will be pairs (A, c) ∈ EP (Y ). We will often denote C-fields by Cˇ = (A, c), and we will call c the “little c-field.” ˇ Given the “gauge potential” Cˇ = 3.2. The gauge equivalence class of C. (A, c) ∈ EP (Y ) we will now describe its gauge equivalence class. The principle we use is that the holonomy, or coupling of the C-field to an elementary membrane contains all the gauge invariant information in C. The holonomy of Cˇ = (A, c) ∈ EP (Y ) around Σ is defined to be " Z # 1 (3.5) χˇA,c (Σ) := exp 2πi CS(A) − CS(g) + c 2 Σ Here CS(A) and CS(g) are Chern-Simons invariants associated to the gauge field A and the metric g, normalized by 4 1 F2 (3.6) dCS(A) = trF 2 := Tr248 60 8π 2 1 dCS(g) = trR2 := − (3.7) Tr11 R2 16π 2 Note that 1 (3.8) [trF 2 ]DR = aIR [trR2 ]DR = (p1 (T Y ))IR 2 It follows immediately that the fieldstrength of the character is 1 (3.9) ω(χˇA,c) = G = trF 2 − trR2 + dc 2 R Note that the normalizations are chosen such that exp[2πi CS(A)] is Σ R well-defined. However, exp[πi Σ CS(g)] has a sign ambiguity when regarded

2A more elementary way to understand this is to use the obstruction theory arguments of [12]. 3This is somewhat unnatural and will lead to problems with gluing of manifolds and hence with locality. One can easily modify the definition below to include triples (P, A, c) where P is an E8 bundle in the isomorphism class determined by a. The discussion that follows is not changed in any essential way, except that one must account for bundle isomorphisms when discussing equivalences of (P, A, c) with (P 0 , A0 , c0 ). We have suppressed this refinement here in the interest of simplicity and brevity. R 4In general CS(A) is not well defined as a differential form, and only exp[2πi Σ CS(A)] is well-defined. See [13]. For a recent account see [14], sec.1.2. Below we will have occasion R to use the relative Chern-Simons invariant CS(A, A0 ) := [0,1] trF 2 defined by integrating along a straightline between the two connections. This is well-defined as a differential form.

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

as a function on Z3 (Y ) × Met(Y ). This sign ambiguity is cancelled by the sign ambiguity of the worldvolume fermion determinant: q / S(N ) ) (3.10) Det(D

It is the metric dependence required to cancel the sign ambiguity of (3.10) which leads to χˇA,c being a shifted character. Previous authors have proposed that there should be an identification of the C field with an E8 Chern-Simons form. See, for examples, [15], and [16]. However, a formula such as C = CS(A) is unsatisfactory for many reasons. A similar idea based on OSp(1|32) was proposed in [17]. Related ideas, in which the 3-form gauge potential of 11-dimensional supergravity should be considered as a composite field, have been explored in [18]. The model we have just described, ( which was inspired by the IR/ZZ index theory of Lott [19], and was first announced in [20]) fits into this circle of ideas, but we emphasize that the E8 connection plays a purely auxiliary role, at least on manifolds without (spatial) boundaries. This becomes more clear when one considers alternative formulations which do not involve E8 at all. 3.3. The C-field gauge group. Now that we have defined “gauge potentials” we seek a gauge group G so that ˇ 41 (Y ). (3.11) EP (Y )/G = H 2

λ

The fiber of the group orbit should be defined by (3.12)

(A, c) ∼ (A0 , c0 )

⇔

χˇA,c = χ ˇA0 ,c0

This condition is easily solved: (A, c) ∼ (A0 , c0 ) iff there exists α ∈ Ω1 (adP ) and ω ∈ Ω3ZZ (Y ) such that : (3.13) (3.14)

A0 = A + α

α ∈ Ω1 (adP )

c0 = c − CS(A, A + α) + ω

ω ∈ Ω3ZZ (Y )

To prove this, note that equality of fieldstrengths implies c0 −c = CS(A0 , A)+ ω for some closed 3-form ω. Then equality of the holonomies shows that ω must have integral periods. ¿From (3.12,3.13,3.14) we might conclude that the “C-field gauge group” is: (3.15)

?

G = Ω1 (adP ) × Ω3ZZ (Y ).

The right hand side of (3.15) is indeed a group, with group law (3.16)

(α1 , ω1 )(α2 , ω2 ) = (α1 + α2 , ω1 + ω2 + d(trα2 ∧ α1 ))

and does satisfy (3.11). Nevertheless, it is not precisely the gauge group we need. This can be understood in two ways, one physical and one mathematical.

The M-theory 3-form and E8 gauge theory

53

Let us first explain the physical point of view. In electromagnetism, if ∂Y = X, the worldline of a charge particle can end on a point P ∈ X. The coupling of the charged particle to the background gauge potential Z (3.17) exp[i eA] γ

is not gauge invariant. Now note that gauge transformations in Maxwell ˇ 1 (Y ), since H ˇ 1 (Y ) is just the theory can be thought of as elements χˇ ∈ H group of U (1)-valued functions on Y . In this view, the gauge transformation by χˇ of the “open wilson line” (3.17) is: Z Z (3.18) exp[i eA] → χ(P ˇ ) exp[i eA]. γ

γ

In M theory, the worldvolume of a membrane Σ can end on a 2-cycle σ. By analogy with electromagnetism we should define C-field gauge transformations to act on such “open membrane Wilson lines” as: (3.19)

exp[2πi

Z

Σ

C] → χ(σ) ˇ exp[2πi

Z

C]

Σ

R where exp[2πi Σ C] is short for (3.5), and χ(σ) ˇ is a U (1)-valued function on Z2 (X). Moreover, the existence of a fieldstrength for C shows that there must exist a fieldstrength for χ. ˇ Indeed, suppose σ = ∂Σ, and C → C + x with x ∈R Ω3ZZ (Y ). Then, if Σ ⊂ X, consistency with (3.19) demands χ(σ) ˇ = 3 ˇ exp[2πi Σ ω]. But this is the defining property of χˇ ∈ H (X) ! Since we could choose any 10-dimensional subspace X in Y in this discussion, we conclude ˇ 3 (Y ). that the gauge transformation parameter should be regarded as χˇ ∈ H In this way we conclude that the proper definition of the gauge group should be ˇ 3 (Y ) (3.20) G := Ω1 (adP ) × H Recall that ˇ 3 (Y ) → Ω3 (Y ) → 0 0 → H 2 (Y, U (1)) → H ZZ

(3.21)

and thus we have a nontrivial extension of the naive gauge group (3.15). The gauge group (3.20) acts on EP (Y ) via (3.22)

(α, χ) ˇ · (A, c) = (A + α, c − CS(A, A + α) + ω(χ)) ˇ

The group law is (3.23)

(α1 , χ ˇ1 )(α2 , χ ˇ2 ) = (α1 + α2 , χ ˇ1 + χˇ2 + χˇb )

where χˇb is a topologically trivial character with b = Tr(α2 α1 ). It is convenient to introduce some terminology. We will refer to gauge transformations in the connected component of the identity in (3.20) as small gauge transformations. Otherwise, the gauge transformation is referred to as a large gauge transformation. We will refer to gauge transformations in H 2 (Y, U (1))

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as micro gauge transformations since they leave (A, c) unchanged. 5 As we shall see, they nevertheless have a crucial physical effect. This formalism makes clear the physical role of the E8 gauge field. It is a kind of topological field theory since we can shift A to any other connection A0 ∈ A(P (a)), and hence A is only constrained by topology. 6 3.4. C-fields and categories. Let us now turn to a more mathematical justification of the definition (3.20). Quite generally, bosonic fields with internal symmetry should be viewed as objects of a groupoid —a category with all arrows invertible—and equivalent groupoids provide alternative models for the same physical object. In this section we demonstrate an equivalence of our model with another possible model for the C-field. To begin, since we have a G-action on a space EP (Y ) we can form an associated groupoid. We regard this groupoid as a category. The objects of this category are elements of EP (Y ). The morphisms Mor(Cˇ1 , Cˇ2 ) are the group elements taking Cˇ1 to Cˇ2 . We denote this category by EP (Y )//G. Note from (3.22) that the objects Cˇ in the category have automorphism group given by the flat characters of degree three: H 2 (Y, U (1)). Now, in [9] M. Hopkins and I. Singer have formulated a theory of “differential cochains” and “differential cocycles” which refines the theory of differential cohomology. Differential cocycles may be regarded as one definition of what is meant by gauge potentials for abelian p-form gauge fields.7 In the framework of [9] a “shifted differential character” is the equivalence class of a differential cocycle which trivializes a specific differential 5-cocycle related to W5 (Y ). The Hopkins-Singer theory can therefore be applied to the C-field of M-theory. To be more specific, 8 the cohomology class w4 (Y ) ∈ H 4 (Y, ZZ2 ) defines a differential cohomology class, w ˇ4 , because we can include H 4 (Y, ZZ2 ) ,→ ˇ 4 (Y ). The characteristic class of the flat character w H 4 (Y, IR/ZZ) ,→ H ˇ4 is the integral class W5 (Y ) ∈ H 5 (Y, ZZ), given by the Bockstein homomorphism applied to w4 (Y ). This class is interpreted as the background magnetic charge induced by the topology of Y , and this class must vanish in order to be able to formulate any (electric) C-field at all. On a spin manifold, W5 (Y ) = 0, since the class λ is an integral lift of w4 (Y ). When W5 (Y ) = 0 we may refine 5Note

that some micro gauge transformations can also be large gauge transformations! The characteristic class a(χ) ˇ for such gauge transformations will be torsion. 6In Donaldson-Witten theory there is also a symmetry under arbitrary shifts of the gauge potential, δA = ψ. In that case ψ is nilpotent. 7For some abelian p-form gauge fields, such as the Ramond-Ramond fields in Type II superstring theory, the quantization law is not in terms of ordinary cohomology so gauge fields are defined in terms of “differential functions” and “generalized differential cohomology theories.” 8This paragraph assumes some familiarity with [9].

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the differential cohomology class w ˇ4 to a differential cocycle by defining (3.24)

ˇ 5 = (0, 1 λ(g), 0) ∈ Zˇ 5 (Y ) ⊂ C 5 (Y, Z) × C 4 (Y, R) × Ω5 (Y ) W 2

1 2 where λ(g) = − 16π 2 TrR is functorially attached to the metric g. We then define a C-field to be a differential cochain Cˇ = (¯ a, h, ω) ∈ C 4 (Y, ZZ)×C 3 (Y, IR)× 4 ˇ 5: Ω (Y ) trivializing W

(3.25)

ˇ 5. δ Cˇ = W

Written out explicitly this means that

(3.26)

δ¯ a=0 1 δh = ω − a¯R + λ(g), 2 dω = 0.

We refer to these as shifted differential cocycles, and denote the space of such cochains by Zˇ 41 λ(g) . This is a principal homogeneous space for Zˇ 4 (Y ) and 2 hence may be considered as the set of objects in a category. Now, finally, we may explain the mathematical motivation for the choice of gauge group (3.20). It is only with this choice that we have the crucial theorem Theorem. There is an equivalence of the categories EP (Y )//G and Zˇ 41 λ(g) . 2

Proof: To prove this it suffices to establish the existence of a fully faithful functor F : EP (Y )//G → Zˇ 41 λ(g) such that any object in Zˇ 41 λ(g) is isomor2 2 ˇ for some C. ˇ Since both categories are groupoids, the morphic to F (C) phism spaces are principal homogeneous spaces for the automorphism group. Since the automorphism group is independent of object and category, namely H 2 (Y, U (1)), and since set of isomorphism classes of objects is the same, ˇ 41 (Y ) it simply suffices to establish the existence of a functor. namely H λ 2 Begin by choosing a Hopkins-Singer 4-cocycle (¯ c0 , h0 , ω0 ) on the classifying 4 space BE8 where [¯ c0 ] is a generator of H (BE8 ; ZZ), and ω0 is determined from the universal connection Auniv on EE8 by ω0 = trF (Auniv )2 . Now, there exists a map γ : P → EE8 which classifies the connection: γ ∗ (Auniv ) = A. 9 Let γ¯ : Y → BE8 be the induced classifying map on the base space Then we 9For

a nice proof, see [21], section 2. One needn’t rely on this, however. Instead, we introduce an equivalent category which includes a classifying map; see [14], section 3.1.

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define our functor on objects by: F (A, c) = (¯ cHS , hHS , ωHS ) where c¯HS = γ¯ ∗ (c0 ) (3.27)

hHS = γ¯ ∗ (h0 ) + c 1 ωHS = γ¯ ∗ (ω0 ) + dc = trF 2 (A) + dc − λ(g) 2

and one checks that (¯ cHS , hHS , ωHS ) is a shifted differential cocycle. ♠ In particular, and crucially, the automorphism group of a C-field (analogous to the constant gauge transformations in nonabelian gauge theory) is the same in both categories, namely H 2 (Y, U (1)). Finally, given the E8 model for the C-field a natural question one may ask is how E8 gauge transformations, that is, bundle automorphisms of P , are related to the C-field gauge transformations. The answer involves a construction which will prove useful later. Suppose (3.28)

(A, c) → (Ag , c)

g ∈ Aut(P )

is an E8 gauge transformation. Every transformation of this type is equivalent to a C-field gauge transformation (α, χ) ˇ acting on (A, c). We first find α, trivially it is (3.29)

α = Ag − A := g −1 DA g.

It follows that ω(χ) ˇ = CS(A, A+α) = CS(A, Ag ). A natural way to construct a character with this fieldstrength is to use g to construct the twisted bundle over X × S 1 (3.30)

Pg := (P × [0, 1])/(p, 0) ∼ (pg, 1)

and take the vertical connection dt∂t + dX + A where (3.31)

A = (1 − t)A + tAg

We can then set χ ˇ = χˇ(g,A) where (3.32)

"

χ ˇ(g,A) (σ) := exp 2πi

Z

σ×S 1

CS(A)

#

It is straightforward to check that (3.33)

1 ω(χˇ(g,A) ) = CS(A, Ag ) = − tr(g −1 DA g)3 + db(g, A) 3

where b(g, A) is globally well-defined. It is rather interesting to note that Aut(P ) is not a subgroup of G, since (α, χ) ˇ depends on A. Of course, since it is a group acting on EP (X) it does define a sub-groupoid. We will come back to this point in section 12 below.

The M-theory 3-form and E8 gauge theory

57

3.5. A third model for the C-field. There is a different approach to differential cohomology theory based on differential function spaces [9]. This motivates a different model for the C-field which will be described in some detail in the following. Of particular importance is the filtration on differential function spaces. While this approach will be less familiar to physicists, because gauge transformations are replaced by morphisms in a category, the approach has an appealing flexibility. As motivation for the construction, let us begin with a simpler example, ˇ 2 (Y ), which is isomorphic to namely the group of differential 2-characters H the group of equivalence classes of line bundles with connection on Y . One can construct a cocycle category for this group as follows. We take the objects to be pairs (L, A) consisting of a line bundle with connection on Y . The space of morphisms between two objects (L, A), (L0 , A0 ) consists of equivalence classes of pairs (L, A) on Y × ∆1 so that10 (3.34)

(L, A)|Y ×{0} = (L, A)

(L, A)|Y ×{1} = (L0 , A0 ).

Here ∆1 is the standard 1-simplex. Two pairs (L0 , A0 ), (L1 , A1 ) satisfying the same boundary conditions (3.34) are said to be equivalent if they are homotopy equivalent relative to the boundary conditions. This means that there exists a pair (L, A) on Y × ∆1 × I which restricts to (Li , Ai ) on Y × ∆1 × {i}, for i = 0, 1. Moreover, (L, A) should also restrict to (L, A), (L0 , A0 ) on the two boundary components Y × (∂∆1 ) × I. Composition of morphisms is defined by concatenation of paths. One can check that the construction sketched above yields a groupoid, but ˇ 2 (Y ). Two objects it is easy to see that this is not a cocycle category for H 0 0 (L, A), (L , A ) are equivalent if they are connected by a morphism (L, A). Then the parallel transport associated to the connection A determines an isomorphism φ : L → L0 . For a general connection A, φ is not compatible with the connections A, A0 . Therefore the equivalence classes of objects are in one to one correspondence to line bundles on Y up to isomorphism, which is not the answer we want. ˇ 2 (Y ) in this manner we have to refine our groupoid In order to obtain H structure so that the resulting equivalence relations are compatible with connections. This is a special case of a general construction described in detail in [9]. The main idea is to impose an extra condition on morphisms by keeping only pairs (L, A) so that (3.35)

F 1,1 (A) = 0

where F 1,1 (A) denotes the component of the curvature with one leg along ∆1 . The effect of this condition is that the parallel transport along ∆1 defined by A 10More

precisely, a morphism is a pair (L, A) together with isomorphisms (3.32) all up to the equivalence relation below. Also, note that the equivalence relation on morphisms and the composition of morphisms can be defined by working on Y × ∆2 , where ∆2 is the standard 2-simplex.

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becomes horizontal. Then the isomorphism φ : L → L0 preserves connections φ∗ A0 = A, and we obtain the desired set of equivalence classes. Note that for consistency we have to impose a similar filtering condition on the pairs (L, A) introduced below (3.34). Namely, we require all components with legs along I × ∆1 to vanish (3.36)

F 1,1 (A) = F 0,2 (A) = 0.

Note that this construction is valid even if Y is a manifold with boundary since we did not have to invoke integration by parts at any stage. The same will be true for the category of C-fields constructed below. (In physical applications one might wish to impose boundary conditions.) In the following we will produce a cocycle category E for C-fields on a manifold Y proceeding in a similar way. The objects are triples (P, A, c) as in section 3.1. Hence P is a principal E8 bundle on Y , A is a connection on P and c ∈ Ω3 (Y ). The “curvature” of a triple is the closed 4-form G defined by 1 G = trF 2 − trR2 + dc 2 The space of morphisms between two objects (P, A, c) and (P 0 , A0 , c0 ) consists of equivalence classes of triples (P, A, γ) on Y × ∆1 so that (3.37)

(3.38)

(P, A, γ)|Y ×{0} = (P, A, c)

(P, A, γ)|Y ×{1} = (P 0 , A0 , c0 ).

Denoting the curvature of (P, A, γ) by G, we impose a filtering condition G 3,1 = 0, where G (4−k,k) , k = 0, 1 denotes the component of G of degree k along ∆1 . Since dG = 0, this implies that G is pulled back from Y . (This means that G is “gauge invariant,” as it should be.) Two triples (P0 , A0 , γ0 ), (P1 , A1 , γ1 ) satisfying the same boundary conditions (3.38) are said to be equivalent if they are homotopy equivalent relative to the boundary. This means that there exists a triple (P, A, c) on Y ×∆1 ×I with obvious restriction properties, as explained below (3.34). For consistency we have to impose a filtering condition G3,1 = G2,2 = 0 analogous to (3.36). One can check that this construction defines a groupoid E. We claim this is a correct cocycle category for the C-field. In the remaining part of this section we will verify this claim by showing that i) the equivalence classes of objects are in one to one correspondence with shifted differential characters, and ii) the automorphism group of any object (P, A, c) is isomorphic to H 2 (Y, IR/ZZ). Let us start with (i). Two objects (P, A, c) and (P 0 , A,0 , c0 ) are equivalent if they are connected by a morphism. Suppose (P, A, γ) represents such a morphism. Since G is pulled back from Y , using the boundary conditions (3.38) we find that G = G = G0 . Moreover, the parallel transport associated to the connection A determines an isomorphism φ0 : P → P 0 . Therefore P, P 0 have the same characteristic class a ∈ H 4 (Y, ZZ). We conclude that we have a well

The M-theory 3-form and E8 gauge theory

59

defined map Ee → A41 λ (Y )

(3.39)

2

where Ee denotes the set of equivalence classes of objects of E and A41 λ (Y ) 2

denotes the set of pairs (a, G) ∈ H 4 (Y, ZZ) × Ω4ZZ+ λ (Y ) subject to the com2

patibility condition aIR −

λIR 2

= [G]DR . Note that A41 λ (Y ) is a principal 2

homogeneous space over the group A4 (Y ) of pairs (a, G) subject to the unshifted compatibility condition aIR = [G]DR . This map is clearly surjective. This is an encouraging sign since the group of shifted differential characters ˇ 1 λ is expected to surject onto this space. In order to finish the identificaH 2 tion, we should show that the fiber of this map is isomorphic to the torus H 3 (Y, IR)/H 3 (Y, ZZ). The fiber over a point (a, G) ∈ A41 λ (Y ) consists of isomorphism classes of 2 triples (P, A, c) with fixed (a, G). Up to isomorphism, a triple satisfying this condition can always be taken of the form (P0 , A0 , c) for some fixed (P0 , A0 ) with a(P0 ) = a. Therefore it suffices to determine the set of isomorphism classes of triples of the form (P0 , A0 , c) with fixed (a, G). Since only the threeform c is allowed to vary, it is straightforward to check that these triples are parameterized by the space of closed three-forms z ∈ Ω3cl (Y ). Two triples (P0 , A0 , c), (P0 , A0 , c + z) are isomorphic if they are connected by a morphism (P, A, γ) in the groupoid. Now we make use of the filtering condition G 3,1 = 0, which yields (3.40)

dt γ (3,0) + dY γ (2,1) + (trF (A)2 )(3,1) = 0.

Integrating this relation along ∆1 , and using the boundary conditions (3.38), we find Z Z (3.41) z = −dY β − η β= γ η= (trF (A)2 ). ∆1

∆1

At this point note that the bundle with connection (P, A) on Y × ∆1 satisfies identical boundary conditions along the two boundary components Y × {0} and Y × {1}. By a standard gluing argument, we can construct a bundle with e A) e over Y × S 1 . Then we have connection (P, Z e 2 ), (3.42) η= (trF (A) S1

which is a closed form with integral periods. In conclusion, the three-forms c and c+z parameterizing isomorphic triples (P0 , A0 , c), (P0 , A0 , c+z) with fixed (a, G) differ by closed forms with integral periods. However Ω3cl (Y )/Ω3ZZ (Y ) ' H 3 (Y, IR)/H 3 (Y, ZZ), hence we obtain the expected result. This shows that the gauge equivalence classes of C-fields in this model are indeed shifted differential characters.

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To conclude this section, let us compute the automorphism group of an arbitrary object (P, A, c) of E. Using the boundary conditions (3.38) and a standard gluing argument, we can easily show that Aut(P, A, c) consists of equivalence classes of triples (P, A, γ) on Y × S 1 subject to the condition G 3,1 = 0. Moreover, S 1 is equipped with a base point ∗ ∈ S 1 , and (P, A, γ) restricts to (P, A, c) on Y × {∗}. As noted below (3.38), since G is closed, it follows that it is a pull back from Y , G = π ∗ G, where π : Y × S 1 → Y is the canonical projection, and G is the curvature of (P, A, c). An important observation is that triples (P, A, γ) on Y × S 1 satisfying G 3,1 = 0 form themselves a groupoid M. The morphism space between two such triples (P, A, γ) and (P 0 , A0 , γ 0 ) consists of triples (P, A, c) on Y ×S 1 ×∆1 which restrict to (P, A, γ) and respectively (P 0 , A0 , γ 0 ) on the two components of the boundary. Two triples (P0 , A0 , c0 ) and (P1 , A1 , c1 ) are said to be equivalent if they are homotopy equivalent relative to boundary conditions. Furthermore, we have to impose a filtering condition of the form G3,1 = G2,2 = 0, where Gk,4−k denotes the component of G of degree k along Y . Note that M is in fact a cocycle category for C-fields on Y × S 1 subject to a filtering condition on the curvature. The automorphisms of (P, A, c) are classified by equivalence classes of objects of M so that (P, A, γ)|Y ×{∗} = (P, A, c). Proceeding by analogy with E, the equivalence classes of objects of M are in one to one correspondence ˇ 4 (Y × S 1 ) so that ω(χ) to differential characters χˇ ∈ H ˇ 3,1 = 0. In particular, this implies that ω(χ) ˇ is pulled back from Y . Adding the extra condition ˇ 4 (Y ) is the charac(P, A, γ)|Y ×{∗} = (P, A, c) fixes χ| ˇ Y ×{∗} = ρˇ, where ρˇ ∈ H ter determined by (P, A, c). Using the exact sequence (3.43)

ˇ 4 (Y × S 1 ) → Ω4 (Y × S 1 ) → 0 0 → H 3 (Y × S 1 , IR/ZZ) → H ZZ

and the K¨ unneth formula H 3 (Y × S 1 , IR/ZZ) ' H 3 (Y, IR/ZZ) ⊕ H 2 (Y, IR/ZZ), we see that ρˇ fixes the component in the first summand, but not the second. It follows that the characters (P, A, γ) are parametrized by H 2 (Y, IR/ZZ). In conclusion, Aut(P, A, c) ' H 2 (Y, IR/ZZ) for any object (P, A, c). This is in agreement with our previous discussion in section 3.3. We have so far given two different constructions of cocycle groupoids for C-fields which have identical equivalence classes of objects and automorphism groups. Therefore the two groupoids must be equivalent. 4. The definition of the C-field measure for Y without boundary 4.1. Witten’s definition. In [6] Witten gave a definition of the phase factor Φ(C) in (1.3) using E8 gauge theory in 12-dimensions. We will review his definition, and then show how to recast it in intrinsically 11-dimensional terms. The 11-dimensional formulation will then be in a form suitable to generalization to the case when Y has a boundary.

The M-theory 3-form and E8 gauge theory

61

Suppose P (a) → Y admits an extension (4.1)

PZ (aZ ) → Z

where Z is a bounding spin manifold (and hence ∂Y = ∅). The existence of such an extension follows from Stong’s theorem [22]. Suppose, moreover, that CˇZ = (AZ , cZ ) extends CˇY = (AY , cY ) ∈ EP (Y ) to EPZ (aZ ) (Z) (such extensions always exist if PZ (aZ ) exists). Then Witten’s definition is ) ( Z 1 3 GZ − GZ I8 (gZ ) (4.2) ΦW (CˇY ; Y ) = exp 2πi Z 6

where

1 GZ = trFZ2 − trRZ2 + dcZ 2 and I8 (g) is defined on a manifold M by 1 2 2 4 TrR − 4TrR . (4.4) I8 (g) = (4π)4

(4.3)

I8 (g) is a closed form, depending on a metric, which represents 1 (p2 (T M) − λ(T M)2 ) 48 in H 8 (M, IR), where we recall that λ = p1 /2. One needs to check that (4.2) is independent of the extensions aZ , PZ , CˇZ . This is almost true thanks to some remarkable identities involving E8 index theory. To check independence of extensions it suffices to check that the right hand side of (4.2) is equal to 1 on a closed 12-manifold Z. 11 / V (aZ ) coupled to the bundle V (aZ ) associated Consider the Dirac operator D to PZ (aZ ) via the 248 of E8 , and endowed with connection AZ . The index density is formed from (4.5)

(4.6)

I8 :=

ˆ Z) / V (aZ ) ) := Tr248 exp[FZ /2π]A(g i(D

ˆ Z ) is the usual Dirac index where FZ is the fieldstrength of AZ , while A(g density formed from the curvature 2-form R, computed using gZ . Using properties of E8 we find that 1 (4.7) Tr248 exp (FZ /2π) = 248 + 60αZ + 6αZ2 + αZ3 3 1 2 2 where αZ = 60 Tr248 F /8π represents (aZ )IR in DeRham cohomology. Now, ˆ eR/2π − / RS ) denote the 12-dimensional gravitino index density 12 ATr let i(D 11It

is quite crucial that we extend the entire triple (AY , cY , PY ), and not just the fieldstrength GZ . Otherwise Φ can vary continuously with a choice of extension. 12This is not the gravitino density of a 12-dimensional theory, but rather that appropriate to an 11-dimensional theory.

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

ˆ Then, extracting the 12-form part, we have the crucial identity 1 + 8A. (12) 1 1 1 / A ) + i(D / RS ) i(D = G3Z − GZ I8 (gZ ) 2 4 6 # " (4.8) 1 1 1 2 − d cZ ( GZ − I8 (gZ )) − cZ dcZ GZ + cZ (dcZ )2 2 2 6 and thus, on a closed 12-manifold Z the RHS of (4.2) is 1 1 / V (aZ ) ) + I(D / RS ) (4.9) exp 2πi I(D 2 4

where I is the index. In 12-dimensions the index is always even, and hence (4.9) / RS )/2 . Thus, (4.2) is independent of extension, up simplifies to a sign (−1)I( D / RS )/2 . to the sign (−1)I( D 4.2. Intrinsically 11-dimensional definition. We would now like to recast the definition (4.2) into intrinsically 11-dimensional terms. This is readily done using the APS index theorem: Z /E) = / E ) − ξ(D /E) (4.10) I(D i(D Z

for the Dirac operator coupled to a bundle with connection E. Here 1 / E ) := (η(D / E ) + h(D / E )) (4.11) ξ(D 2 This is the standard invariant involving the η invariant and the number of / E ). zeromodes h(D Motivated by (4.2,4.8, and 4.10) we define iπ ˇ / RS ) + 2πiIlocal / V (a) ) + ξ(D (4.12) Φ(C; Y ) := exp iπξ(D 2

The total derivative in (4.8) leads to the elementary local factor Z 1 1 1 2 2 (4.13) Ilocal = c G − I8 (g) − cdcG + c(dc) 2 2 6 Y / RS ) is short for The expression ξ(D

(4.14)

/ T ∗ Y ) − 3ξ(D) / ξ(D

As a function of the C-field, (4.12) is gauge invariant, that is, Φ(A, c; Y ) = Φ(A0 , c0 ; Y ) for (A0 , c0 ) = (α, χ) ˇ · (A, c). Thus it is a U (1)-valued function on 4 ˇ H 1 λ (Y ). 2 Let us now comment briefly on Φ as a function of the metric g. Note that we have dropped a sign-factor in passing from (4.2) to (4.12): (4.15)

/ RS )/2 Φ ΦW = (−1)I( D

The M-theory 3-form and E8 gauge theory

63

/ T ∗ Y and D / have zero modes, and Let D be the locus of metrics g such that D suppose g ∈ Met(Y ) − D. On Met(Y ) − D , Φ is well-defined as a number. However, it does not extend continuously to the full space of metrics Met(Y ). This difficulty was essentially solved in [6]. One should consider Φ as a continuous section of a complex line bundle associated to a principal ZZ2 bundle T . Then the gravitino path integral, which we will denote schematically as / RS ) is a section of a real line bundle associated to a canonically isoPfaff(D morphic principal ZZ2 bundle, and will have (in general) a first order zero along / RS ) · Φ becomes a well-defined the locus of zeromodes. 13 The product Pfaff(D smooth diffeomorphism-invariant function on the gauge equivalence classes of metrics and C-fields. ˇ for 3-form gauge potentials deThere is a standard formal measure µ(C) fined by the the metric g on Y together with the Faddeev-Popov procedure applied to small C-field gauge transformations. Taking this into account, the well-defined measure for the C-field path integral, after integrating out the gravitino, is R ˇ − 2`13 Y G∧∗G Pfaff(D ˇ Y ). / RS ) · Φ(C; (4.16) µ(C)e 5. The C-field measure when Y has a boundary Now we are ready to reap some of the benefits of the E8 model for the C-field. Using this formalism we will completely solve problem 1 of the introduction. When we try to formulate the C-field measure in the case where ∂Y = X is nonempty the same formula (4.12) holds. In particular we continue to take iπ ˇ Y ) = exp iπξ(D / A ) + ξ(D / RS ) + 2πiIlocal (5.1) Φ(C; 2

/ A )] However, there is now a conceptually important distinction: the factor exp[iπξ(D is a section of a U (1) bundle with connection over the space of C-fields on / RS )] is X. As in the case when Y has no boundary, the factor exp[ 12 iπξ(D more subtle and must be combined with the gravitino determinant. We hope to discuss this elsewhere [23]. In this paper we consider a fixed metric g ∈ Met(Y ) − D. It is well-known in Chern-Simons theory that in the presence of a boundary / E )] should be viewed as a section of a the exponentiated invariant exp[iπξ(D line bundle. In our case Φ is valued in a principal U (1) bundle (5.2)

Q → EP (X).

This bundle can be defined by the following property: Each extension CˇY of CˇX ∈ EP (X), defined for some Y with ∂Y = X, produces an element: (5.3) Φ(CˇY ; Y ) ∈ Q ˇ CX

13The

definition of this principal ZZ2 bundle and the canonical isomorphism will be explained in detail in [23].

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and two such extensions satisfy the “gluing law” Φ(CˇY ; Y ) (5.4) = Φ(CˇY − CˇY 0 ; Y Y¯ 0 ) 0 ˇ Φ(CY 0 ; Y ) where Y¯ denotes orientation reversal. Alternatively, we may define QCˇX more directly as follows. Suppose P → X is a fixed E8 bundle of characteristic class aX . For any extension PY of P define EPY (CˇX ) to be the set of all extensions CˇY of CˇX . 14 We may then define the fiber QCˇX to be the set of all U (1)-valued functions F on EPY (CˇX ), for some Y , such that the ratio F (CˇY )/F (CˇY 0 ) is the right hand side of (5.4). This space of functions is a principal homogenous space for U (1) and varies smoothly with CˇX . Thus the fibers define a smooth U (1) bundle Q → EP (X). The circle bundle Q we have just defined carries a canonical connection. Recall that a connection is simply a law for lifting of paths in the base space to the total space satisfying the natural composition laws. A path p(t) = (AX (t), cX (t)), 0 ≤ t ≤ T in EP (X) defines a C-field on the cylinder Cˇp ∈ EP (X × [0, T ]). So (5.5)

Φ(Cˇp ; X × [0, T ]) ∈ Q(A,c)(T ) ⊗ Q∗(A,c)(0) = Hom(Q(A,c)(0) , Q(A,c)(T ) )

defines the appropriate parallel transport. Given the explicit formula (5.1) we can give an explicit formula for the connection. Choose an extension (Y, PY ) of (X, P ). Since every cotangent vector (δA, δc) to EP (X) has an extension to a cotangent vector of EPY it suffices to evaluate the covariant derivative on a section provided by an infinitesimal family of extensions CˇY . The main variational formula is: Z ∇Φ(CˇY ; Y ) 1 2 G − I8 2tr(δAF ) + δc (5.6) = 2πi Φ(CˇY ; Y ) Y 2

where 2tr(δAF ) + δc is a form of type (1, 3) on EP (Y ) × Y . It is now a simple matter to compute the curvature. Identifying the tangent space T EP (X) = Ω1 (adP ) ⊕ Ω3 (X) evaluation on tangent vectors (αi , χi ) gives the curvature 2-form Ω of Q: Z (5.7) Ω((α1 , ξ1 ); (α2 , ξ2 )) = 2πi G ∧ 2tr(α1 F ) + ξ1 ∧ 2tr(α2 F ) + ξ2 X

The theory of determinant line bundles shows that the associated line bundle L to Q may be regarded as a determinant line bundle. More precisely, / adP ), the Pfaffian line bundle for adjoint E8 fermions on the L = PFAFF(D boundary X. But, we emphasize, we are not introducing physical E8 fermions on the boundary.

˜ spin (K(ZZ, 4)) = ZZ2 ⊕ ZZ2 in general we cannot Ωspin Z2 ⊕ ZZ2 ⊕ ZZ2 and Ω 10 (pt) = Z 10 extend P → X. However, in the physical situations of interest there will always be such an extension since X is the boundary of the theory defined on the spacetime Y . 14Since

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The line bundle we have defined can also be defined by considering families of Dirac operators with boundary conditions along the lines of [24]. This definition fits in more naturally with the inclusion of the Pfaffian of the gravitino operator and will be discussed in [23]. It is also possible to formulate the line, together with its connection, in the categorical approach of sections 3.4 and 3.5. 6. The action of the gauge group on the physical wavefunction and the Gauss law In this section we will study the action of the C-field gauge group on the wavefunctions defined by the C-field path integral. Fix an extension PY of P to Y . For CˇX ∈ EP (X) recall that EPY (CˇX ) is the set of all extensions CˇY of CˇX . Then we set, formally, Z Z 1 ˇ ˇ (6.1) Ψ(CX ) = µ(CY ) exp[− 3 G ∧ ∗G] Φ(CˇY ; Y ) 2` ˇ ˇ EPY (CX )/G(CX ) Here G(CˇX ) is the group of C-field gauge transformations on Y which fixes CˇX . That is, we integrate over all isomorphism classes of extensions CˇY of CˇX . As we have seen, Φ(CˇY ; Y ) is valued in QCˇX and hence Ψ(CˇX ) is valued in the line associated to QCˇX . We will denote this line LCˇX . 6.1. The Gauss Law. Physically meaningful wavefunctions must be gauge invariant. In our case, the wavefunction is a section of a line bundle L → EP (X). In order to formulate gauge invariance we will need to define a lift of the group action: L (6.2)

G

→

↓ EP (X)

L ↓

G

→

EP (X)

Then the condition of gauge invariance is simply (6.3) g · Ψ(CˇX ) = Ψ(g · CˇX ) for all g ∈ G, CˇX ∈ EP (X). This condition is known as the Gauss law. Formally, any wavefunction defined by a path integral with gauge invariant measure, such as (6.1) automatically defines a gauge invariant wavefunction. This is essentially because the path integral integrates over gauge copies of fields and thus projects onto gauge invariant quantities. Of course, this projection might vanish, so the Gauss law should be considered, in part, as a condition on CˇX such that nonvanishing gauge invariant wavefunctions can be supported on CˇX .

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We will now give a precise definition of the group lift (6.2). We will then use that to derive an important consequence of the Gauss law. 6.2. Definition of the G action on L. The main result of this section is the construction of a well-defined lift (6.2). The full construction is rather lengthy. We confine ourselves to sketching the most important part, namely defining the action of gauge transformations of the form (0, χ) ˇ on L. We will construct the group lift using the natural connection on L defined above. To do this, we introduce some standard paths in EP (X). For any ξ ∈ Ω3 (X) define the linear path (6.4) pC,ξ ˇ (t) = A, c + tξ connecting Cˇ to Cˇ + ξ. (It is convenient to write Cˇ + ξ := (A, c + ξ) for a 3-form ξ. ) ˇ 3 (X) then we will define Now, if Ψ ∈ LCˇ and χ ˇ∈H (6.5)

ˇ χ) (0, χ) ˇ · Ψ := ϕ(C, ˇ ∗ U (pC,ω( ˇ χ) ˇ )·Ψ

where ϕ is a phase. The reason we must introduce the phase factor ϕ in (6.5) is that the connection on L has curvature. Indeed, an elementary computation shows that we have: (6.6)

iπ U (pC,ω( ˇ χ ˇ χ ˇ χ ˇ1 +χ ˇ2 ) ) = U (pχ ˇ1 C,ω( ˇ2 ) )U (pC,ω( ˇ1 ) )e

R

X

G∧ω(χ ˇ1 )∧ω(χ ˇ2 )

where χˇ1 Cˇ denotes the gauge transform of Cˇ by χˇ1 . It follows that to define a group theory lift we will need to find functions ˇ χ) ϕ(C, ˇ such that: (6.7)

ˇ χ ˇ χˇ2 )ϕ(C, ˇ χˇ1 )e+iπ ϕ(C, ˇ1 + χˇ2 ) = ϕ(χˇ1 C,

R

X

G∧ω(χ ˇ1 )∧ω(χ ˇ2 )

.

We now construct a cocycle satisfying the relation (6.7). ( In the following section we will provide some physical motivation for this choice of ϕ. ) We use the data CˇX and χˇ to construct the character on the closed 11-fold Y = X ×S 1 defined by (6.8) π ∗ [CˇX ] + π ∗ (tˇ) · π ∗ (χ) ˇ 1

2

1

ˇ 1 (S 1 ) is the canonical Here πi are projections (henceforth dropped) and tˇ ∈ H character associated with S 1 ∼ = U (1). This character (6.8) has fieldstrength G + dt ∧ ω(χ), ˇ characteristic class a + [dt]a(χ) ˇ and holonomy (6.9)

χ ˇ(A,c) (Σ3 )e

2πit

R

Σ3

ω(χ) ˇ

on cycles of type Σ3 × {t} and holonomy χ(σ ˇ 2 ) on cycles of type σ2 × S 1 . We will refer to the character (6.8) as the twist of [CˇX ] by χ. ˇ Choosing 1 the Neveu-Schwarz, or bounding spin structure, S− , we define the correction factor in terms of the the M-theory phase (4.12): (6.10)

ϕ(CˇX , χ) ˇ := Φ([CˇX ] + tˇ · χ; ˇ X × S−1 )

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ˇ 41 (Y ) for closed spin 11Note that since Φ is defined as a function on H λ 2 manifolds Y equation (6.10) makes sense. Also note that, since we are working / RS )/2] must be with a fixed metric a definite choice of the sign of exp[iπξ(D made here. The idea behind (6.10) is essentially stated in [25], footnote 19. An easy argument shows that (6.10) indeed satisfies the requisite cocycle law. The cocycle (6.10) can be expressed in terms of η invariants of Dirac operators by using the construction of equations (3.28 - 3.33) above. A similar construction allows one to define the group lift for the entire group G. We need to check that (6.10) really satisfies (6.7). The proof is a standard cobordism argument. We consider a differential character on the spin 11ˇ + tˇ · χˇ1 , [C] ˇ + tˇ · manifold (X × S−1 ) ∪ (X × S−1 ) ∪ (X × S−1 ) restricting to [C] ˇ − tˇ· (χˇ1 + χ χ ˇ2 , [C] ˇ2 ) on the three components. We then choose the extending spin 12 manifold to be Z = X × ∆ where ∆ is a pair of pants bounding the three circles with spin structure restricting to S−1 on the three components. To be explicit we can choose ∆ to be the simplex {(t1 , t2 ) : 0 ≤ t1 ≤ t2 ≤ 1} with identifications ti ∼ ti + 1. Now we extend the differential character as (6.11)

ˇ + tˇ1 · χ [C] ˇ1 + tˇ2 · χ ˇ2

which clearly restricts to the required character on the boundary. The field strength is GZ = GX + dt1 ∧ ω1 + dt2 ∧ ω2 . For such extensions, ΦWitten = Φ because the gravitino index is trivial. We can therefore use (4.2) to show that the product of phases around the boundary is Z Z 1 ˜3 ˜ 1 (6.12) exp 2πi G − GI8 = exp −2πi G ∧ ω 1 ∧ ω2 2 X ∆×X 6 where 21 is the area of ∆. This proves the desired cocycle relation (6.7). ˇ χ) The following property of the cocycle ϕ(C, ˇ will be useful in what follows. If χˇb is a topologically trivial character then

(6.13)

ˇ χˇ + χˇb ) = ϕ(C, ˇ χ)e ϕ(C, ˇ −2πi

R

X

b( 12 G2 −I8 )

This can be proved using the variational formula (5.6), or by using a cobordism argument, since for a topologically trivial χˇb we may fill in S−1 with a disk and continue the character as χˇrb where r is the radius on the disk. 6.3. A physical motivation for formula (6.10). The choice of cocycles defining the group lift (6.2) is not unique. Therefore we would like to justify the choice (6.10). This definition of the group lift is necessary if we want to represent path integrals in certain twisted topological sectors as traces over Hilbert space. Suppose Ψ(Cˇ1 , Cˇ2 ) is the path integral on the cylinder X × [0, 1]. Then, it is the kernel of the operator e−βH where H is the Hamiltonian and β is the ˇ C) ˇ length of the cylinder. By integrating over the diagonal of the kernel Ψ(C,

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

we represent a trace on Hilbert space as a path integral over X × S 1 : (6.14)

Za (X × S 1 ) = TrHa (X) e−βH

On both the left and right hand side we are working within a topological sector defined by a ∈ H 4 (X, ZZ). On the right hand side we have the trace over the Hilbert space of wavefunctions with support on C-fields of characteristic class a; on the left-hand side the path integral is the integral over isomorphism classes of C-fields with Cˇ ∈ EP (a) (X × S 1 ), where P (a) is simply pulled back from X to X × S 1 . We would now like to twist this construction using the group G(X). That is, we would like a path integral representation of (6.15)

TrHa (X) U (g)e−βH

where U (g) is the unitary operator implementing the action of g on the Hilbert space. Therefore, we would like to glue together the ends of a cylinder with boundary conditions related by an element g ∈ G(X). Thus we consider ˇ C) ˇ for Cˇ ∈ EP (a) (X). This is an element of a line L∗ˇ ⊗ L ˇ . Now, to Ψ(g C, gC C give a lift of the group action to L is to give a specific isomorphism (6.16)

L∗Cˇ ⊗ LgCˇ ∼ =C

Such an isomorphism is necessary if we are to integrate over the boundary values Cˇ to produce a path integral over C-fields on X × S 1 in the topological sector a + [dt]a(χ). ˇ The path integral is, after all, just a complex number. If we consider simple field configurations, such as those with isomorphism class [CˇX ] + tˇ · χˇ on the cylinder then there is no phase factor in the evolution on the cylinder, since the evolution is through a path of gauge transformations. On the other hand, the phase factor in the path integral is (6.10) and this phase can only enter through the choice of isomorphism (6.16). Here again the E8 formalism proves to be very useful since it allows one to make precise the notation of a “loop L in the space of C fields on R” used in [25]. To do this one relates a group transformation (α, χ) ˇ to an E8 bundle automorphism (3.28) and uses this to define a twisted bundle with connection on X × S 1 as in (3.30,3.31). 7. The definition of C-field electric charge In electromagnetism the electric current satisfies (7.1)

d ∗ F = je .

If je has compact support (or falls off sufficiently rapidly) one then defines n−1 [je ] ∈ HDR,cpt (X) as the electric charge of the source. On a closed manifold X, this must be zero, since ∗F provides an explicit trivialization.

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ˇ is cubic and hence the theory In M-theory, the Chern-Simons term Φ(C) is nonlinear. The equation of motion (in Lorentzian signature) is: 1 1 d ∗ G = G2 − I8 (g) 3 ` 2 Thus, background metrics and C-fields induce electric charge. As an element of DeRham cohomology 1 8 (7.3) [ G2 − I8 ] ∈ HDR (X) 2 is the induced charge. Nevertheless, (7.3) is not a suitable expression for the electric charge induced by a C-field and metric. The quantization law (2.12) on the C-field implies, via Dirac quantization, that the C-field electric charge is valued in integral cohomology H 8 (X, ZZ). 15 Thus, in order to describe the electric charge induced by the self-interactions of C (and gravity) we need to define a precise integral lift of (7.3). We will do this in the next section by considering the micro gauge transformations of section 3.3. (7.2)

7.1. The action of automorphisms on L. In section 6.2 we have defined a group action (0, χ) ˇ : LA,c → LA,c+ω(χ) ˇ is flat, i.e., χˇ ∈ H 2 (X, U (1)) ˇ . When χ then ω(χ) ˇ = 0 and hence (0, χ) ˇ · (A, c) = (A, c). These are the automorphisms of the objects (A, c) in the categorical approach. If Ψ ∈ LA,c then (7.4)

(0, χ) ˇ · Ψ = ϕ(CˇX , χ) ˇ∗Ψ

and ϕ(CˇX , χ) ˇ can be a nontrivial phase. If Ψ is a gauge invariant wavefunction nonvanishing at CˇX then the invariance under the automorphisms of CˇX requires ϕ(CˇX , χ) ˇ = 1 for all flat χ. ˇ We will now regard flat characters as 2 elements χ ˇ ∈ H (X, IR/ZZ). The cocycle law (6.7) then implies that on flat characters, ϕ(CˇX , ·) is a homomorphism (7.5)

ϕ(CˇX , ·) : H 2 (X, IR/ZZ) → U (1)

Poincare duality now implies the existence of an integral class, ΘX (CˇX ) ∈ H 8 (X, ZZ) such that (7.6)

ϕ(CˇX , χ) ˇ = exp[2πihΘX (CˇX ), χi] ˇ

Equation (7.6) defines, mathematically, an important integral class ΘX (CˇX ). Note that if a gauge invariant wavefunction is nonvanishing at CˇX then (7.7) ΘX (CˇX ) = 0. This is the first nontrivial consequence of the Gauss law. 15Moreover,

recall that the quantization law on G follows from the existence of elementary membranes. Indeed, the elementary membrane has C-field electric charge given by the isomorphism H2 (X, ZZ) ∼ = H 8 (X, ZZ).

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Since ΘX (CˇX ) is an integral class it only depends on the characteristic class a of CˇX (and the spin structure of X). Therefore, we will sometimes denote it as ΘX (a). ˇ is C-field electric charge. To interpret ΘX (a) let us “insert” 7.2. ΘX (C) a membrane wrapping a cycle σ ∈ Z2 (X) in the boundary X. Let us conˇ Relative to the sider the gauge invariance of the new wavefunction Ψσ (C). wavefunction without the insertion of σ the flat gauge transformations χˇ with χ ˇ ∈ H 2 (X, IR/ZZ) act on the wavefunction with an extra phase (7.8) χ(σ) ˇ = exp 2πihP D[σ], χi ˇ

Membranes carry C-field electric charge. Comparison of (7.8) with (7.6) shows that we should interpret ΘX as the C-field charge induced by the background metric and C-field. An important consistency check on our argument is the following. Using (6.13) and comparing with the definition (7.6) shows that 1 (7.9) [ΘX ]IR = [ G2 − I8 ]DR 2 The above arguments lead to a simple extension of the Gauss law: In the presence of membranes wrapping a spatial cycle σ ∈ Z2 (X), a gauge invariant wavefunction can have support at Cˇ only if ˇ + P D([σ]) = 0 (7.10) ΘX (C)

We have discussed a necessary condition for the existence of a nonvanishing gauge invariant wavefunction. The E8 formalism is very useful for giving the full statement of the Gauss law, and provides an interesting interpretation of the quantization of “Page charges.” This will be discussed elsewhere [23]. ˇ 8. Mathematical Properties of ΘX (C) ˇ is a subtle integral cohomology class defined by E8 ηThe class ΘX (C) invariants, and not much is known about it. Here we collect a few known facts. As we have explained above, restriction to topologically trivial and flat Cfield gauge transformations χˇb , for db = 0 shows that in DeRham cohomology we have (7.9). Since [G]IR = aIR − 21 λIR , we learn that 16 1 (8.1) [ΘX (a)]IR = aIR (aIR − λIR ) + 30Aˆ8 2 2 (To derive this note that 30Aˆ8 = 18 λ2 − I8 = 7λ 48−p2 .) There are three basic facts about the integral class ΘX (a). First, hΘX (a), χi ˇ is a spin cobordism invariant of (X, a, α), where a ∈ H 4 (X) and α ∈ H 2 (X, IR/ZZ). incidentally, that this implies that there is a natural integral lift of 30Aˆ8 defined on spin 10-folds. One easily checks that on CP 5 , 30Aˆ8 = (xIR )4 , where x generates H 2 (CP 5 ; ZZ), so 30 is the smallest multiple of Aˆ8 that has an integral lift. 16Note,

The M-theory 3-form and E8 gauge theory

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Second, from this and a computation of spin bordism groups one can prove that ΘX is a quadratic refinement of the cup product: (8.2)

ΘX (a1 + a2 ) + ΘX (0) = ΘX (a1 ) + ΘX (a2 ) + a1 a2

Third, Θ is “parity invariant,” i.e. it satisfies the identity (8.3)

ΘX (a) = ΘX (λ − a).

ˇ 41 via In order to prove (8.3) we define a parity transformation on H λ 2

(8.4)

χˇP (Σ) = (χ(Σ)) ˇ

∗

The parity transform takes (8.5)

a→λ−a G → −G.

Under this transformation the M-theory phase transforms as (8.6)

ˇ P ; Y ) = (ΦW ([C]; ˇ Y )∗ ΦW ([C]

ˇ then [CˇZ ]P extends [C] ˇ P , and G([CˇZ ]P ) = This follows since, if [CˇZ ] extends [C] −G([CˇZ ]). Now we simply note that the integral in (4.2) is odd in GZ . Applying this observation to the case Y = X × S 1 we see that ∗ (8.7) ΦW ([CˇX ]P − tˇ · χ; ˇ X × S−1 ) = ΦW ([CˇX ] + tˇ · χ; ˇ X × S−1 )

where we have made a parity transformation t → −t on the circle. Now, (8.3) follows from (8.7). The cobordism invariance of hΘX (a), χi ˇ and the bilinear identity (8.2) allows us to compute ΘX (a) in several simple examples. To choose but one example take X = L(3, p1 )×L(3, p2 )×S 4 , where L(3, p) is any three-dimensional Lens space S 3 /ZZp . Let v be a generator of H 4 (S 4 , ZZ). If bi ∈ H 2 (L(3, pi )) = ZZpi (we omit pullback symbols) then (8.8)

˜ + b1 b2 ) = Θ(v) ˜ ˜ 1 b2 ) + vb1 b2 Θ(v + Θ(b

˜ X (a) := ΘX (a) − ΘX (0). Now Θ(b ˜ 1 b2 ) = 0 because we can fill in where Θ 4 S = ∂B5 and extend the Cheeger-Simons character over this manifold. At ˜ the same time we can extend the class α ∈ H 2 (X, U (1)). Similarly, Θ(v) = 0. Here we fill in one of the two Lens spaces (any oriented 3-manifold has a bounding oriented 4-manifold). So, we conclude: (8.9)

˜ + b1 b2 ) = vb1 b2 Θ(v

where we used that the spin bordism group is trivial in 3 dimensions. Equation (8.9) is potentially relevant to G2 compactifications and 5-brane physics. It would be helpful for topological investigations of M-theory if further methods were developed to compute the subtle integral class ΘX (a).

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9. Φ as a cubic refinement, with applications to integration over flat C-fields 9.1. The sum over flat C-fields as a subintegration in the path integral for Ψ(CˇX ). In this section we make some comments on the nature of the wavefunction Ψ(CˇX ) associated to a manifold X with boundary Y . The sum over the gauge copies of CˇY in the path integral automatically projects onto gauge invariant wavefunctions. Therefore, the wavefunction will vanish unless ΘX (CˇX ) = 0. We restrict attention to these components. The wavefunction will be a sum over topological sectors. These are labelled by extensions a ˜ of the characteristic class of CˇX . Let ι : X ,→ Y , then we ∗ sum over ker ι ⊂ H 4 (Y, ZZ). By the long exact sequence this is equivalent to a sum over H 4 (Y, X; ZZ)/δH 3 (X; ZZ)

(9.1)

where δ is the connecting homomorphism. For each extension a ˜ of a ∈ H 4 (X; ZZ) we choose an extending bundle PY of characteristic class a ˜. We can fix the Ω1 (adP ) gauge symmetry by choosing an extension (A˜0 , c˜0 ) of (AX , cX ). All other extensions are given by (A˜0 , c˜0 + cY ) with cY ∈ ker ι∗ ⊂ Ω3 (Y ). Let us denote this space by Ω3 (Y, X). The integral (6.1) reduces to an integral over Ω3 (Y, X)/Ω3 (Y, X)ZZ. In the path integral we integrate over the compact space of harmonic forms H3 (Y, X)/H3 (Y, X)ZZ

(9.2)

where H3 (Y, X) := ker ι∗ restricted to H3 (Y ), and the addition of such fields yields no cost in action. That is, the real part of the Euclidean action is unchanged by the addition of such fields. In the sum over (9.1) one can (noncanonically) split the integration over the torsion subgroup and a lattice. The sum over the torsion subgroup can be combined with the integral over (9.2) to produce an integral over ker ι∗ applied to H 3 (Y, U (1)). Using the topological considerations of this paper one can make some exact statements about the nature of this subintegration. In the next section we spell out these statements. 9.2. The cubic refinement law. In order to study the sum over flat C-fields we will need a result which is interesting and important in its own right,17 namely the interpretation of the M-theory phase as a cubic refinement of a ˇ 4 (Y ). trilinear form on H The product of differential characters on Y together with evaluation on Y leads to a trilinear form (ˇ a1 a ˇ2 aˇ3 )(Y ) ∈ U (1). 18 The cubic refinement law 17Indeed, 18The

this is the starting point for [8]. product of differential characters is described in [9].

The M-theory 3-form and E8 gauge theory

73

states that (9.3) ˇ +a ˇ +a ˇ + aˇ2 ; Y )Φ([C] ˇ +a Φ([C] ˇ1 + a ˇ2 + a ˇ3 ; Y )Φ([C] ˇ1 ; Y )Φ([C] ˇ3 ; Y ) = (ˇ a1 a ˇ2 a ˇ3 )(Y ) ˇ ˇ ˇ ˇ Y) Φ([C] + a ˇ1 + a ˇ2 ; Y )Φ([C] + a ˇ1 + a ˇ3 ; Y )Φ([C] + aˇ2 + a ˇ3 ; Y )Φ([C]; Recall that Φ descends to shifted differential characters, so it makes sense ˇ 4 (Y ), while [C] ˇ is a shifted character. In order to prove (9.3) to take aˇi ∈ H ˇ note that if C, a ˇi all extend simultaneously on the same spin 12-fold then we may use Witten’s definition (4.2) as an integral in 12 dimensions. The cubic refinement law follows from the simple algebraic identity 1 1 1 1 (a + x + y + z)3 − (a + x + y)3 − (a + y + z)3 − (a + x + z)3 6 6 6 (9.4) 6 1 1 1 1 + (a + x)3 + (a + y)3 + (a + z)3 − a3 = xyz 6 6 6 6 It thus follows that if we can simultaneously extend the differential characters ˇ and a [C], ˇi then we have (9.3). It follows from the E8 model for the C-field that, if we can extend the characteristic classes of the characters then we can extend the entire character. Therefore we consider the purely topological ˇ together with the classes ai . The extenproblem of extending the class a(C) sions of the individual classes exist by Stong’s theorem. Next, consider the obstruction to the existence of an extension of an 11-manifold together with a pair of classes (Y, a1 , a2 ). The obstruction to finding an extension of (9.5)

(Y, a1 , a2 ) + (Y, 0, 0) − (Y, a1 , 0) − (Y, 0, a2 )

is measured by the group Ωspin 11 (K(Z, 4) ∧ K(Z, 4)). Similarly, the obstruction for triples (Y, a1 , a2 , a3 ) lies in Ωspin 11 (K(Z, 4) ∧ K(Z, 4) ∧ K(Z, 4)) and that for spin 4 quartets lies in Ω11 (∧ K(Z, 4)). All of these groups can be shown to vanish by a simple application of the Atiyah-Hirzebruch spectral sequence. 9.3. Summing over the flat C-fields. We can now apply the cubic refinement law to learn some facts about the sum over flat C-fields on Y . For simplicity we will consider the case of ∂Y = ∅. As we discussed, evaluating the sum over flat C-fields reduces to evaluation of the integral Z (9.6) [dCˇf ]Φ(Cˇ + Cˇf ; Y ) H 3 (Y,U (1))

where [dCˇf ] is the natural measure given by the Riemannian metric. Recall that H 3 (Y, U (1)) is a disjoint union of connected tori. The connected component of the identity is H 3 (Y, ZZ) ⊗ U (1), and the quotient by this subgroup is isomorphic to the torsion group HT4 (Y ). 19 To evaluate this we first integrate 19We

will denote the torsion subgroup of H p (Y, ZZ) by HTp (Y ).

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over the subgroup of topologically trivial flat C-fields, writing the sum (9.6) as: X Z (9.7) [dCˇf ]Φ(Cˇ + Cˇf ; Y ) HT4 (Y )

H 3 (Y )⊗U (1)

Using the variational formula for the topologically trivial flat fields we learn that the integral over the topologically trivial flat characters H 3 (Y ) ⊗ U (1) vanishes unless 1 ˜2 (9.8) [ G − I8 ]DR = 0 ∈ H 8 (Y ; IR) 2 Thus, the sum over flat C-fields projects onto fields allowing a solution to the equation of motion. When (9.8) is satisfied the function of flat characters Cˇf → Φ(Cˇ + Cˇf ; Y ) descends to an interesting U (1)-valued function (9.9)

ˇ + Cˇf ) φ¯[C] ˇ (aT ) := Φ(C

of torsion classes aT ∈ HT4 (Y ). The way Φ(Cˇ + Cˇf ; Y ) depends on aT is not obvious since changing aT changes the isomorphism class of the E8 bundle. It is here that the cubic refinement law (9.3) becomes quite useful, since it follows that the function φ¯[C] ˇ is a cubic refinement of the U (1)-valued trilinear form (9.10)

exp[2πiha1 a2 , a3 i]

on HT4 (Y ). Here we have used the torsion pairing HT8 (Y ) × HT4 (Y ) → Q/ZZ. In conclusion the sum over flat fields projects onto fields satisfying (9.8) and for such fields we have a further projection onto topological sectors such that X (9.11) φ¯[C] ˇ (a) a∈HT4 (Y )

is nonzero. The sum in (9.11) is a sum of exponentiated cubic forms on a finite abelian group, and is hence a kind of “Airy function” generalization of Gauss sums. If we specialize further to the case Y = X × S+1 with Ramond, or nonbounding, spin structure on S 1 we can go further, and make contact with the results of [7, 15]. In this case HT4 (Y ) ∼ = HT3 (X) ⊕ HT4 (X) and hence the sum over torsion classes may be arranged as a sum X X (9.12) φ¯[C] ˇ (hdt + a) h∈HT3 (X) a∈HT4 (X)

and it is fruitful to study the sum over HT4 (X) for fixed values of h. ˇ is also pulled back from X. Then, as noted in [7] Let us assume that [C] ˇ 4 (X) be a differential character we have a dramatic simplification. Let a ˇ∈H

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with characteristic class a (not necessarily torsion). Then Φ(Cˇ + a ˇ; Y ) ˇ ˇ (9.13) = eiπ[f (a(C)+a)−f (a(C ))] ˇ Φ(C; Y ) / V (a) on X. Moreover, where f (a) is the mod 2 index of the Dirac operator D f (a) is a quadratic refinement of a bilinear form: Z (9.14) f (a1 + a2 ) = f (a1 ) + f (a2 ) + r2 (a1 )Sq 2 r2 (a2 ). X

where r2 denote reduction modulo two. It now follows that φ¯[C] ˇ (hdt + a) 4 defines a quadratic refinement as a function of a ∈ HT (X). Indeed, the cubic refinement law reduces to the quadratic refinement law: ¯ ˇ (hdt) φ¯[C] ˇ (hdt + a1 + a2 )φ [C] 3 (9.15) = exp 2πiha , (Sq + h)a i 1 2 ¯ ˇ (hdt + a2 ) φ¯[C] ˇ (hdt + a1 )φ[C] where we have used the bilinear identity for the E8 mod 2 index proved in [7], and again have used the torsion pairing HT4 (X) × HT7 (X) → Q/ZZ. Thus, for fixed values of h we are averaging a quadratic form over a finite abelian group in (9.12). We now need a little lemma from group theory. Let A, B be finite abelian groups, with a perfect pairing (9.16)

A × B → Q/ZZ

which we shall write as ha, bi. Suppose we have a homomorphism ϕ : A → B such that (9.17)

Q(a1 , a2 ) = ha1 , ϕ(a2 )i

is a symmetric bilinear form. Let fQ : A → Q/ZZ be a quadratic refinement of this bilinear form, that is: (9.18)

fQ (a1 + a2 ) = fQ (a1 ) + fQ (a2 ) + ha1 , ϕ(a2 )i

Note first that, when restricted to ker ϕ, fQ is a character, and hence there is P ∈ B/Imϕ such that the restriction of fQ to ker ϕ is given by (9.19)

fQ (a) = ha, P i

a ∈ ker ϕ

Moreover, P 6= 0 implies fQ (a) 6= 0 for some a. Now, we have the key statement: The sum X (9.20) S(fQ ) := e2πifQ (a) a∈A

is nonzero if and only if P = 0. We now apply the above discussion with A = HT4 (X), B = HT7 (X) and ϕ = Sq 3 + h and Φ(Cˇ + hdt + a) ˇ (a) = e2πifC,h (9.21) ˇ Φ(C + hdt)

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3 where fC,h ˇ (a) is a quadratic refinement of the quadratic form ha1 , (Sq +h)a2 i on HT4 (X). When restricted to a ∈ ker(Sq 3 + h) we have fC,h ˇ (a) = ha, PC,h ˇ i. It follows that the path integral vanishes due to the sum over flat fields unless there exists some C-field Cˇ0 , such that

(9.22)

fCˇ0 ,h (a) = ha, P0 i = 0

for all a ∈ ker(Sq 3 + h)|HT4 . Suppose such a C-field exists. We may ask what other C-fields contribute. Applying once more the cubic refinement law we find (9.23)

3 ˇ − a(Cˇ0 ))i fC,h ˇ0 ,h (a) + ha, (Sq + h)(a(C) ˇ (a) = fC

ˇ and a(Cˇ0 ) need not be torsion. Thus we conclude where a is torsion, but a(C) ˇ which contribute to the path integral must that the topological sectors a(C) satisfy: (9.24)

ˇ − a(Cˇ0 )) = 0 (Sq 3 + h)(a(C)

mod(Sq 3 + h)HT4 (X)

This simplifies (considerably) the discussion of the “Gauss law” in [7] and generalizes it to arbitrary torsion h-fields. Using a similar strategy one can easily derive an SL(2, ZZ) “equation of motion” for the torsion components of the C-field on 11-manifolds of the type Y = X9 × T 2 where T 2 carries the RR (a.k.a. odd, or nonbounding) spin structure. (The reader should compare our discussion with sec. 11 of [7].) Choose coordinates t1 , t2 on T 2 with ti ∼ ti + 1 and consider C-fields ˇ · tˇ2 where Cˇ0 , gˇ, h ˇ are all pulled back from of the type Cˇ = Cˇ0 + gˇ · tˇ1 + h ˇ can be X9 and, for clarity, we will drop all pullback symbols. The fields gˇ, h interpreted in type IIB string theory with ω(ˇ g) = Gr the fieldstrength of the ˇ RR 3-form and ω(h) = Hns the NSNS 3=form fieldstrength. The sum over the topologically trivial flat C-fields imposes (9.8), which in turn shows that (9.25)

[G0 ∧ Hns ]DR = [G0 ∧ Gr ]DR = [Gr ∧ Hns ]DR = 0

where G0 = ω(Cˇ0 ). The equations (9.25) are indeed standard consequences of the supergravity equations of motion. However, once these equations are satisfied there is a further constraint on the characteristic classes ag := a(ˇ g) ˇ and ah = a(h). Note that these classes need not be torsion classes, although, by (9.25) ag ah is a torsion class. Combining the cubic refinement law (9.3) with the bilinear identity (9.14) we find (9.26)

ˇ · tˇ2 + Cˇf ; Y ) Φ(Cˇ0 + gˇ · tˇ1 + h = haf , Sq 3 (ag + ah ) + ag ah ieiπf (af ) ˇ ˇ ˇ ˇ Φ(C0 + gˇ · t1 + h · t2 ; Y )

where Cˇf is a flat character on X9 , af = a(Cˇf ) and f (af ) is the mod 2 index on X9 × S+1 . By the bilinear identity eiπf (af ) = haf , P i where P depends only

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on the topology and spin structure of X9 . In particular, it is independent of ˇ Thus, we arrive at the SL(2, ZZ) invariant equation of motion: Cˇ0 , gˇ, h. (9.27)

Sq 3 (ag ) + Sq 3 (ah ) + ag ah = P

While this does not fully resolve the puzzle in section 11 of [7], it does constitute progress. 10. Application 1: The 5-brane partition function In addition to the electrically charged membranes, M-theory has magnetically charged 5-branes. In [25, 26] Witten has analyzed in detail topological considerations concerning the 5-brane partition function. In particular, he stated topological conditions necessary for the construction of a nonzero partition function. In this section we make contact with his work and the related work of Hopkins and Singer [9]. In particular we interpret a certain anomaly cancellation condition of Witten’s in terms of the class ΘX (a). As a preliminary, let us discuss some geometrical facts. The worldvolume of the 5-brane is denoted by W . We assume W is compact and oriented and embedded in an 11-dimensional spacetime ι : W ,→ Y . We may identify a tubular neighborhood of W in Y with the total space of the normal bundle N → W . The unit sphere bundle of radius r, X = Sr (N ) is then an associated S 4 bundle π : X → W , and we may construct an 11-manifold Yr with boundary X by removing the disk bundle of radius r, Yr = Y − Dr (N ). If X is oriented, compact, and spin, while W is orientable and compact, then one can show that the Euler class of the normal bundle vanishes and hence the Gysin sequence simplifies to give a set of short exact sequences (10.1)

π∗

π

0 → H k (W, Z) → H k (X, Z) →∗ H k−4 (W, Z) → 0.

¿From this we conclude that π ∗ : H 3 (W, Z) → H 3 (X, Z) is an isomorphism. Moreover, (10.2) H 4 (X, Z) ∼ = H 4 (W, Z) ⊕ Z The isomorphism is noncanonical, indeed, a choice of splitting is defined by 4 a choice of global angular form υ ∈ Hcpt (N ) with π∗ (υ) = 1. The general degree four class on X can then be written (10.3)

a = π ∗ (¯ a) + kυ.

with k ∈ Z. Finally, it follows from the above that (10.4) H 3 (X, U (1)) ∼ = H 3 (W, U (1)). 10.1. Statement of Anomaly inflow. The 5-brane is a magnetic source for the C-field. This means that if S 4 is a small sphere linking W in Y then Z (10.5) G=k S4

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for a “5-brane of charge k.” Here k is a nonzero integer. In the E8 model for the C-field this means that there is an E8 instanton on the 4-sphere of instanton number k. Consequently, the bundle P (a) → Yr cannot be smoothly prolonged to a bundle over Y . There are two ways to approach this difficulty. First, one may note that it is natural to associate the magnetic ˇ ) ∈ Zˇ 5 (Y ) to a single 5-brane wrapping current in differential cohomology δ(W W and view the C-field as a 4-cochain giving rise to this class. A second point of view, which we shall adopt here, proceeds by dividing up the physical system into two subsystems, the brane and the exterior. More precisely, we divide up Y into a small tubular neighborhood Dr (W ) of W and the complement Yr of this neighborhood. The two regions overlap on the 4sphere bundle X. The exterior of the 5-brane, Yr is referred to as the “bulk” and is described by 11-dimensional supergravity. In particular, the C-field path integral over Yr is the wavefunction (10.6)

Ψbulk ∈ Γ(L → Met(X) × EP (X))

discussed throughout this paper. In order to define the “partition function of M-theory with a 5-brane wrapping W ” we must define the 5-brane partition function ZM 5 together with a well-defined pairing (10.7)

hZM 5 , Ψbulk i

which can be integrated over, for example, the space of C-fields on X to produce the full partition function. The existence of a well-defined, gauge invariant function (10.7), is the so-called anomaly inflow cancellation statement. Note, in particular, that ZM 5 must be a section of a dual line bundle to L. Note that, since the 5-brane is a brane, ZM 5 should only depend on local data near W . In terms of the metric, it should be a function of the induced metric on W and its first few normal derivatives, such as the induced connection on the normal bundle N W . Moreover, ZM 5 should only depend on the “C-fields on W .” For this reason we should consider the limit that the radius r of the 4-sphere bundle goes to zero in discussing (10.7). The gauge equivalence classes of “C-fields on W ” will be shifted differential ¯ ∈H ˇ 41 (W ). In order to make the anomaly inflow statement characters [C] λ 2 W precise we must relate C-fields on W to C-fields on X. To that end we choose a C-field Cˇ0 so that we can map ˇ 41 (X) ˇ 41 (W ) → H (10.8) i:H 2

λW

2

λX

¯ = [Cˇ0 ] + π ∗ [C]. ¯ The choice of basepoint Cˇ0 is not canonical. Note via i[C] that it should be an unshifted character since π ∗ (λW ) = λX . The map (10.8) is compatible with the isomorphism (10.9)

J 3 (W ) → J 3 (X)

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where J 3 (X) is the connected component of the identity in H 3 (X, U (1)). Therefore, the construction of the line L over J 3 (X) can be used to construct a line L−1 over J 3 (W ). In section 3.2 of [25] (see especially p. 125) Witten uses the anomalyinflow statement to determine the line of which ZM 5 should be a section. He then notes that Hodge ∗ defines a complex structure on J 3 (W ) such that the curvature form of L−1 Z (10.10) ω(x1 , x2 ) := −2π x1 x2 X

is of type (1, 1). In this complex structure L−1 can be given the structure of a holomorphic line bundle which in turn gives J 3 (W ) the structure of a principally polarized abelian variety. The partition function ZM 5 is then identified, up to a metric-dependent constant, with the unique holomorphic section of this line bundle. The condition of holomorphy is interpreted as the condition of self-duality of the 3-form fieldstrength of the 5-brane. 10.2. Intrinsic and extrinsic definitions of the 5-brane partition function. In [25, 26], Witten has in fact given two definitions of the 5-brane partition function, which we will refer to as the “intrinsic” and “extrinsic” definitions. The extrinsic definition is the one based on anomaly inflow, as reviewed in the previous section. The intrinsic definition proceeds only from the data of a compact oriented Riemannian 6-manifold W . In the intrinsic definition we consider J 3 (W ) as a complex manifold with complex structure induced by ∗. We seek to define a holomorphic line bundle LM 5 → J 3 (W ) with Hermitian metric and curvature given by (10.10). In order to define the holomorphic line one must find a function (10.11)

ΩW : H 3 (W, ZZ) → ZZ2

which satisfies the cocycle relation: (10.12)

ΩW (x1 + x2 ) = ΩW (x1 )ΩW (x2 ) exp[iπ

Z

x1 x2 ] W

Different choices of ΩW correspond to lines LM 5 differing by tensoring with a flat holomorphic bundle of order 2. In [25, 26] Witten constructs such a function ΩW , when W is Spinc , making use of the Spinc structure. 20 It follows from (10.12) that ΩW is a homomorphism HT3 (W, ZZ) → ZZ2 ,→ Q/ZZ and hence by Poincar´e duality ΩW (x) = hθ, xi, for x ∈ HT3 (W ), is pairing with a distinguished element θ ∈ HT4 (W ). In [26], eqs. 5.5 et. seq., Witten then points out that in constructing a theta function as a sum over H 3 (W, ZZ), if θ 6= 0, the sum will 20This

construction is given in section 5.2 of [26]: One extends W × S 1 to a Spinc 8-fold R∂B = W × S 1 , and simultaneously extends x[dt] to a class z. Then ΩW (x) := exp[iπ B8 (z 2 + λz)], where 2λ = p1 − α2 and r2 (α) = w2 (B8 ).

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vanish, indicating the presence of an anomaly. Thus, Witten’s anomaly cancellation condition is θ = 0. In the next section we will interpret the class θ in terms of ΘX (a). 10.3. The decoupling conditions. It is widely believed that 5-brane theories exist independently of M-theory as distinguished (2, 0)-superconformal field theories in six dimensions. The “simple” theories are expected to be classified by ADE gauge groups and reduce, upon compactification on a circle and upon taking a long distance limit, to 5-dimensional nonabelian gauge theories with 16 supersymmetries. Nevertheless, the only known definition of the 5-brane theory is that given by taking a decoupling limit from the theory of the 5-brane in 11-dimensional supergravity. Therefore, we should only expect to be able to give an “intrinsic definition” to the 5-brane partition function when the brane can be consistently decoupled from the bulk. We will now interpret Witten’s torsion anomaly condition θ = 0 as a necessary condition for decoupling the 5-brane from the bulk. Let us now recall ˇ as C-field charge. If ΘX (C) ˇ 6= 0 then we cannot the interpretation of ΘX (C) decouple the 5-brane from the bulk. Effectively, open 2-branes must end on the 5-brane to satisfy charge conservation, and these spoil decoupling. Thereˇ = 0 is a necessary condition for fore, physical reasoning implies that ΘX (C) decoupling of the M5 brane from the bulk, i.e. a necessary condition for the existence of a nonzero “5-brane partition function.” In order to state the decoupling condition in terms intrinsic to W let us consider the integration over the fiber: π∗ (ΘX (a)) ∈ H 4 (W6 , ZZ). We use (10.3) to conclude that that a = π ∗ (¯ a) + υ for a single brane. Then we may split (noncanonically) π∗ (ΘX (a)) = 0 into its torsion and nontorsion components. In DeRham cohomology π∗ (ΘX (a)) = 0 implies that a¯ = 0. This is the well-known condition that the “C-field on W ” must have a fieldstrength which can be written as (10.13)

¯ = dh G

for some globally well-defined 3-form h. The form h is interpreted as the fieldstrength of the chiral 2-form field on W . When (10.13) is satisfied π∗ (ΘX (a)) is a torsion class. As we have seen, it is precisely this class which obstructs the definition of a well-defined line bundle L → J 3 (X) ∼ = J 3 (W ). When ΘX (a) = 0 we can identify (10.14)

ΩW (x) = ϕ(Cˇ0 , χ). ˇ

Here χ ˇ is a character such that ω(χ) ˇ = x. Of course, such lifts are ambiguous by flat characters, but, precisely because ΘX (a) = 0, this ambiguity drops out of the right hand side of (10.14). This is in accord with the argument in [26], section 5.3 . (The latter argument relied on several topological restrictions on the normal bundle of W in Y . )

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In general, the anomaly inflow argument implies that, for α ∈ H 2 (W, U (1)) we should identify hα, θi = ΩW (β∗ α), where β∗ is the Bockstein homomorˇ = hα, π∗ ΘX (C)i, ˇ phism, with Φ(X × S−1 , Cˇ + tˇπ ∗ α) = hπ ∗ (α), ΘX (C)i and ˇ thus we conclude that θ = π∗ (ΘX (C)). 11. Application 2: Relation of M-theory to K-theory In the previous section we have shown that if π∗ (ΘX ) 6= 0 then there must be 2-branes ending on a 5-brane. In type II string theory there is an analogous statement for D-branes. If a D-brane wraps a worldvolume W in isolation, i.e. with no other D-branes ending on it, then it is necessary that: (11.1)

(Sq 3 + [H])P D(W ) = 0.

This condition is closely related to the K-theoretic classification of D-branes, as explained in [7, 15, 27, 28, 29]. If Y11 is S 1 -fibered over a 10-manifold U10 , then we expect a relation between M theory on Y and type II string theory on U10 . Therefore, in this situation we expect a relation between ΘX (a) and the left hand side of (11.1). We will now show that this is indeed the case. For simplicity, suppose Y11 = U10 × S 1 . Then ∂Y11 = X10 , ∂U10 = V9 , so X10 = V9 × S 1 . A characteristic class of CˇX on X10 can be written as: (11.2)

a = π ∗ (¯ a) + π ∗ ([H])[dx11 ]

where a¯ ∈ H 4 (V9 ; ZZ), [H] ∈ H 3 (V9 ; ZZ), and π : X10 → V9 is the projection. The first result is that if X10 = V9 × S 1 has Ramond (nonbounding) spin structure on S 1 and [H] = 0 then ˜ X (a)) = Sq 3 (¯ (11.3) π ∗ (Θ a) Thus the Gauss law and the K-theoretic restriction on the RR fluxes are indeed closely related. To prove (11.3) we compute the pairing of the left hand side with α ∈ 2 H (V9 , U (1)). This is (11.4) hΘX (a), π ∗ (α)i = Φ(p∗ Cˇ + tˇp∗ (π ∗ α), ˇ X × S−1 ) ˇ 4 (X10 ), p : X10 × S 1 → X10 , and the coordinate on S 1 is t. where Cˇ ∈ H − − If Cˇ is pulled back from V9 then we may regard the 11-manifold Y = X10 × S−1 instead as Y = X 0 × S+1 , with X 0 = V9 × S−1 . Then we can apply to result of [7] to obtain ˜ X (a), αi = eiπ[f (¯a+dtβ(π∗ α))−f (dtβ(π∗ α)] (11.5) hπ∗ Θ where β : H 2 (V9 , U (1)) → HT3 (V9 ) is the Bockstein and f (a) is the E8 mod 2 index on X 0 = V9 × S−1 . Here we have used [7] eq. 8.24 and the fact that β(α) is flat, hence the local term cannot contribute because the local density does not contain dt. Now we use the bilinear identity of [7]. f (¯ a) = 0 since the S−1 is a bounding spin structure. Therefore (11.6)

˜ X (a), αi = e hπ∗ Θ

iπ

R

1 V9 ×S−

a ¯Sq 2 dtβ(α)

= hSq 3 (¯ a), αi

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where in the second equality we used the pairing H 7 (V9 , ZZ) × H 2 (V9 , U (1)) → U (1). This establishes (11.3). It is also of interest to know π∗ (ΘX (0)). The style of argument above shows that (11.7)

hα, π∗ (ΘX (0))i = eiπf (β(α)dt)

This formula shows that π∗ (ΘX (0)) is at most two-torsion. There seems to be no elementary formula for Rthis mod two index. If V9 = X8 × S+1 then one can show that 21 f (bdt) = r2 (b)Sq 2 r2 (b) for b ∈ H 3 (X8 , ZZ). The latter expression is, in general, nonzero (e.g. for X8 = SU (3) and b a generator of H 3 (SU (3); ZZ) ), and hence we conclude that π∗ (ΘX (0)) is in general nonzero. Now let us turn to the inclusion of an [H] flux. It is straightforward to show (11.8)

π∗ (ΘX (a)DR ) = [H]DR ∧ a ¯DR

There are two ways to lift this to a statement about the integral classes. First, using the bilinear identity (8.2) (with a2 → a2 − a1 , together with (11.3) we find that if ai = π ∗ a¯i + π ∗ [H][dx11 ] then (11.9)

π∗ (ΘX (a1 ) − ΘX (a2 )) = (Sq 3 + [H])(¯ a1 − a¯2 )

Moreover, using (8.3) we also have (11.10)

¯ π∗ (2ΘX (a)) = [H](2¯ a − λ)

Now, (11.10) and (11.9) together amount to the “moral” statement 1¯ (11.11) π∗ (ΘX (a))“ = 00 (Sq 3 + [H])(¯ a − λ) 2 (it is only a moral statement because the division by 2 is illegal in the presence of 2-torsion). Hence, the Gauss law is in harmony with the K-theoretic classification of RR fluxes. The above arguments extend the results of [7, 15] to manifolds with boundary. To complete the story one should demonstrate that the C-field wavefunction is related to the RR flux wavefunction in the expected manner. This is an interesting problem, but we leave it for future work. (Some preliminary remarks appear in [30].) 12. Application 3: Comments on spatial boundaries One of the primary motivations for developing the E8 formalism for the C-field is the desire to make precise the intuition that the C-field is related to some kind of topological gauge theory in the bulk of 11 dimensions, but 21Let

+ denote the nonbounding, − the bounding spin structure on S 1 . Note that f−− (bdt1 ) = f−− (bdt2 ) = 0 where the subscript denotes the spin structure. Note that under a diffeomorphism (t1 , t2 ) → (t1 , t1 + t2 ) we have f+− (bdt2 ) = f−− (bdt1 + bdt2 ). Now apply the bilinear identity.

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which becomes a dynamical theory upon the introduction of boundaries. In this section we make some simple comments on one realization of that idea. Suppose now we have a spatial boundary ι : X ,→ Y . The adjective “spatial” means that we will be regarding X as a Euclidean spacetime, rather than as a temporal slice. We need to choose boundary conditions on the Cfield. One natural choice of boundary condition on Cˇ = (A, c) is the condition (12.1)

ι∗ (c) = 0.

That is, we restrict attention to the C-fields: (12.2)

EP (Y, X) := {(A, c) ∈ EP (Y )|ι∗ (c) = 0}

Now, to define a physical theory, we need to describe the gauge symmetry. It is quite standard for boundary conditions to break a symmetry group G to a subgroup H. It is thus natural to consider the subgroup of G(Y ) of elements that preserve (12.2). In the present case, the group elements (α, χ) ˇ ∈ ∗ ∗ G(Y ) which preserve the entire set (12.2) must satisfy ι (α) = ι (ω(χ)) ˇ = 0. Such group transformations leave “too many” physical degrees of freedom. In particular, any two connections on ι∗ (P ) would be considered to be gauge inequivalent. It is here that the categorical viewpoint of section 3.4 becomes quite useful. Let Pˆ = ι∗ (P ). There is an obvious restriction functor r : EP (Y )//G(Y ) → EPˆ (X)//G(X). The simplest way to characterize the category defined by the boundary condition (12.1) is that it is the subcategory of EP (Y )//G(Y ) which maps under r to a category with the morphisms of the form (A, 0) → (Ag , 0), ˆ To say this a little more formally, we noted in section 3.4 for g ∈ AutP. above that ordinary bundle automorphisms of Pˆ define a subgroupoid of EPˆ (X)//G(X), in spite of the fact that Aut(Pˆ ) is not a subgroup of G(X). Let us fix a functor I from the standard gauge theory groupoid A(Pˆ )//Aut(Pˆ ) to EPˆ (X), by choosing I(A) = (A, 0) on objects. The groupoid of C-fields and gauge transformations defined by (12.1) is the fiber product EP (Y, X): r

(12.3)

EP (Y )//G(Y ) → EPˆ (X)//G(X) ↑ ↑I ˆ EP (Y, X) → A(P )//Aut(Pˆ )

The situation is best described as follows: This boundary condition breaks the topological gauge symmetry G. It is not possible to describe it as the breaking of a gauge group, but we can Rview it as a symmetry breaking of groupoids. Note in particular that while Y trF ∧ ∗F is not gauge R invariant, and hence there are no propagating gauge modes in the interior, X trF ∧ ∗F is gauge invariant and therefore, with the above notion of a groupoid of fields, we can define a gauge invariant theory with dynamical E8 gauge fields on the boundary.

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12.1. Relation to heterotic M-theory. Now that we have explained one way in which dynamical gauge fields on the boundary can be related to the topological gauge theory of the E8 gauge field in the interior we will indicate how it can be compatible with the outstanding example of M-theory on a manifold with spatial boundary, namely the Horava-Witten model [4, 5] of heterotic M-theory. Consider the E8 model of the C-field on an 11-manifold of the type X ×[0, 1] and impose boundary conditions ι∗L (c) = ι∗R (c) = 0. We learned in the previous section that we indeed find dynamical E8 gauge fields on each boundary. However, because the 11-dimensional spacetime provides a homotopy of the left and right connections the E8 bundles on the boundaries necessarily have characteristic classes aL = aR . Thus, this theory resembles the nonsupersymmetric model introduced by Fabinger and Horava [31]. The above simple observation raises a challenge to using the E8 formalism to describe the Horava-Witten model. We may overcome this difficulty as follows. Note that M-theory should be formulated without use of an orientation, but our formulation breaks parity automatically since a → λ − a under orientation reversal. We may give a parity invariant formulation of the M-theory C-field by passing from Y to Yd , the orientation double cover of Y , and defining a Cfield to be a parity invariant E8 cocycle on Yd . Let σ be the nontrivial deck transformation on Yd , a parity invariant E8 cocycle is one such that the shifted differential character satisfies: ˇ = [C] ˇP (12.4) σ ∗ ([C]) where we recall that the parity transform was defined in (8.4). If Y is orientable, this amounts to saying that a C-field is a pair {(A1 , c1 ), (A2 , c2 )} such that (12.5)

χ ˇ(A1 ,c1 ) = χˇ∗(A2 ,c2 )

The gauge group is G(Yd ) × G(Yd ). This gauge invariance preserves (12.5). So, again, no degrees of freedom are added when Y has no boundary. However, on a manifold with boundary this formulation leads to a natural boundary condition which again leads to dynamical gauge fields on the boundary. On X × [0, 1] we take (12.6)

i∗L (c1 ) = 0

i∗R (c2 ) = 0

Then, we get a dynamical gauge field AL with characteristic class aL on the left boundary and another one AR with aR on the right boundary. Note that ι∗L ([G]) = aL − 12 λ, while ι∗R ([G]) = −(aR − 12 λ) and hence aL + aR = λ, as expected. It is important to note that with our boundary conditions ι∗ (G) = ±(trF 2 − 1 trR2 ), on left- and right- boundaries, respectively, while dG = 0 throughout 2 the 11-manifold. This is to be contrasted with the equation one often finds

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in the literature on heterotic M-theory, namely, 1 1 (12.7) dG = δ(x) trF (AL )2 − trR2 ± δ(π − x) trF (AR )2 − trR2 . 2 2 In our view, such a Bianchi identity implies that the boundary carries nontrivial magnetic current, and a proper formulation of the C-field will involve a different geometrical construction from what we have used. It is also worth noting that since Φ is valued in the Pfaffian line bundle the E8 gauge anomalies cancel (locally) in a way which is manifest. The gravitational anomalies are more subtle, but we expect that the same will be true for them. We hope to discuss this elsewhere [23]. (See [32] for a different point of view.) 13. Conclusions and future directions In this paper we have given a precise mathematical formulation of the E8 model for the M-theory C-field. We have used it to write the Chern-Simons term of the 11-dimensional supergravity action and we have used it to describe the Gauss law for the C-field in precise terms, applicable to topologically nontrivial situations. In particular we have shown how to define the C-field electric charge induced by the nonlinear interactions of the C field, and by gravity, as an integral cohomology class. We have also given other applications to a clarification of the topological conditions for the existence of the 5-brane partition function and to the relation of M-theory flux quantization to IIA Ktheoretic flux quantization. Finally, we have sketched how one may formulate the topological field theory of the E8 gauge field to allow for dynamical gauge fields on the boundary of an 11-manifold. Many related problems remain open and issues remain to be resolved. We will survey some of these problems here. • Gravitino determinant. There are several nontrivial technical issues related to the gravitino path integral. We hope to address these elsewhere [23]. • Derivation of the K-theoretic classification of RR fields. The K-theoretic formulation of RR fields should be generalized to manifolds with boundary and the wavefunction of the C-field in IIA supergravity compared to the wavefunction of C-field in M theory. We made some progress in section 11 but the story remains to be completed. • Is the E8 formalism really necessary ? One of the virtues of the E8 formalism is that it allows us to define the action of 11-dimensional supergravity, and the Gauss law for wavefunctions of C-fields on manifolds with boundary. Nevertheless, the E8 gauge field plays a purely topological role and appears, in some sense, to be a “fake.” As we have mentioned, the category EP (X)//G is equivalent to the cateogry Zˇ 41 λ (X) of Hopkins-Singer cocycles. Indeed, 2 there is an alternative construction of the C-field and M-theory action which

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makes no use of E8 gauge fields [8]. What remains to be seen is whether any of these formalisms is really useful for physical investigations, and whether they lead to a useful reformulation of M-theory. • Electric-Magnetic duality. The E8 formalism is seemingly very asymmetric between 5-branes and 2-branes. Formally, one would expect a dual formuˇ 7 (X). lation of the theory in terms of Cheeger-Simons characters [CˇD ] ∈ H These objects would define the holonomies of the C-field on 5-brane worldvolumes. However, there is no obvious E8 model for a dual object CˇD . Since the theory is nonlinear, this duality transformation is not obvious. • Parity invariance. A basic axiom of M theory is that it is parity invariant. This is what allows us to gauge parity and produce chiral theories such as the heterotic string. In this paper, we have assumed that Y is an oriented 11-manifold and we have used the orientation heavily in writing integrals of differential forms, and in defining spinor bundles and η invariants. A very interesting open problem is the generalization of our formalism to unoriented and nonorientable 11-manifolds. • Anomaly cancellation. In spite of the shortcommings noted above, we believe that the present formalism should have many future applications to anomaly cancellation issues connected to 5-branes, G2 singularities, and frozen singularities in M-theory [33]. For example, the present formalism leads to a substantial simplification of the anomaly cancellation for normal bundle anomalies of the M5-brane described in [34]. Acknowledgements: We would like to thank J. Harvey, M. Hopkins, D. Morrison, G. Segal, and E. Witten for useful discussions on this material. G.M. and D.F. would like to thank H. Miller and D. Ravenel for the invitation to speak at the Newton Institute conference on elliptic cohomology. Portions of this work were carried out by G.M. at the Aspen Center for Physics. Portions of this work were carried out by D.F. and G.M. at the Isaac Newton Institute, Cambridge and at the KITP, Santa Barbara. The work of E.D. and G.M. is supported in part by DOE grant DE-FG02-96ER40949. The work of D.F. is supported in part by NSF grant DMS-0305505. References [1] An online version is available at http://online.kitp.ucsb.edu/online/mp03/moore1. [2] D. S. Freed, “K-theory in quantum field theory,” Current Developments in Mathematics 2001, International Press, Somerville, MA, 41-87 [arXiv:math-ph/0206031]. [3] D. S. Freed, “Dirac charge quantization and generalized differential cohomology,” Differ. Geom. VII (2000) 129–194 [arXiv:hep-th/0011220]. [4] P. Horava and E. Witten, “Eleven-Dimensional Supergravity on a Manifold with Boundary,” Nucl. Phys. B 475, 94 (1996) [arXiv:hep-th/9603142].

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[5] P. Horava and E. Witten, “Heterotic and type I string dynamics from eleven dimensions,” Nucl. Phys. B 460, 506 (1996) [arXiv:hep-th/9510209]. [6] E. Witten, “On flux quantization in M-theory and the effective action,” J. Geom. Phys. 22, 1 (1997) [arXiv:hep-th/9609122]. [7] D. E. Diaconescu, G. W. Moore and E. Witten, “E(8) gauge theory, and a derivation of K-theory from M-theory,” Adv. Theor. Math. Phys. 6, 1031 (2003) [arXiv:hepth/0005090]. [8] D. Freed and M. Hopkins, to appear. [9] M. J. Hopkins and I. M. Singer, “Quadratic functions in geometry, topology, and M-theory,” arXiv:math.at/0211216. [10] J. Evslin, “From E(8) to F via T,” arXiv:hep-th/0311235. [11] V. Mathai and H. Sati, “Some Relations between Twisted K-theory and E8 Gauge Theory,” arXiv:hep-th/0312033. [12] E. Witten, “Topological Tools In Ten-Dimensional Physics,” Int. J. Mod. Phys. A 1, 39 (1986). [13] S. S. Chern and J. Simons, “Characteristic forms and geometric invariants,” Ann. Math. 99 ( 1974) pp. 48–69. [14] D. S. Freed, Classical Chern-Simons Theory II, Houston J. Math. 28 (2002), no. 2, 293–310. [15] D. E. Diaconescu, G. W. Moore and E. Witten, “A derivation of K-theory from Mtheory,” arXiv:hep-th/0005091. [16] D. Morrison, Talk at Strings2002, http://www.damtp.cam.ac.uk/strings02/avt/morrison/. [17] P. Horava, “M-Theory as a Holographic Field Theory,” hep-th/9712130. [18] S. Melosch and H. Nicolai, “New canonical variables for d = 11 supergravity,” Phys. Lett. B 416, 91 (1998) [arXiv:hep-th/9709227]. [19] J. Lott, “R/Z index theory,” Comm. Anal. Geom. 2 (1994) 279. [20] G. W. Moore, “Some comments on branes, G-flux, and K-theory,” Int. J. Mod. Phys. A 16, 936 (2001) [arXiv:hep-th/0012007]. [21] J. Dupont, R. Hain and S. Zucker, Regulators and characteristic classes of flat bundles, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 47–92, Amer. Math. Soc., Providence, RI, 2000, alg-geom/9202023. [22] R. Stong, “Calculation of Ωspin 11 (K(Z, 4))” in Unified String Theories, 1985 Santa Barbara Proceedings, M. Green and D. Gross, eds. World Scientific 1986. [23] To appear. [24] X. Dai and D. S. Freed, “Eta-invariants and determinant lines,” J. Math. Phys. 35, 5155 (1994) [Erratum-ibid. 42, 2343 (2001)] [arXiv:hep-th/9405012]. [25] E. Witten, “Five-brane effective action in M-theory,” J. Geom. Phys. 22, 103 (1997) [arXiv:hep-th/9610234]. [26] E. Witten, “Duality relations among topological effects in string theory,” JHEP 0005, 031 (2000) [arXiv:hep-th/9912086]. [27] J. M. Maldacena, G. W. Moore and N. Seiberg, “D-brane instantons and K-theory charges,” JHEP 0111, 062 (2001) [arXiv:hep-th/0108100]. [28] J. Evslin and U. Varadarajan, “K-Theory and S-Duality: Starting Over from Square 3,” hep-th/0112084; Journal-ref: JHEP 0303 (2003) 026. [29] G. Moore, “K-theory from a physical perspective,” arXiv:hep-th/0304018. [30] G. W. Moore and E. Witten, “Self-duality, Ramond-Ramond fields, and K-theory,” JHEP 0005, 032 (2000) [arXiv:hep-th/9912279]. [31] M. Fabinger and P. Horava, “Casimir effect between world-branes in heterotic Mtheory,” Nucl. Phys. B 580, 243 (2000) [arXiv:hep-th/0002073]. [32] A. Bilal and S. Metzger, “Anomaly cancellation in M-theory: A critical review,” Nucl. Phys. B 675, 416 (2003) [arXiv:hep-th/0307152].

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[33] J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D. R. Morrison and S. Sethi, “Triples, fluxes, and strings,” Adv. Theor. Math. Phys. 4, 995 (2002) [arXiv:hep-th/0103170]. [34] D. Freed, J. A. Harvey, R. Minasian and G. W. Moore, “Gravitational anomaly cancellation for M-theory fivebranes,” Adv. Theor. Math. Phys. 2, 601 (1998) [arXiv:hepth/9803205]. Department of Physics, Rutgers University, Piscataway, New Jersey, 088550849 and Department of Mathematics, University of Texas at Austin

ALGEBRAIC GROUPS AND EQUIVARIANT COHOMOLOGY THEORIES J.P.C. GREENLEES Abstract. Many important cohomology theories are associated to algebraic groups, and this is clearest when equivariant theories are considered. The purpose of this article is to describe some familiar examples and to speculate on the existence of further examples.

Contents 1. Introduction. 2. K-theory and the multiplicative group. 3. The shape of a cohomology theory. 4. The non-split torus. 5. Elliptic cohomology and elliptic curves. 6. T-equivariant elliptic cohomology. 7. Shapes from projective varieties. 8. Rational equivariant cohomology theories. 9. The model for the circle group G = T. 10. Reflecting the group structure of the elliptic curve. References

89 91 93 96 97 99 100 103 105 107 109

1. Introduction. 1.A. The context. This article discusses a connection between the world of algebraic geometry and the world of algebraic topology. In the world of algebraic geometry, an algebraic group G is a functor associating a group G(k) to a commutative ring k. In the world of algebraic topology, a G-equivariant cohomology theory associates a graded abelian group EG∗ (X) to a space X with G-action, where G is the ambient compact Lie group of equivariance. Sometimes an algebraic group G corresponds to an equivariant cohomology theory EG∗ (·) in a way we will describe. Going from topology to geometry is the more elementary process, and we will describe several well-known examples: in fact when G is the circle group T, suitable cohomology theories ET∗ (·) give rise to one dimensional algebraic The author is grateful to M.Ando, M.J.Hopkins, H.R.Miller and N.P.Strickland for collaborations and discussions which have informed this work.

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groups G by evaluation on the appropriate spaces. More striking is the fact that we can sometimes reverse the process and move from geometry to topology. From the easy direction, we shall see that a one dimensional algebraic group G suggests the values of an equivariant cohomology theory ET∗ (·) on equivariant spheres. Rather remarkably, the algebraic model for rational Tequivariant cohomology theories [11] shows that, for rational theories, the values on spheres are very close to determining the cohomology theory, and in favourable cases they determine it completely. Going one step further, one can often construct a cohomology theory from its putative values on spheres, and this is often the case for values arising from algebraic groups. To put flesh on this philosophy we need some examples. 1.B. Some familiar examples. In the most familiar case, the multiplicative group Gm is associated to K-theory, with the association being clearest for KT∗ (·) where T is the circle group. This is discussed in detail in Section 2, and if the introduction seems too abstract, the reader should read Section 2 first. It is also familiar that the additive group Ga is associated to ordinary cohomology with the association being clearest for the T-equivariant Borel theory H ∗ (ET ×T (·)). Similarly, there is at least a formal group associated to any complex oriented theory E ∗ (·), with the association being clearest for the T-equivariant Borel theory E ∗ (ET ×T (·)). When the formal group is the formal completion of an ellipic curve around the identity we obtain elliptic cohomology theories, but again the association between the cohomology theory and the ellipitc curve is clearest for the T-equivariant version of the theory [19, 15]. It is the new behaviour of this T-equivariant elliptic theory, and the possibility of adapting its construction that led to the author writing this article. 1.C. Some putative examples. We will speculate on higher dimensional examples, at least when both the algebraic group and the compact Lie group of equivariance are abelian and connected. In this case it seems that if an algebraic group G of dimension d is associated to a cohomology theory EG∗ (·) then the compact Lie group G is of dimension d, and in general one might expect their ranks to agree. However we will see that it is very rare for EG∗ (·) to be complex orientable, so that these theories will be of an unfamiliar sort. 1.D. Some motivation. One reason for this investigation is to understand familiar objects better. Equivariant versions of elliptic cohomology behave in a way which is unfamiliar to topologists. To develop appropriate intuition, we need to put this into context. One might even hope that the new context would help us to find a truly geometric definition. History has shown that cohomology theories arising from algebraic groups are significant, having applications to index theory [32, 31], rigidity [25, 2, 3] and to geometric representation theory [10, 9]. This alone makes it worth seeking new ones.

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In any case, the strength of the connection with algebraic geometry was already the motivation for the introduction of equivariant elliptic cohomology: applying the functor to natural objects in representation theory gives interesting algebras. Similar constructions are straightforward for other cohomology theories based on algebraic groups. 2. K-theory and the multiplicative group. We will describe properties of the multiplicative group, K-theory and the relationship between them in some detail because the underlying phenomena are well known, and several of the issues arise in this case. 2.A. The multiplicative group and the multiplicative formal group. The multiplicative group Gm is the functor on commutative rings l defined by Gm (l) = Units(l) = Rings(Z[z, z −1 ], l). The first equality is the definition, and the second shows that Gm is represented by the ring Z[z, z −1 ]. We also say that Z[z, z −1 ] is the ring of functions on Gm and write OGm = Z[z, z −1 ]. The group operation is given by multiplication of units Units(l)×Units(l) = Rings(Z[z, z −1 ]⊗Z[z, z −1 ], l) −→ Rings(Z[z, z −1 ], l) = Units(l), and this induced by the ring homomorphism ∆ : Z[z, z −1 ] −→ Z[z, z −1 ] ⊗ Z[z, z −1 ] z 7−→ z ⊗ z. The identity element of Units(l) is 1 ∈ l and therefore the identity e of Gm is represented by the augmentation Z[z, z −1 ] −→ Z setting z = 1. Accordingly, the ideal of functions vanishing at e is the principal ideal (y) where y = 1 − z. We say that y is a coordinate at the identity. In terms of y, the coproduct then takes the more familiar form y 7−→ 1 ⊗ y + y ⊗ 1 − y ⊗ y. We may then complete around the identity to form the multiplicative formal group (Gm )∧e with ring of functions (Z[z, z −1 ])∧(y) = Z[[y]]. 2.B. T-equivariant K-theory and T-equivariant Borel K-theory. We can easily recover the multiplicative group from T-equivariant K-theory and the multiplicative formal group from T-equivariant Borel K-theory, where T is the circle group. Just as the multiplicative formal group is only an approximation to the multiplicative group itself, so Borel K-theory K ∗ (ET×T X) is only an approximation to the true equivariant theory KT∗ (X) for a Tspace X. Just as the multiplicative formal group is a completion of the multiplicative group, so the Atiyah-Segal completion theorem says Borel Ktheory is a completion of equivariant K-theory.

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The cohomology theory associated to the group Gm is T-equivariant Ktheory for the circle group T. Indeed, the ring of functions on Gm is realized by KT0 = R(T) = Z[z, z −1 ]. The coproduct is realized by the multiplication map µ : T × T −→ T in the sense that the coproduct is µ∗

0 KT0 −→ KT×T = KT0 ⊗ KT0 .

The cohomology theory associated to the formal group (Gm )∧e is Borel Ktheory for the circle group T. Indeed, the ring of functions on the formal group (Gm )∧(e) is realized by its value on a point K 0 (BT) = K 0 (ET ×T pt) = (KT0 )∧J = R(T)∧(e(z)) = (Z[z, z −1 ])∧(y) = Z[[y]], where the first equality is the Atiyah-Segal completion theorem [5] with J = ker(R(T) −→ Z) the augmentation ideal, and the other equalities hold since J = (e(z)) and the Euler class e(z) = 1 − z = y. The coproduct is realized by the multiplication map Bµ : BT × BT = B(T × T) −→ T since ˆ 0 (BT). K 0 (BT × BT) = K 0 (BT)⊗K 2.C. Reconstructing T-equivariant K-theory from the multiplicative group. To fully justify saying that K-theory is associated to Gm we would need to reverse the process of making the multiplicative group from K-theory. In general it is not known how to make such a construction, but it can be done rationally using the algebraic model for rational T-spectra [11]. We return to this in Section 9.C below. 2.D. A-equivariant formal groups and A × T-equivariant Borel Ktheory. There is an intermediate stage between the multiplicative group and its completion at the identity: we may complete Gm along a subgroup. This is known as an equivariant formal group [6, 14, 29]. More precisely, we choose a homomorphism ζ : A∗ −→ Gm of algebraic groups, where A is a finite abelian group and A∗ = Hom(A, S 1 ) is the dual group. The equivariant formal group is the formal neighbourhood (Gm )∧ζ of the image of ζ equipped with the map A∗ −→ (Gm )∧ζ . To obtain an interesting construction, we need some points of finite order. Thus we want ζ to be specified by a map A∗ −→ Units(k) for a ring k with more units than Z. Accordingly, we change base to the version of the multiplicative group Gm |k over k: this is the functor defined on k-algebras l and has ring of functions k[z, z −1 ]; by the Yoneda lemma, ζ is now given by a homomorphism ζ : A∗ −→ Units(k) = Gm |k (k) of ordinary groups. The homomorphism ζ is represented by the Hopf algebra map θ : k[z, z −1 ] −→ A∗ kQ with αth component θα (z) = 1 − ζ(α) for α ∈ A∗ . Thus θ has kernel ( α∈A∗ (1 − ζ(α))) and the ring of functions on (Gm |k )∧ζ is the completion O(Gm |k )∧ζ = k[z, z −1 ]∧(Qα∈A∗ (1−ζ(α))) .

The universal example of this is when k = R(A) = Z[A∗ ] and ζ(α) = α.

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This algebraic intermediate stage is also realized topologically, by using a suitable A × T-equivariant Borel version of A-equivariant K-theory. To 0 explain this, we start by observing that KA×T = R(A × T) = R(A)[z, z −1 ] is the ring of functions on the multiplicative group Gm |k over k = R(A), and the coproduct is still induced by µ : T × T −→ T. We can now construct the ring of functions on the A-equivariant formal group by using a Borel theory. This time we need the universal T-free A-space EA T, which is the total space of the universal bundle over the classifying space BA T for Aequivariant line bundles. We may then take the Borel theory KA0 (EA T ×T X) where X is an A × T-space. Again the Atiyah-Segal completion theorem [1] gives a connection with rings of functions on the equivariant formal group: the ring of functions on (Gm |k )∧ζ when k = R(A) is 0 0 0 KA0 (BA T) = KA×T (EA T) = (KA×T )∧(e(CA⊗z)) = (KA×T )∧(Qα (1−αz)) .

The components of the fixed point set (BA T)A correspond to the A-equivariant line bundles over a point (i.e., to points of A∗ ), and the resulting A-map A∗ −→ BA T induces the structure map A∗ −→ (Gm |k )∧ζ of the equivariant formal group. 3. The shape of a cohomology theory. Before we turn to other examples we must be more precise about what we mean by a cohomology theory being associated to an algebraic group. In this section we consider the example of K-theory in some detail. The description is phrased so that it can be easily transposed to give general principles in the following sections. 3.A. The definition of the shape. Non-equivariantly, the spheres S n play a dual role. They are the building blocks for spaces and are also invertible as spectra. In G-equivariant homotopy theory, nice spaces are built from the suspended homogeneous spaces S n ∧ G/K+ . These are not invertible unless K = G, and the group theory means that the cohomology groups E˜G∗ (S n ∧ ∗ G/K+ ) = E˜K (S n ) can be complicated as modules over the coefficient ring EG∗ . On the other hand, the one point compactification S V of a representation V is invertible as a spectrum, and E˜G∗ (S V ) is often rather simple. For this reason, we may view the values on representation spheres S V as a rather basic invariant of the homology or cohomology theory. Partly because of variance, it is simpler to concentrate on homology, but the analogue for cohomology is also useful. Definition 3.1. The shape of an equivariant homology theory E˜∗G (·) is the composite functor ˜ G (·) E

∗ AbGp RepC (G) −→ G-spectra −→

where RepC (G) is the category of unitary representations of G and inclusions.

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For example, if E˜∗G (·) is complex oriented then E˜∗G (S V ) is a shifted copy of the coefficient group E∗G . However, even in this case the shape includes useful information. 3.B. Equivariant K-theory. We concentrate on the circle group G = T. In this case, any complex representation is a sum of the one dimensional representations z n for integers n, where z is the natural representaton of T on C. Applying K-theory to the inclusion 0 −→ V gives a diagram ˜ T (S 0 ) = 0 − 7 → K 0 ↓ ↓ T ˜ V − 7 → K0 (S V ) ∼ =

R(T) = ↓ χ(V ) ∼ R(T) =

R(T ) ↓ iV 1 · R(T ). χ(V )

˜ 0T (S V ) ∼ The isomorphism K = R(T), and the fact that the inclusion S 0 −→ S V induces multiplication by the Euler class χ(V ) come from equivariant Bott periodicity. The last isomorphism has replaced multiplication by the Euler class χ(V ) by an inclusion, at the price of making 1/χ(V ) the generator of the free R(T )-module. For a 1-dimensional representation α we have χ(α) = 1−α, and χ(V ⊕ W ) = χ(V )χ(W ), so that we may easily calculate χ(V ) in general. In particular for V = z n then we find χ(z n ) = 1 − z n , which is familiar as the function on the multiplicative group Gm whose vanishing defines the n subgroup Gm [n] := ker(Gm −→ Gm ) of points of order dividing n. Writing OGm (Gm [n]) for the ideal of functions permitted a simple pole on Gm [n], we may summarize the discussion by the calculation ˜ 0T (S z n ) = 1 · R(T ) = OGm (Gm [n]). K 1 − zn Thus the shape of K-theory is specified by functions on the multiplicative group; more precisely, if V T = 0 we have ˜ 0T (S V ) = OGm (D(V )) K P P for a suitable divisor D(V ); indeed if V = n6=0 an z n then D(V ) = n6=0 an Gm [n]. Suspension by the trivial complex representation (i.e., double real suspension) corresponds to the tangent bundle (Ω1Gm )∨ . It is convenient to prove the dual statement that desuspension by corresponds to Ω1Gm . Since we have ˜ 0 (S W ) = K ˜ T (S −W ), we work in cohomology to prove K ˜ 0 (S 2 ) ∼ K = Ω1Gm . First 0 T T note that S 2z is obtained from S z by attaching a single T-free 3-cell, so that we have a cofibre sequence Σ2 T+ −→ S z −→ S 2z . ˜ 0 . Remembering that we are now working in cohomology, Now apply K T 0 z ˜ (S ) = J and K ˜ 0 (S z ) = J 2 , where J = OGm (−(e)) is the ideal of functions K T T vanishing at the identity, and we obtain the a short exact sequence ˜ 0 (Σ2 T+ ) ←− J ←− J 2 ←− 0. 0 ←− K T

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Hence the value of the non-equivariant theory on S 2 is the conormal bundle at the identity (i.e., the cotangent space at the identity since we are dealing with a point): ˜ 0 (S 2 ) = K ˜ 0 (Σ2 T+ ) ∼ K = J/J 2 = ωe1 . Accordingly, double suspension in cohomology is given by tensoring with ωe1 , and in particular the equivariant theory is the module of K¨ahler differentials ˜ 0 (S 2 ) = ω 1 ⊗K 0 K 0 ∼ K = Ω1 T

e

T

Gm

as claimed, where the final isomorphism is given by using the group structure on Gm to trivialize the module, or in more concrete terms, by using the function z. The fact that Ω1Gm is trivializable corresponds to the fact that KT∗ is 2-periodic. Since the K-theory of all complex spheres is in even dimensions, this describes all of K-theory. 3.C. Complex orientable theories. The above discussion also applies to complex orientable cohomology theories E ∗ (·) and their associated formal b Since different notation may be more familiar to some readers, we groups G. provide a brief comparison. The relevant equivariant cohomology is the Borel theory ET∗ (X) = E ∗ (ET ×T X), so we have ET∗ = E ∗ (BT) and

E˜T∗ (S V ) = E˜ ∗ (BTV ), where BTV denotes the Thom space of the bundle associated to the representation V . The ring E ∗ (BT) is complete for the skeletal topology, and is b and since we have a therefore the ring of functions on a formal scheme G, ∗ b completed K¨ unneth theorem for E (BT × BT), multiplication on T gives G the structure of a formal group. Now consider the ideal J = ker(ET∗ = ET∗ (pt) −→ ET∗ (T) = E ∗ ) of functions vanishing at the identity, which is principal by complex orientabilb and the coefficient ring is the ity. A generator y of J is a coordinate on G, ∗ ∗ power series ring ET = E [[y]]. The cofibre sequence T+ −→ S 0 −→ S z ˜ ∗ (BT z ). Since shows that y is the restriction of a Thom class in E˜T∗ (S z ) = E T n ∗ zn z = n z we see that a Thom class for BT is given by [n](y), the pull back of y along the nth power map Bn : BT −→ BT. In these terms, and remembering we are working in cohomology, we recover the usual formula b ˜ ∗ (S z n ) = E˜ ∗ (BT z n ) = ([n](y)) = O b (−G[n]). E T

G

Specializing, this gives ˜ ∗ (S z ) = E˜ ∗ (BT z ) = (y) = J = O b (−(e)), E T G

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and hence the behaviour under untwisted suspensions E˜ ∗ (S 2 ) ∼ = J/J 2 ∼ = ωe1 and E˜T∗ (S 2 ) ∼ = E˜ ∗ (BT+ ∧ S 2 ) ∼ = Ω1Gb . 3.D. What is special about K-theory and the multiplicative group? In algebra and topology there are several very special features of the examples described in this section. They are rather obvious in these cases, but the analogues become more interesting as we change the algebraic group and the cohomology theory: we will argue that the algebraic and topological special features correspond precisely. We concentrate on K-theory for simplicity, and we have in mind G = A × T for a finite abelian group A. • The ideal of functions on Gm vanishing at the identity is principal, and KG∗ (·) is complex orientable. • Gm is affine and KG∗ is in even degrees. • The group Gm is one dimensional and KG∗ is 2-periodic. We will see in Section 4 that the first property fails for the non-split torus, in Section 5 that the second property fails for the elliptic case, and in Section 7 that the third fails for abelian varieties of dimension ≥ 2. 4. The non-split torus. We describe a one dimensional algebraic group whose ideal of functions vanishing at the origin is not principal. There is a corresponding cohomology theory which is therefore not complex orientable. 4.A. The non-split torus and its formal group. Let C be a 1-dimensional torus over a Q-algebra k, with ring of functions O. The standard torus is the multiplicative group C = Gm with O = k[z, z −1 ], but there is also a non-split (non-deploy´e) form Gnd with O = k[a, b]/(a2 + b2 = 1), which we consider here. This is related to Gm since they become isomorphic when we adjoin a square root i of −1. We then have z = a + ib, and the usual formula (a1 + ib1 )(a1 + ib1 ) = (a1 a2 − b1 b2 ) + i(a1 b2 + a2 b1 ) gives the coproduct a 7−→ a ⊗ a − b ⊗ b b 7−→ a ⊗ b + b ⊗ a. Now the counit sets a = 1 and b = 0 so that the ideal of functions vanishing at the identity is (1 − a, b). This is not a principal ideal. Furthermore, even if we complete at the identity, the ideal (1 − a, b) is still not principal. Thus even the Borel theory of a theory associated to the non-split torus will not be complex orientable. One way to obtain Gnd is as a quotient of Gm |k(i) . Indeed, k[a, b]/(a2 + b2 = 1) is the fixed ring of k(i)[z, z −1 ] under the action of the group of order 2 acting by Galois automorphisms on k(i) and exchanging z with z −1 .

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4.B. Constructing the associated cohomology theory. Rationally, one may construct a T-equivariant cohomology theory associated to Gnd by the method of Section 9. It would be interesting to have a bundle theoretic construction. Complex orientable theories ET∗ (·) have the property that for any complex representation V the ET∗ -module E˜T∗ (S V ) is free on one generator. The cohomology theory associated to Gnd is therefore not complex orientable since its value on S z is the non-principal ideal of functions vanishing at the identity. 5. Elliptic cohomology and elliptic curves. We now replace the multiplicative group by an elliptic curve C over k. This makes both the algebra and the topology more complicated, but probably the biggest change is that C is no longer affine. For general background on elliptic curves and algebraic geometry see [28, 20]. 5.A. Elliptic curves and their formal completions. Because C is not affine we must replace the ring of functions by the sheaf of functions O = OC . Since all regular functions on C are constant, if we want to recover C from a knowledge of functions, we need to permit some poles. These are measured by considering formal sums of points of C, known as divisors. Given a divisor P D = P n(P )P on C we may form the line bundle O(D), of functions which have poles of order ≤ n(P ) at P . The Riemann-Roch theorem states that if D has degree ≥ 1 then P

dimk H 0 (C; O(D)) = deg(D),

where deg(D) = P n(P ). Thus in particular if we concentrate on the identity e and consider a multiple D = ne, the sheaf O(ne) allows functions with a pole of order n ≥ 1 at the identity e, and we obtain n independent functions since dimk H 0 (C; O(ne)) = n. Thus if n = 1 we only obtain the constants, for n = 2 we obtain a new function, call it x, and for n = 3 we obtain a new function, call it y. For n = 4 we may use x2 as the new function, and for n = 5 we may use xy. However for n = 6 we have two potential new functions, y 2 and x3 . There is therefore a linear relation amongst the 7 functions so far, and this may be put into the Weierstrass form y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 , for suitable a1 , a2 , a3 , a4 , a6 in k. It turns out that this implies all relations, and defines C. Indeed, we may consider the graded ring k(C, e) := {H 0 (C; O(ne))}n≥0 , and the above discussion my be restated briefly in geometric language as C = Proj(k(C, e)). In the Weierstrass form, we find there is one point at infinity, and we may choose this as the identity of the group C.

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5.B. Reflecting the group structure on functions. For an elliptic curve C the graded ring k(C, e) is not a Hopf algebra. This is because the pullback of a line bundle L, such as O(ne), along the multiplication map C × C −→ C does not decompose as a tensor product. However, for any line bundle L we may form M = M(L) = p∗1 L ⊗ p∗2 L, and then if we define ξ : C × C −→ C × C by ξ(x, y) = (x + y, x − y), the theorem of the square shows that ξ ∗ M ∼ = M⊗2 . This gives ξ ∗ : H 0 (C; L) ⊗ H 0 (C; L) −→ H 0 (C; L⊗2 ) ⊗ H 0 (C; L⊗2 ), and hence the multiplication on C is reflected in the functions. The properties satisfied by the function ξ are fairly complicated [24]. On the other hand, we may construct a formal group by completing C at the identity, and since C is a smooth curve the resulting representing ring is isomorphic k[[t]]. Since C × C is a smooth surface, if we complete it at its identity we obtain a ring of functions k[[1 ⊗ t, t ⊗ 1]]. Thus, the group operation on C gives a coproduct on k[[t]], and when we choose a suitable coordinate, this takes the familiar form of an elliptic formal group law. 5.C. Elliptic cohomology theories and elliptic formal groups. An elliptic spectrum in the sense of Ando-Hopkins-Strickland [4] is a triple (E, C, φ) where E is a 2-periodic even spectrum, necessarily complex orientable, an elliptic curve C over E 0 and an isomorphism φ between the formal group E 0 (BT) of E and Ce∧ . In the present terms, the T-equivariant Borel theory of E is associated to the formal group Ce∧ . 5.D. T-equivariant elliptic cohomology theories and elliptic curves. Since C is not determined by a ring of functions, it is now rather less clear what it means for a T-equivariant cohomology theory to be associated to an elliptic curve. However the way the graded ring k(C, e) was used in reconstructing C suggests that the sections of various line bundles should occur as values of the cohomology theory. The way the multiplication on C was reflected in ξ : C × C −→ C × C suggests that the T × T equivariant theory for C × C should play a role in describing this structure. All of this structure should reduce to elliptic formal groups in the Borel theory. In Section 10 we make precise the properties it is natural to hope for. Such a cohomology theory has been constructed [15] by the methods described in Section 8 below. 5.E. A-equivariant elliptic cohomology theories and elliptic formal groups. Just as we constructed A-equivariant formal groups from Gm for a finite abelian group A, we may construct them by completing around the image of ζ : A∗ −→ C. Strickland has recently shown [30] that provided 2 is inverted, these correspond to complex oriented A-equivariant cohomology

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theories. Devoto earlier constructed a version for the Jacobi quartic when the group order is inverted [7] and investigated its role in index theory. 6. T-equivariant elliptic cohomology. If we are looking for other interesting shapes in the sense of Definition 3.1, it is natural to consider other ways of getting a graded abelian group from a representation, and we might consider replacing the triangulated category of spectra by a different one, such as the category of sheaves on an algebraic variety. 6.A. Sheaves on a curve. As with K-theory, we expect to associate to an invertible (i.e., one dimensional) representation α, an invertible sheaf (i.e., a line bundle) Lα = OC (D(α)) on the curve C. Sums of representations should correspond to sums of divisors or tensor products of bundles: α 7 → − Lα = OC (D(α)) V ⊕W − 7 → LV ⊗ LW = OC (D(V ) + D(W )). One also hopes to make the tensor product of representations correspond to some structure in the category of sheaves, but this seems less straightforward. One can then obtain a potential shape of a homology theory E˜∗T (·) by taking sheaf cohomology of the associated line bundle ˜ T (S V ) = H i (C; OC (D(V ))) E −i

for suitable values of i. Since C is a curve, the sheaf cohomology on the right will only be non-zero for i = 0, 1, so we expect to deal with other values of i by twisting the sheaf with the line bundle L = OC (D()) associated to the trivial representation of T on C; just as in the case of the multiplicative group, we expect L to have a rather different character to the line bundles associated to representations with no fixed points. When C is an elliptic curve, the identity element e provides a preferred divisor, and the example of the multiplicative group suggests we should choose D(z) = e. We then observe that z n = n∗ z and hence choose n

D(z n ) = n∗ e = C[n] = ker(C −→ C), and this too is analogous to the case of the multiplicative group. Again the example of the multiplicative group suggests the trivial representation should be associated to the tangent sheaf L = (Ω1C )∨ . However, since C is a group, this tangent sheaf is trivializable. Thus we have the association 0 z zn V

7−→ 7−→ 7−→ 7−→ 7−→

L 0 = OC L = (Ω1C )∨ Lz = OC (e) Lz n = OC (C[n]) LV = OC (D(V )); P P as with K-theory and Gm , if V = n6=0 an z n , we take D(V ) = n an C[n].

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6.B. An associated cohomology theory. In fact one may associate a Tequivariant cohomology theory to an elliptic curve. Theorem 6.1. [15] If C is an elliptic curve over a field of characteristic 0 then there is a T-equivariant cohomology of the shape described in Subsection 6.A. In particular, it is 2-periodic and for i = 0, 1 we have E˜ T (S V ) = H i (C; O(D(V ))) −i

and

E˜Ti (S V ) = H i (C; O(−D(V ))).

˜ T (S nz ) = k n for n ≥ 1, so that the theory is not complex For example E 0 orientable, and E˜iT (S 0 ) = k for all i, so that the theory is not concentrated in even degrees. The construction depends on an auxiliary piece of structure: a coordinate divisor, which is the divisor of a function te vanishing to the first order at e with all zeros and poles at points of finite order. For instance if the points of order 2 are P, P 0 , P 00 then e + P − P 0 − P 00 is a coordinate divisor. From the function te one can construct functions ts vanishing to the first order at points of exact order s for each s ≥ 2. Indeed, the space of functions vanishing at the ns points of exact order s and with a pole of order ≤ ns at e is one dimensional, and we may normalize by requiring that ts /tne s has the value 1 at e. In fact, the connection between elliptic curves and T-equivariant cohomology goes a bit further. The representing spectrum EC is a commutative ring spectrum, and we may consider modules over it. Inverting equivariant equivalences (which are equivalences in T-fixed points and in T[n]-fixed points for all n ≥ 1), we obtain its homotopy category Ho(T-EC-mod). On the other hand we may consider sheaves of modules over OC ; here we only consider sheaves as taking values on open sets which are the complements of complete sets of points of specified finite order. Inverting strong equivalences (which are equivalences in all H ∗ (C; O ⊗ (·)) and in H ∗ (C; O(C[n]) ⊗ (·)) for all n ≥ 1), we obtain its derived category D(OC -mod). These algebraic and topological derived categories are equivalent. Theorem 6.2. [15] There is an equivalence of triangulated categories D(OC -mod) ' D(T-EC-mod). 7. Shapes from projective varieties. Encouraged by the success described in the previous section, we seek suitable shapes associated to curves of higher genus, and discuss the properties of a cohomology theory with such a shape. At present the author does not know if cohomology theories with this shape exist, but for suitable curves, one can construct what would be T-equivariant restrictions of them. The geometry is too intricate to be described here in detail, but we give an outline.

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7.A. Sheaves on a Jacobian. If C is a curve of genus g, we may consider shapes associated to the Jacobian J = Jac(C) (for background see [23, 22, 21]). If the genus is 1 then C = J, and we are reduced to the case of an elliptic curve, as discussed in Subsection 6.A above. Suppose C is a curve over the complex numbers. The genus g of C may be calculated either topologically via the singular homology H1 (C; Z) ∼ = Z2g 0 1 ∼ g or geometrically, via the space H (C; ΩC ) = C of holomorphic 1-forms. We may construct the Jacobian Jac(C) as follows. A topological 1-chain Z on R C defines a linear form on the holomorphic 1-forms by integration ω 7−→ Z ω; this defines an embedding H1 (C; Z) −→ H 0 (C; Ω1C )∨ , and Jac(C) is the quotient. The most important features of J are (1) that it is a group, (2) that it has a preferred divisor Θ (the theta divisor), uniquely specified up to translation and (3) that it has the structure of an algebraic variety. Any other principally polarized abelian variety of dimension g would do as well for the general discussion, although special properties of the abelian variety will probably required to actually construct such a theory. Since J is of dimension g, one might expect the associated group of equivariance to be a torus Tg of dimension g. The group of torsion points of J is Jtors ∼ = (Q/Z)2g as an abelian group, whereas (Tg )∗ ∼ = Zg , so we may choose a complete level structure (Tg )∗ ⊗ (Q/Z × Q/Z) ∼ = Jtors . For the Jacobian we have a natural isomorphism Jtors = H1 (C; Q/Z), so it is natural to choose the factors of Tg to correspond to a decomposition of H1 (C; Z) into a sum of 2-dimensional hyperbolic forms; in more concrete terms we choose copies of Q/Z × Q/Z to correspond to pairs of curves on C with a single point of intersection. For some purposes, it may be more appropropriate to use the torus T = Hom(N S(J), T) dual to the NeronSeveri group as the group of equivariance. Here N S(J) := im(H 1 (J; O× ) −→ H 2 (J; Z)) is a free abelian group. Its rank is called the Picard number, and this is ≤ g 2 . With this structure available, we may attempt to define a suitable shape. It turns out that the divisors n∗ Θ and n2 Θ define isomorphic line bundles. However, the divisor n∗ Θ is usually not decomposable. It is therefore better to replace it by a sum of translates of Θ. Indeed, if z is a generating one dimensional representation, our level structure provides P an embedding z˜ : Q/Z × Q/Z −→ Jtors , and we may consider Θz,n := na=0 Θz˜(a) , where Θx denotes the translation of Θ by x. The divisor Θz,n is of degree n2 , and linearly equivalent to n2 Θ since {˜ z (a) | na = 0} is a subgroup of Jtors with the property that the sum of its elements is the identity. This leads to the

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following association. 0 z zn V

7 → − 7−→ 7−→ 7−→ 7−→

L0 L Lz Lz n LV

= = = = =

OJ ? OJ (Θ) OJ (Θz,n ) O(D(V ))

The constraints on the choice of L are suggested by the discussion in Subsection 7.B. Note in particular that (Ω1J )∨ is not a line bundle for g ≥ 2, so we need another candidate. Since J is of dimension g, the sheaf cohomology of these line bundles will be in degrees between 0 and g, so we now only expect a spectral sequence g E i,2j = H i (J; OJ (D(V )) ⊗ Lj ) ⇒ E˜ T (S V ), 2

∗

where odd rows are zero. The cohomology of all these sheaves is well-known, and for g = 2, a spectral sequence of this form necessarily collapses for some natural choices of L . This leads to predictions for the values of the cohomology theory on spheres. 7.B. Further constraints. We argue here that if there is a cohomology theory associated to an abelian surface, it cannot be 2-periodic. Suppose then that C is a curve of genus 2, and that ET∗ 2 (·) is a cohomology theory associated to it. Since g = 2, we may identify the theta divisor Θ with the curve C embedded in J by the Abel-Jacobi map (once we have picked a basepoint on C). We choose a circle subgroup T and restrict attention to the T-equivariant cohomology theory. Now we consider the two cofibre sequences S 0 −→ S z −→ ΣT+ and S z −→ S 2z −→ Σ3 T+ , where the second is obtained by smashing the first with S z . The point to note is the untwisting T+ ∧ S z ' T+ ∧ S 2 that has been used to identify the last term in the second cofibre sequence. These cofibre sequences should correspond to the exact sequences O −→ O(C) −→ Q1 and O(C) −→ O(2C) −→ Q2 of sheaves over J, where the second is obtained by tensoring the first with O(C). Here we note that Q1 and Q2 are supported on C. Indeed, we have Q1 ∼ = i∗ q1 and Q2 ∼ = i∗ q2 for suitable sheaves q1 and q2 on C. We even expect q1 and q2 to be line bundles on C. Since the second exact sequence is obtained from the first by tensoring with O(C), we find Q2 ∼ = Q1 ⊗ O(C) ∼ = i∗ (q1 ⊗ O(C)|C ). On the other hand, by the untwisting we have Q2 ∼ = Q1 ⊗ L ∼ = i∗ (q1 ⊗ (L )|C ).

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This gives

L |C ∼ = OJ (C)|C ∼ = Ω1C , where the final equality is the adjunction formula. An alternative argument adapts the discussion in Subsection 3.B to show L |C is the normal bundle of C in J; we see that this is Ω1C , since the canonical bundle on J is trivial. Since C is of genus 2, the line bundle Ω1C is of degree 2, and H 0 (C; (Ω1C )⊗n ) depends on n (it is of dimension 2 for n = 1 and of dimension 2n − 1 for n ≥ 2). This shows that even the non-equivariant form E ∗ (·) = ET∗ (T × (·)) of the theory will not be 2-periodic. 8. Rational equivariant cohomology theories. So far we have only presented evidence for a one-way connection between algebraic groups and equivariant cohomology theories. In the one dimensional case, suitable cohomology theories ET∗ (·) give rise to one dimensional algebraic groups G by evaluation on suitable spaces. On the other hand, we have argued that a one dimensional algebraic group G suggests the values of an equivariant cohomology theory ET∗ (·) on spheres (i.e., the shape of the theory). In this section we show that the shape of a cohomology theory is very close to determining a cohomology theory over the rationals, and that it does determine a cohomology theory in favourable cases. 8.A. Algebraic models for categories of rational cohomology theories. The idea is that for any compact Lie group G there is an abelian category A(G) modelling rational G-equivariant cohomology theories. As usual, it is worth working with the category of objects representing the cohomology theories, namely rational G-spectra. This category admits the structure of a Quillen model category, and its homotopy category gives the cohomology theories. Thus homotopy equivalence classes of G-spectra correspond to Gequivariant rational cohomology theories, and all cohomology theories are of the form EG∗ (X) = [X, E]∗G . On a practical level, we want to be able to calculate in this homotopy category, but if we understand the category completely we can also construct interesting new cohomology theories. The idea is that objects of A(G) should be a rather small, and based on information easily accessible from the cohomology theories they represent. One natural way to construct simpler cohomology theories from EG∗ (·) is to take Borel cohomology of fixed points. Indeed, if X is a G-space we may form the H-fixed point space X H ; the group WG (H) = NG (H)/H acts on this, as does the maximal torus T WG (H) of its identity component. We may therefore form the Borel theory E˜T∗ WG (H) (ET WG (H)+ ∧ X H ). At least when G is abelian, this is closely related to the shape of the cohomology theory. Indeed, each of the two steps can be performed (up to extension) using the

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shape. For the first step, we use the formal localization theorem, which states that in suitable circumstances X H may be obtained by inverting formal Euler classes. To explain this, we suppose H is normal, and consider the infinite sphere S ∞V (H) = lim S V . We say that G has enough representations to → V H =0

detect H if (S ∞V (H) )K is contractible when K does not contain H. When H is normal in G and G has enough representations to detect H, obstruction theory gives [T H , X H ]G/H = [T, X ∧ S ∞V (H) ]G as required. For the second step, we note that for any torus T we may form the universal free T -space ET from spheres. More precisely, if we choose a direct product decomposition T = T×· · ·×T and z1 , z2 , . . . , zr are the natural representations of the factors, we have ET ' S(∞z1 ) × S(∞z2 ) × · · · × S(∞zr ). The factors are related to spheres relevant to the shape by the cofibre sequences S(∞z)+ −→ S 0 −→ S ∞z , so that we obtain a model for ET+ like the stable Koszul complex. Conjecture 8.1. For a compact Lie group G there is an abelian category A(G) and a Quillen equivalence G-spectra/Q ' dgA(G) such that (1) A(G) is abelian (2) InjDim(A(G)) = rank(G) (3) The category consists of sheaves of modules over a space of closed subgroups of G; the object corresponding to a cohomology theory EG∗ (·) has fibre over H built from the Borel theory ET∗ WG (H) (ET WG (H)+ ∧ X H ). The additional structure is built from these Borel theories using their relation under localization and inflation. (4) The model of EG∗ (·) is built from its shape and a little extra structure. 8.B. Consequences of the conjecture. Note immediately that if the conjecture holds we have an equivalence of homotopy categories Ho(G-spectra/Q) ' D(A(G)) as triangulated categories. This reduces to algebra the problem of classifying rational equivariant cohomology theories and the process of calculation with them. Furthermore, it provides a universal homology theory π∗A(G) : G-spectra −→ A(G) and an Adams spectral sequence A(G) Ext∗,∗ (X), π∗A(G) (Y )) ⇒ [X, Y ]G ∗ A(G) (π∗

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for calculation. Finally, because of the injective dimension of A(G), the Adams spectral sequence is only non-zero on s-line for 0 ≤ s ≤ r, so the calculation is very accessible. 8.C. Status of the conjecture. • G finite. The conjecture is true. Indeed, all rational cohomology theories are ordinary. This follows easily from three facts: (1) rational cohomology is rational homotopy, (2) the rational cohomology of a finite group is trivial, and (3) rational Mackey functors are all injective. The idempotents in the rationalized Burnside ring allow us to express this in terms of representation theory, so that Y QWG (H)-mod. A(G) = (H)

• The circle group G = T. Again the theorem is true. Indeed, [11] constructs A(G) and shows that there is a triangulated equivalence of homotopy categories. Shipley [27] upgraded this to a Quillen equivalence. The model is briefly described in Section 9 below. • The groups G = O(2), SO(3) and their double covers. In this case the equivalence of homotopy categories is proved in [12, 13]. • The tori G = Tg . The Adams spectral sequence exists [16], and in [17, 18] we work towards showing that the Quillen equivalence holds. 9. The model for the circle group G = T.

9.A. The category A(T). The objects of A(T) are sheaves over the discrete set F ∪ {T}, where F is the set of finite subgroups. It is thus natural that the structure sheaf should be the constant sheaf Q[c], with global sections Y R = map(F, Q[c]) = Q[c]. n

Next we need to discuss suspension. If w : F −→ Z≥0 is a function zero almost everywhere, we may form Y Σw R = Σ2w(n) Q[c]. n

For example, if W is a complex representation with W T = 0 we might take w(n) = dimC (W T[n] ), and Σw would correpond to suspension by W ; this explains the factor of 2. The inclusion R −→ Σw R corresponds to the inclusion S 0 −→ S W , and hence we think of it as division by a universal Euler class. Now form [ t= Σw R = E−1 R, w

where the last equality refers to the idea that the map R −→ Σw R is multiplication by an Euler class, so that in forming the union we have inverted the multiplicative set of Euler classes.

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The objects M of A(T) are of the form β

M = (N −→ t ⊗ V ), where N is an R-module, and β is a map of R-modules which becomes an isomorphism when E is inverted. The morphisms are diagrams M = (N −→ t ⊗ V ) ↓ θ↓ ↓1⊗φ 0 0 M = (N −→ t ⊗ V 0 ) It is not hard to show that A(T) is abelian and of injective dimension 1. Theorem 9.1. [11, 27] There is a Quillen equivalence T-spectra/Q ' dgA(T). 9.B. Rigid even objects. Consider an object M = (N −→ t ⊗ V ) of A(T). If N is concentrated in even degrees and E-torsion free we have N = ker(t ⊗ V −→ T ) where T isLthe E-torsion module (t ⊗ V )/N . Furthermore, it then turns out that T = n Tn where Tn is a torsion module over Q[c]. We may thus define the object M by using the map M q : t ⊗ V −→ Tn , n

and this always defines an object M provided q is surjective. Now if M represents the homology theory E∗T (·) it turns out that we can very nearly recover M from the shape of E∗T (·): • V = lim E˜ T (S W ) • T =

∗ → W T =0 ˜ T (S 0 ) cok(E ∗

i

−→ lim

→ W T =0

E˜∗T (S W )).

The reason that the shape does not quite determine the object is that the map q also encodes information about division by Euler classes. Summary 9.2. If i is injective, we may almost write down the object ME of A(T ) representing the homology theory E∗T (·) purely in terms of the shape. 9.C. Construction of a cohomology theory from an algebraic group. When C is a 1-dimensional algebraic group we may now describe the construction of a 2-periodic rational T-equivariant cohomology theory ET∗ (·) associated to it. Indeed, since the theory is 2-periodic we need only describe the model in degrees 0 and 1, and we take the odd part to be zero. In degree 0 we take V0 = lim

→ W T =0

˜ T (S W ) = lim E ∗

→ W T =0

H 0 (C; O(D(W )) = KC ,

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where KC is the ring of meromorphic functions with poles only at points of finite order. For the torsion we take M 1 T0 = H 0 (C; KC /OC ) = HChsi (C), s

1 where Chsi consists of the points of exact order s and HChsi (C) is the local cohomology, and therefore consists of the principal parts of functions at points of exact order s. Finally, to define the map q : t ⊗ V0 −→ T0 we must choose some additional structure. For each s ≥ 1 we must choose a function ts vanishing to the first order at points of Chsi and with poles only at points of finite order. The geometry of C may suggest natural ways to make this choice but for now we assume the choice has been made. For instance if G = Gm we may take ts to be the sth cyclotomic polynomial φs (z), and for an elliptic curve we may use the functions constructed from the coordinate divisor in Section 6. Having made this choice, we may define q(cw ⊗ f ) for a meromorphic function f to have as sth component w(s)

qs (cw ⊗ f ) = ts

f.

w(s)

In fact this is legitimate since ts f is regular on Chsi for all but finitely many values of s, and in fact q is determined by the values on these elements. 9.D. The shape of a rigid even spectrum. The sphere S W is modelled by (Σw R −→ t ⊗ Q. ¿From this it is easy to calculate HomA(T) (S W , ME) and ExtA(T) (S W , ME). This gives E˜0T (S W ) = E˜T0 (S −W ) = ker(q : cw ⊗ KC −→ T ), and

T E˜−1 (S W ) = E˜T1 (S −W ) = cok(q : cw ⊗ KC −→ T ). Since functions are regular if they are regular at all points, it is clear that H 0 (C; O(D(W ))) = ker(q : c−w ⊗ KC −→ T ) = E˜ 0 (S W ), T

T as required. The argument for E˜−1 (S W ) involves a little more geometry [15, Section 5].

10. Reflecting the group structure of the elliptic curve. We explain here how the map ξ ∗ of Subsection 5.B should be realized in topology. Since we do not yet have a complete algebraic model of rational T × T-equivariant spectra, we can only show that the structures arising in algebra would be precisely mirrored in topology under the assumption that the model works as expected. Suppose there exists a T × T-equivariant cohomology theory with shape suggested by C × C. Constructing such a theory is significantly easier than constructing a T × T-equivariant theory for an arbitrary abelian surface. To

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the representation w i ⊗ z j of T × T we associate the divisor C[i] × C[j] and extend this to arbitrary representations in the usual way. The theory E(C × C)∗T×T (·) should then come with a spectral sequence T×T

^ H ∗ (C × C; OC×C (D(W ))) ⇒ E(C × C)∗

(S W ).

Since some line bundles have cohomology in degree 2, this does not determine T×T ^ E(C × C)∗ (S W ) in general. However when OC×C (D(W )) has no cohomology in dimension 2 we would find T×T

^ E(C × C)0

(S W ) = H 0 (C × C; OC×C (D(W ))).

Next, the map ξ : C × C −→ C × C is an isogeny with kernel ∆A[2] = {(a, a) | a + a = e}. We also consider the corresponding group homomorphism ξˆ : T × T −→ T × T, ˆ z) = (wz, w/z), which is surjective with kernel defined by ξ(w, ∆T[2] = {(z, z) | z 2 = 1}. To minimize confusion, we identify the second T × T with T × T = (T × T)/∆T[2]. The map ξ should correspond to a map E(C × C) −→ E(C × C) ξi∗ : inf T×T T×T (i for inflation) of T × T-spectra or adjointly, to a map ξf∗ : E(C × C) −→ E(C × C)∆T[2] (f for fixed point) of T × T-spectra. Lemma 10.1. For any representation W of T × T, the map ξf induces ξf∗ : E(C × C)T×T (S W ) −→ E(C × C)0T×T (S W ). 0 Proof: The map ξf∗ induces ξf∗ : [S 0 , S W ∧ E(C × C)]T×T −→ [S 0 , S W ∧ E(C × C)∆T[2] ]0T×T , 0 T×T

^ so it suffices to identify the domain and codomain. By definition E(C × C)0 T×T 0 W [S , S ∧ E(C × C)]0 so we turn to the codomain and calculate

(S W ) =

[S 0 , S W ∧ E(C × C)∆T[2] ]0T×T = [S −W , E(C × C)∆T[2] ]0T×T = [S −W , E(C × C)]0T×T = [S 0 , S W ∧ E(C × C)]0T×T T×T ^ = E(C × C) (S W ). 0

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To model M = p∗1 L ⊗ p∗2 L with L = O(D(W )) for a representation W of T we take W = (W ⊗ 1) ⊕ (1 ⊗ W ). Direct sum of representations corresponds to tensor product of line bundles and to sums of divisors, so if W corresponds to the line bundle L and the divisor D(W ), then W corresponds to the line bundle p∗1 L⊗p∗2 L and the divisor [D(W )×C]+[C×D(W )]. Viewed as a representation of T × T by pullback along ξˆ we find ξˆ∗ (W ) = ξˆ∗ W ⊕ ξˆ∗ W. 1

2

In particular if W = z we find ξˆ∗ (W ) = (w n ⊗ z n ) ⊕ (w n ⊗ z −n ). n

Finally, we need to observe that for any n, the bundles associated to (w n ⊗ z n ) ⊕ (w n ⊗ z −n ) and (w 2n ⊗ 1) ⊕ (1 ⊗ z 2n ) are isomorphic: this is precisely the same argument as showed ξ ∗ M ∼ = M2 above. With L = O(D(W )), we thus expect a commutative diagram ξ∗

H 0 (C; L)⊗2 = H 0 (C × C; p∗1 L ⊗ p∗2 L) −→ H 0 (C × C; p∗1 L2 ⊗ p∗2 L2 ) = H 0 (C; L2 )⊗2 ↓ ↓ T×T

^ E(C × C)0

(S ) W

ξf∗

−→

T×T

^ E(C × C)0

(S W ).

References [1] J.F.Adams, J.-P.Haeberly, S.Jackowski and J.P.May “A generalization of the AtiyahSegal completion theorem.” Topology 27 (1988) 1-6. [2] M. Ando “Power operations in elliptic cohomology and representations of loop groups” Trans. American Math. Soc. 352 (2000) 5619-5666 [3] M. Ando and M. Basterra “The Witten genus and equivariant elliptic cohomology.” Math. Z. 240 (2002) 787-822. [4] M. Ando, M.J.Hopkins and N.P.Strickland “Elliptic spectra, the Witten genus and the theorem of the cube.” Inventiones Math. 146 (2001) 595-687 [5] M.F.Atiyah and G.B.Segal “Equivariant K-theory and completion” J.Diff. Geom. 3 (1969) 1-18. [6] M.Cole, J.P.C.Greenlees and I.Kriz ‘Equivariant formal group laws.’ Proc. LMS 81 (2000) 355-386 [7] J.Devoto “Equivariant elliptic homology and finite groups.” Michigan Math. J. 43 (1996) 3-32 [8] L.G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) “Equivariant stable homotopy theory.” Lecture notes in maths. 1213, Springer-Verlag (1986) [9] V.Ginzburg (notes by V. Baranovsky) “Geometric methods in the representation theory of Hecke algebras and quantum groups.” NATO Sci. Ser. C Math.Phy. Sci. 514, Kluwer (1998) 127-183 (AG/9802004) [10] V.Ginzburg, M. Kapranov, E. Vasserot “Elliptic algebras and equivariant elliptic cohomology” Preprint (1995) (q-alg/9505012)

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[11] J.P.C.Greenlees “Rational S 1 -equivariant stable homotopy theory.” Mem. American Math. Soc. 661 (1999) xii + 289pp [12] J.P.C.Greenlees “Rational O(2)-equivariant stable homotopy theory.” Fieldds Inst. Comm. 19 AMS (1998) 103-110 [13] J.P.C.Greenlees “Rational SO(3)-equivariant cohomology theories.” Contemporary Maths. 271 AMS (2001) 99-125 [14] J.P.C.Greenlees “Equivariant formal group laws and complex oriented cohomology theories.” HHA 3 (2001) 225-263 [15] J.P.C.Greenlees “Rational S 1 -equivariant elliptic cohomology.” Topology (to appear) 67pp. [16] J.P.C.Greenlees “Rational torus equivariant cohomology theories I: calculating groups of stable maps.” (Submitted for publication) 25pp [17] J.P.C.Greenlees “Rational torus equivariant cohomology theories II: the algebra of localization and inflation.” (Submitted for publication) 22pp [18] J.P.C.Greenlees and B.E. Shipley “An algebraic model for rational torus equivariant stable homotopy.” (Submitted for publication) 41pp [19] I. Grojnowski “Delocalized elliptic cohomology.” Yale University Preprint (1994). [20] R. Hartshorne “Algebraic geometry” Springer-Verlag, 1977 [21] H.Lange and Ch.Birkenhake “Complex abelian varieties.” Springer-Verlag (1992) [22] D.Mumford “Abelian varieties.” Oxford University Press (1974) [23] D.Mumford “Curves and their Jacobians.” University of Michigan Press (1975), reprinted as Appendix to Lecture Notes in Mathematics 1358 Springer-Verlag (1999) [24] D.Mumford “On the equations defining abelian varieties, I” Inventiones, (1966) 287354 [25] I.Rosu “Equivariant elliptic cohomology and rigidity.” American J. Math. 123 (2001) 647-677. [26] J.-P.Serre “Algebraic groups and class fields” Springer-Verlag (1988) [27] B.E.Shipley “An algebraic model for rational S 1 -equivariant stable homotopy theory.” Q.J.Math. 54 (2002) 803-828 [28] J.H.Silverman “The arithmetic of elliptic curves.” Springer-Verlag (1986) [29] N.P.Strickland “Multicurves and equivariant cobordism.” Preprint (2002) 55pp. [30] N.P.Strickland “Arithmetic abelian-equivariant elliptic cohomology.” (In preparation) [31] H.Tamanoi “Elliptic genera and vertex operator super-algebras.” Lecture Notes in Mathematics, 1704 Springer-Verlag, Berlin, 1999. vi+390 pp [32] E.Witten “Elliptic genera and quantum field theory.” Comm. Math. Phys. 109 (1987), no. 4, 525–536. Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK. E-mail address: [email protected]

IAN GROJNOWSKI’S “DELOCALIZED EQUIVARIANT ELLIPTIC COHOMOLOGY”

In 1994 Ian Grojnowski distributed a preprint sketching the construction of a model for a complex circle-equivariant elliptic cohomology theory. This preprint has had a great impact on the subject, and we are grateful to Grojnowski for allowing us to reproduce it here. We provide a brief setting for this paper, and mention some of the work which has flowed from it. Since rational cohomology theories are ordinary, it is natural to try to express equivariant rational (or complex) cohomology theories in terms of the ordinary rational cohomology of fixed point data. This idea was put into effect for abelian groups by Paul Baum, Jean-Luc Brylinksi, and Bob MacPherson [BBM85], who expressed complex equivariant K-theory in terms of a sheaf over the group. Grojnowski took a different approach and a more difficult example, giving a construction of a complex circle-equivariant cohomology theory starting from an elliptic curve E over C and taking values in sheaves of algebras over E. (At around the same time, Victor Ginzberg, Mikhail Kapranov, and Eric Vasserot [GKV95] released a sketch of the general properties one should expect of an equivariant elliptic cohomology theory, using a general compact Lie group but omitting an explicit construction.) Ioanid Rosu later [Ros03] closed the circle by providing a model for complexified torus-equivariant K-theory along these lines, taking values in sheaves over C× . Short as it is, Grojnowski’s paper sketches more than just the construction of his circle-equivariant elliptic cohomology. He also constructs the “pushforward” map in his theory, and he indicates how elliptic cohomology is related to representations of loop groups. We focus on subsequent developments which also use these features. In his thesis [Ros01], Rosu interpreted the work of Raoul Bott and Cliff Taubes [BT89] as the construction of a Thom class (or “orientation”) in Grojnowski’s elliptic cohomology whose push-forward is the equivariant Ochanine genus. This gives a proof of the rigidity of the Ochanine genus, fulfilling a proposal for such a proof by Haynes Miller [Mil89]. More recently, Matthew Ando and Maria Basterra [AB02, And03] carried out the analogous program for the Witten genus, and constructed a circle-equivariant analogue of the “string orientation” of Hopkins et al [Hop95, AHS01, Hop02]. Thus the rigidity of the Witten genus is an aspect of the equivariant string orientation in Grojnowski’s theory, just as the modularity of the Witten genus is an aspect of the string orientation of nonequivariant elliptic cohomology. 111

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The push-forward in Grojnowski’s theory can also be used to give a coherent account of a number of formulae in the literature. In their paper in this volume [AF], Ando and Chris French use it to produce the orbifold elliptic genus of [BL02]. More recently, Ando, French, and Ganter [AFG] have used it to give an account of the two-variable elliptic genus [EOTY89, Kri90, H¨oh91, BL02] and the level-N genus [Hir88]. Another very influential feature of Grojnowski’s paper is his introduction of Eduard Looijenga’s work [Loo76]. Grojnowski mentions it only briefly at the end of his paper, but he recognized that it was the key to understanding the relationship between equivariant elliptic cohomology and representations of loop groups, as explained in [And00]. Looijenga’s work expresses in a direct way the relationship between the second Chern class (and so “string structures”) and elliptic curves, and it has played an important role in the development of the subject. Matthew Ando Haynes Miller June, 2006 References [AB02]

Matthew Ando and Maria Basterra. The Witten genus and equivariant elliptic cohomology. Mathematische Zeitschrift, 240(4):787–822, 2002, arXiv:math.AT/0008192. [AF] Matthew Ando and Christopher P. French. Discrete torsion for the supersingular orbifold sigma genus, arXiv:math.AT/0308068. [AFG] Matthew Ando, Christopher P. French, and Nora Ganter. The Jacobi orientation and the two-variable elliptic genus, arXiv:math.AT/0605554. [AHS01] Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. Elliptic spectra, the Witten genus, and the theorem of the cube. Inventiones Mathematicae, 146:595–687, 2001, DOI 10.1007/s002220100175. [And00] Matthew Ando. Power operations in elliptic cohomology and representations of loop groups. Trans. Amer. Math. Soc., 352(12):5619–5666, 2000. [And03] Matthew Ando. The sigma orientation for analytic circle-equivariant elliptic cohomology. Geometry and Topology, 7:91–153, 2003, arXiv:math.AT/0201092. [BBM85] Paul Baum, Jean-Luc Brylinski, and Robert MacPherson. Cohomologie ´equivariante d´elocalis´ee. C. R. Acad. Sci. Paris S´er. I Math., 300(17):605–608, 1985. [BL02] Lev A. Borisov and Anatoly Libgober. Elliptic genera of singular varieties, orbifold elliptic genus and chiral de Rham complex. In Mirror symmetry, IV (Montreal, QC, 2000), volume 33 of AMS/IP Stud. Adv. Math., pages 325–342. Amer. Math. Soc., Providence, RI, 2002, arXiv:math.AG/0007126. [BT89] Raoul Bott and Clifford Taubes. On the rigidity theorems of Witten. J. of the Amer. Math. Soc., 2, 1989. [EOTY89] H. Eguchi, H. Ooguri, A. Taormina, and S.-K. Yang. Superconformal algebras and string compactification on manifolds with SU (N ) holonomy. Nucl. Phys. B, 315, 1989.

Ian Grojnowski’s “Delocalized equivariant elliptic cohomology” [GKV95] [H¨ oh91]

[Hir88]

[Hop95]

[Hop02]

[Kri90] [Loo76] [Mil89]

[Ros01] [Ros03] [Wit87]

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V. Ginzburg, M. Kapranov, and E. Vasserot. Elliptic algebras and equivariant elliptic cohomology. 1995. Preprint. Gerald H¨ ohn. Komplexe elliptische Geschlechter und S 1 -¨ aquivariante Kobordimustheorie (complex elliptic genera and S 1 -equivariant cobordism theory), 1991, arXiv:math.AT/0405232. Bonn Diplomarbeit. Friedrich Hirzebruch. Elliptic genera of level N for complex manifolds. In Differential geometrical methods in theoretical physics (Como, 1987), volume 250 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 37–63. Kluwer Acad. Publ., Dordrecht, 1988. Michael J. Hopkins. Topological modular forms, the Witten genus, and the theorem of the cube. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), pages 554–565, Basel, 1995. Birkh¨ auser. M. J. Hopkins. Algebraic topology and modular forms. In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291– 317, Beijing, 2002. Higher Ed. Press, arXiv:math.AT/0212397. Igor M. Krichever. Generalized elliptic genera and Baker-Akhiezer functions. Mat. Zametki, 47(2):34–45, 158, 1990. Eduard Looijenga. Root systems and elliptic curves. Inventiones Math., 38, 1976. Haynes Miller. The elliptic character and the Witten genus. In Mark Mayowald and Stewart Priddy, editors, Algebraic topology (Northwester University, 1988), volume 96 of Contemporary Math. Amer. Math. Soc., 1989. Ioanid Rosu. Equivariant elliptic cohomology and rigidity. Amer. J. Math., 123(4):647–677, 2001, arXiv:math.AT/9912089. Ioanid Rosu. Equivariant K-theory and equivariant cohomology. Math. Z., 243(3):423–448, 2003. Edward Witten. Elliptic genera and quantum field theory. Comm. Math. Phys., 109, 1987.

DELOCALISED EQUIVARIANT ELLIPTIC COHOMOLOGY I. GROJNOWSKI

Abstract. In this paper we construct an equivariant elliptic cohomology theory over C. As defined, the level k equivariant cohomology of a point with respect to a compact group G is just the span of the characters of c at level k. the loop group LG

In this paper we construct a ‘delocalised’ equivariant elliptic cohomology over C. The construction here suffers from several obvious disadvantages—it is unwieldy to work with, and clearly misses completely the point of elliptic cohomology [8]. However, it suffices for the construction of the elliptic affine algebras (see below), and does produce the ‘correct’ results. The result was inspired by [2] (which in turn is a child of [3]). We essentially define equivariant elliptic cohomology by using the Chern character E(X ×T ET ) → H(X ×T ET ), using the topology of the abelian variety ET to avoid completions. This construction produces non-trivial bundles on ET (for example ES 1 (P1 )), and, compatibly, the Gysin homomorphism we define involves twisting the bundles still further. In section 2.4 we define ES 1 (X), a sheaf over a fixed totally marked elliptic curve E. The definition works by identifying a neighbourhood of the elliptic curve around e ∈ E with its tangent space, and taking the equivariant cohomology of X e , the fixpoints of e on X. In 2.6 this is generalised to an arbitrary compact connected group. In section 2.5 we define the pushforward, or Gysin, homomorphism. This involves a choice of local coordinate on the elliptic curve, i.e. an analytic map s : U ⊆ E → C, where U is a neighbourhood of 1. This data is precisely that of a complex orientation in topology, and defines in a standard manner a homology theory with pushforward maps [1]. When this standard pushforward is applied locally on the elliptic curve, the effect of this is to twist the sheaf ES 1 (X), so that π : X → Y induces a map from a twist of ES 1 (X) to ES 1 (Y ). This behaviour is forced upon us, as elliptic cohomology satisfies a ‘locality’ property: the Mayer-Vietoris exact sequence. As ES 1 (X) is usually a nontrivial sheaf, if we can partition X into contractible pieces, a long exact sequence must necessarily twist some of the cohomology of the pieces in order Date: Feb 1994.

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to produce ES 1 (X). This is indeed what happens (consider, for example ES 1 (P1 )). It is immediate from definitions that all standard properties of a cohomology theory hold; in section 3 we mention a few that are specific to elliptic cohomology. Finally, we consider what happens when we vary the elliptic curve. It turns out that the fixpoints X e(z,τ ) make sense for a point on the universal elliptic curve (no extra marking is necessary). It follows that if the given fixed complex orientation s is defined on a curve with some marking, than so is our cohomology theory. For example, if s is the orientation class of [8], the S 1 -equivariant cohomology is a sheaf over the universal elliptic curve with a marked point of order 2. Taking the formal neighbourhood of 1 on this marked universal curve, we recover the usual elliptic cohomology E(X ×T ET ) of [8]. (See also [11] for an explanation of why this is the ‘wrong’ object.) However, instead of this one may take global sections on the universal curve of tensor products of EG (X) with certain line bundles (see section 3). These global objects are the ‘correct’ definition of a level k elliptic cohomology theory. In sharp contrast to HG (X) and KG (X) they are finite dimensional.

This note is part of a project of the author to construct certain generalisations of quantum groups and Hecke algebras which I call elliptic affine algebras. These algebras depend on a marked elliptic curve as well as a point on the curve (the analog of q in the quantum group), and theta constants occur in the structure coefficents of the algebra. One proceeds by applying the equivariant elliptic cohomology constructed here to certain well known varieties to produce elliptic Hecke algebras, elliptic quantum groups and their finite dimensional representations. The usual case is obtained by applying equivariant K-theory to these same varieties (see [6]). The construction detailed here produces certain ‘twisted’ algebras: coherent sheaves A on an abelian variety, equipped with a multiplication A ⊗ A ⊗ L → A, where L is a certain line bundle. The representations of A are easily described as in [5], even at points of finite order. The unfinished task is then to produce an honest algebra out of A, with a similar representation theory. I believe I now understand how to do this, though at the glacial pace at which I work it will take some time to get the details correct. In any case, the polite interest expressed by the people who have seen this note (which has been circulating since February, 1994) suggest that it may be worth publishing as is.

Finally, it is worth mentioning what we have really done. We have produced a theory such that the level k elliptic cohomology of a point, with respect to a

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group G, is precisely the span of the characters of the level k representations f of LG. We have done this purely finite dimensionally, by a cheap trick. However, it is morally clear how to do this in general. One must define a certain f equivariant vector bundles on the loopspace LX with the semicategory of LG infinite topology. Pushforward maps then become Euler characteristics in semi-infinite cohomology [4]. I believe this is not too difficult to do rigorously. 2. Elliptic Cohomology 2.1. Fix τ ∈ C, Im(τ ) > 0. Let E = Eτ denote the marked elliptic curve C/(Z + τ Z). There is a continuous isomorphism of groups E → S 1 × S 1 , induced from the map C → S 1 × S 1 , x + τ y 7→ (e2πix , e2πiy ). Let T be a compact torus (product of S 1 ’s); Y (T ) = Hom(S 1 , T ) its lattice of cocharacters. Then T ∼ = Y (T ) ⊗Z S 1 ; define ET = Y (T ) ⊗Z E and t = Y (T ) ⊗Z C. We identify t with both the complexified Lie algebra of T , and the tangent space to ET at 1. Write Ot for the sheaf of complex valued analytic functions on t. The map E → S 1 × S 1 gives rise to a continuous isomorphism of groups ET → T × T ; for e ∈ ET denote its image under this map as (e1 , e2 ). If V is a small neighbourhood of 0 ∈ t, there is a neighbourhood U of 1 ∈ ET and an isomorphism exp : V → U with inverse log : U → V . We will write log∗ for the corresponding map from sheaves on V to sheaves on U . Now suppose T acts on a topological space X. Define the fixpoint set X e , for e ∈ ET to be X e1 ,e2 = {a ∈ X | e1 a = e2 a = a}. For H a connected subgroup of T , put X(H) = {a ∈ X | stab(a)0 = H}. Here, stab(a)0 denotes the identity component of the subgroup of T that fixes a. Then for x ∈ t define the fixpoint set X x as X x = ∪H:x∈(LieH)C X(H). If X is compact and T acts smoothly, then for each e ∈ ET there exists an open neighbourhood U of e such that X f ⊆ X e for all f ∈ U . This is essentially a result of Mostow (see [2, 1.3]). Note that for e in a small neighbourhood of 1 ∈ ET we have X e = X log e , and more generally, for f in a small neighbourhood of e we have X f = (X e )log(f −e) . We will systematically use this fixpoint notation (though neither the Abelian variety or the Lie algebra act, we have made perfect sense of their fixpoints). For e ∈ ET let te : ET → ET , e0 7→ e0 + e be the map ‘translation by e’. 2.2. Recall there is a functor, equivariant cohomology, from the category of pairs (G, X), where G is a topological group and X a topological space on which G acts continuously to the category of Z-graded super-commutative complex algebras, (G, X) 7→ HG (X), with the following properties: i) HG (X) is a graded super-commutative algebra over HG = HG (point). If T is a compact torus, then HT = S(t∗ ) canonically, where S(t∗ ) is the algebra of polynomial functions on t, graded so the generators

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are in degree 2. If G is compact connected, T ⊆ G a maximal torus, W = NG (T )/T the Weyl group, then HG = HTW , the W -invariant polynomial functions on t. We often regard HG (X) as a sheaf over Spec(HG ). In the case G is compact connected, we can also regard HG (X) as a W -equivariant sheaf on t. ii) If G is compact and connected, HG (X) is determined by X and the Lie algebra of G. We denote it Hg (X). Further, if G is a general 0 compact group, HG (X) = Hg (X)G/G , and if G → G0 is a homotopy equivalence, where G’ is an arbitrary topological group, the induced map HG (X) → HG0 (X) is an isomorphism of graded algebras. iii) Let T be a compact torus. Then if x ∈ t and U is a sufficiently small neighbourhood of x, the inclusion i : X x ,→ X induces an isomorphism i∗ : Ht (X) ⊗Ht Γ(U, Ot ) ∼ = Ht (X x ) ⊗Ht Γ(U, Ot ). More generally, if x ∈ g is semisimple, where G is a possibly disconnected compact Lie group, then the inclusion i : (Gx , X x ) ,→ (G, X) induces a map HG (X) ⊗HG HGx → HGx (X x ) which becomes an isomorphism when both sides are localised at x ∈ SpecHGx . This is the “non-Abelian localisation” of Atiyah-Segal. iv) If tx : t → t, y 7→ y + x is the translation by x map, then it induces a functor from sheaves on t to sheaves on t, denoted (as always) by t∗x . Then t∗x Ht (X x ) ∼ = Ht (X x ). We indicate the proof. Write t = h ⊕ h0 , where h is the line of multiples of x. Then Ht (X x ) = Hh ⊗C Hh0 (X x ), and tx acts only on Hh . More generally, if x ∈ g is semisimple, then t∗x : HGx (X x ) → HGx (X x ) is an isomorphism, as x is in the center of gx . 2.3. Let O = OET denote the sheaf of complex valued analytic functions on ET ; Γ(U, O) = Γ(U ) its sections over U . Recall that to specify a sheaf A of O-modules on ET it is enough to give a Γ(U, O) module AU for each U in some open cover of ET by sufficiently small sets, and for each U, V with U ∩ V 6= ∅ a Γ(U ∩ V ) isomorphism φU V : Γ(U ∩ V ) ⊗ΓU AU → Γ(U ∩ V ) ⊗ΓV AV such that if U, V, W are such that U ∩ V ∩ W 6= ∅, then φV W φU V = φU W . Clearly it also suffices to only give this glueing data φU V when V ⊆ U if the open cover is closed under finite intersection. Similarly, if A is a sheaf of Z/2-graded super-commutative O-algebras on × ET , then elements λU V ∈ Γ(U ∩ V ) ⊗ΓU A× U (where AU denotes the commutative group of invertible elements in the ring AU ) such that λV W λU V = λU W defines an element [λ] of H 1 (ET , A× ) and a sheaf A[λ] , the “twist” of A by λ. Here, Γ(U, A[λ]) = AU and the glueing isomorphisms are φ0U V = λU V φU V . The isomorphism class of A[λ] depends only on the class of [λ] in H 1 (ET , A× ) (A notational warning: t∗e , log∗ refer to the pullback of sheaves. On the other hand, if π : X → Y is a map, we also write π ∗ to denote pullback

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in cohomology. These two uses are married in the definition below, most particularly in 2.6). 2.4. Now define ET (X), a Z/2 graded sheaf of supercommutative algebras over O. If e ∈ ET , and U is a sufficiently small neighbourhood of e, define Γ(U, ET (X)) = t∗e log∗ {Ht (X e ) ⊗Ht Γ(Ot , log(t−e U )}. Write Ht (X e )U −e for Ht (X e ) ⊗Ht Γ(Ot , log(t−e U ). If f ∈ U , and V ⊆ U is a small enough neighbourhood of F , define φU V : Γ(U, ET (X)) ⊗ΓU ΓV → Γ(V, ET (X) as the composition of the following isomorhisms Γ(U, ET (X)) ⊗ΓU ΓV ∼ = t∗ log∗ {Ht (X e )U −e ⊗Γ(log(t U,O )) Γ(log(t−e V ), Ot )} e

−e

t

i∗ ∼

→= te log∗ {Ht ((X e )log(f −e) )V −e )} ∼ Ht (X f )V −e } = t∗ log∗ {t∗ f

log(e−f )

∼ = t∗f log∗ {Ht (X f )V −f } = Γ(V, ET (X)) where i denotes the inclusion (X e )log(f −e) = X f ,→ X e . We have i∗ is an isomorphism by localisation in equivariant cohomology (2.2,iii), and the last line is an isomorphism by (2.2,iv). It is clear that φU V satisfy the cocycle condition, and so by the discussion above we have defined a sheaf over ET . Similarly, if π : X → Y is a T -map, then π ∗ : Ht (Y e ) → Ht (X e ), e ∈ ET , induces a map of O-algebras, also denoted π ∗ , π ∗ : ET (Y ) → ET (X). (This is a map of sheaves by naturality of π ∗ and by (2.2,iv) above; the diagram chase is omitted). We remark that if L is a T -equivariant local system on X, or even a complex in DT (X), the derived category of T -equivariant sheaves on X, then the same procedure serves to define elliptic cohomology with coefficents in L, ET (X, L) and π ∗ : ET (Y, π ∗ L) → ET (X, L) (see [10]), as the localisation theorem (2.2,ii) is still true in this case. 2.5. Consider the local ring at 1 of ET (X); ET (X)1 @ > exp >> Ht (X)0 = Ht (X) ⊗Ht (Ot )0 ,→ H(X ×T ET ), where BT = ET /T is the classifying space of T . In the case X is a point, we may define s(x) = s exp = exp∗ (s) ∈ H(BS 1 ) as an orientation class, and regard it as an element of Γ(U, ES 1 ) for sufficiently small U . Here s : U ⊆ E → C is a local coordinate. The usual machinary of algebraic topology means that from this data we get Gysin morphisms π!E : Ht (X)0 → Ht (Y )0 for π : X → Y a proper weakly complex oriented map [1], as well as Todd classes s(x)/x and a Riemann-Roch isomorphism relating π!E and π!H , where π!H is the usual Gysin morphism in cohomology. Now let π : X → Y be a proper weakly complex oriented map. If e ∈ ET , denote π ˜ : X e → Y e the induced map. This is still a proper weakly complex oriented map.

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The map π defines a cohomlogy class λ(π) = [π] ∈ H 1 (ET , ET (X)× ) as follows. Let e ∈ U , f ∈ V ⊆ U , e 6= f be points on ET and small neighbourhoods containing them. Let i : X f ,→ X e be the inclusion. Then f f ∗ E i∗ iE ! : Ht (X )U −e → Ht (X )U −e is well defined, and i i! 1, which we write e f f e as e(X /X ), the Euler class of X ,→ X gives an invertible element of Ht (X f )V −f . (It is invertible as for each connected component X 0 of X f , the normal bundle in X e to X 0 does not contain the trivial T -bundle, and the orientation s(x) is a local coordinate on E; i.e. an analytic function with an isolated simple zero at 0 ∈ t). Then if π ˜ : X f → Y f is the induced map, π ˜ ∗ e(Y e /Y f ) · e(X e /X f )−1 is f an invertible element of Ht (X )V −e , so applying t∗e log∗ (i∗ )−1 to it gives an element λU V ∈ Γ(U ∩ V ) ⊗ΓU Γ(U, ET (X)× ). It is clear that if W ⊆ V , W a neigbourhood of f 0 that λV W λU V = λU W and so (λU V ) defines a cohomology class, λ(π). Now we define π! : ET (X)λ(π) → ET (Y ) by defining, for e ∈ U , U sufficiently small, Γ(U, π! ) := t∗e log∗ π ˆ!E , where π ˆ : X e ,→ Y e . It follows from the definitions and the “excess intersection formula” in a generalised cohomology theory that this is well defined (again we leave the diagram chase to the reader), and a map of ET (Y )-modules. 2.6. More generally, suppose G is a compact connected Lie group, T ,→ G the maximal torus, W = NG (T )/T the Weyl group. We define EG (X), a Z/2graded sheaf on ET /W as follows. Write p : ET → ET /W for the canonical projection. Then if U ⊆ ET is a small open neighbourhood, e ∈ U is such that W e U = U , wU ∩ U = ∅ if w 6∈ W e we define we

Γ(pU, EG (X)) = (⊕w∈W/W e t∗we log∗ {HGwe (X we ⊗HtW we Γ(log(t−we U ), Ot )W })W e ∼ = t∗ log∗ {HGe (X e ) ⊗ W e Γ(log(t−e U ), Ot )W }. e

Ht

e

Observe that HGe = HtW , and that neighbourhoods of this form cover ET , as W is finite. If V ⊆ U is a neighbourhood of f (so W f ⊆ W e ) such that W f V = V and wV ∩ V = ∅ if w 6∈ W f , define φU V : Γ(pU, EG (X)) ⊗ΓU W e f (ΓV )W → Γ(pV, EG (X)) as the composition of the obvious maps and the maps induced by i : (Gf , X f = (X e )log(f −e) ) ,→ (Ge , X e ) and translation t∗log(e−f ) as in (2.4). Then φU V is an isomorphism of rings over Γ(pV, ET /W ) = f (ΓV )W by non-Abelian localisation (2.2,iii), and the cocycle condition is satisfied. Similarly, if f : (G, X) → (H, Y ) is a map of (compact connected groups, spaces) inducing a map TG → TH of the maximal tori of G to that of H, and hence a map f˜ : ETG /WG → ETH /WH , we get an induced map of sheaves f ∗ : f˜∗ EH (Y ) → EG (X). Further, if h : (H, Y ) → (K, Z) is another such map of (groups,spaces) we have h∗ f ∗ = (f h)∗ . The obvious diagram chases required to verify this are omitted.

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Finally, by repeating word for word the discussion in (2.5) we see that for a proper weakly complex oriented map of G-spaces π : X → Y we have a cohomology class λ(π) ∈ H 1 (ET /W, EG (X)× ) and maps of Z/2-graded OET /W modules π! : EG (X)λ(π) → EG (Y ). (This is not a ring homomorphism!). One may check that this is functorial in the appropriate sense; i.e. that if π 0 : Y → Z is also a proper weakly complex oriented map of G-spaces, then 0 π ∗ λ(π 0 ) · λ(π) = λ(π 0 π) and (π 0 π)! : EG (X)λ(π π) → EG (Z) is the composite of π!0 and the map induced from π! . 3. Remarks 3.1. We leave to the reader the task of making a systematic list of the properties of EG . Most follow immediately from the definitions and the corresponding properties of HG , and are obvious analogues of the usual properties of a cohomology theory. The following remarks are some indications of properties more particular to EG . 3.2. Let X, X 0 be smooth projective G-spaces, with H odd (X) = H odd (X 0 ) = 0. Then the natural morphism EG (X) ⊗EG EG (X 0 ) → EG (X × X 0 ) is an isomorphism if and only if the centralizer of every pair of commuting semisimple elements of G is connected (see [7]). (This is immediate from the definition of EG , as with these hypotheses on X, X 0 such a Kunneth theorem holds in equivariant cohomology for arbitrary connected G). Essentially the only groups G with this property are products of GLn ’s and tori. Note that a Kunneth theorem holds in equivariant K-theory if and only if the centraliser of every semisimple element is connected; i.e. if and only if G is simply connected, by a theorem of Steinberg. The usual proof of this fact (Kazhdan-Lusztig, Hodgkins) relies on another theorem of Steinberg; clearly the ‘delocalised’ technique we use gives a different proof. 3.3. Let G be a compact group, with a fixed invariant non-degenerate symmetric bilinear form on g. This data defines a line bundle L on ET with this form as its Chern class [9]. For example, if G is simple and simply connected, and L has degree the order of the center of G, then the Weyl denominator for the affine Kac-Moody algebra b g is a section of ΓLg , where g is the dual Coxeter number for G [9].. One may then consider a “level k” elliptic cohomlogy of X as Γ(EG (X) ⊗ Lk ). 3.4. Modularity. Let H = {τ ∈ C | Imτ > 0}. Recall the group SL2 Z acts on C × H by ( ac db ) · (z, τ ) = (z/(cτ + d), (aτ + b)/(cτ + d)). Hence SL2 Z acts on t × H also. Denote by e(z, τ ) the image of (z, τ ) ∈ t × H in ET τ = t ⊗C Eτ . Then if X is a T -space, the fixpoints X e(z,τ ) depend only on the orbit of (z, τ ) under SL2 Z. (This is clear for ( −1 1 ) and ( 1 11 ), which generate SL2 Z). Hence the modular properties of EG (X) depend only on the modular properties of s(z, τ ), the local coordinate around 1. We can thus regard equivariant elliptic cohomology as defined on the moduli of marked elliptic curves, with marking

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determined by the chosen s. For example, for s the orientation class of [8], EG (X) is defined over the curve S = H/Γ0 (2). (In such a case, all sections Γ should be interpreted as sections over the universal marked curve.) References [1] J. F. AdamsStable homotopy and generalised homology University of Chicago press Chicago lectures in mathematics 1974 [2] J. Block and E. Getzler Equivariant cyclic homology and equivariant differential forms preprint 1993 [3] P.Baum, J. L. Brylinski and R. MacPherson Cohomologie ´equivariante d´elocalise C.R. Acad. Sci. Paris, Serie I 3001985605–608 [4] P. Etingof and I. FrenkelCentral extensions of current groups in two dimensions Communications in Math Physics [5] I. GrojnowskiRepresentations of affine Hecke algebras (and affine quantum GL n ) at roots of unity International Math. Research Notes 41994215–217 [6] I. GrojnowskiAffinizing quantum algebras: from D-modules to K-theory Preprint, 1994 [7] M. J. Hopkins, N. J. Kuhn and D. C. RavenelGeneralised group characters and complex oriented cohomology theories preprint [8] P. S. Landweber (Ed.) Elliptic curves and modular forms in algebraic topology Springer LNM 13261988 [9] E. LooijengaRoot systems and elliptic curves Invent. Math38197617–32 ´ [10] G. LusztigCuspidal local systems and graded Hecke algebras I Inst. Hautes Etudes Sci. Publ. Math 1988 67 145–202 [11] H. MillerThe elliptic character and the Witten genusContemp. Math961989281–289 Yale University, New Haven, CT 06520

ON FINITE RESOLUTIONS OF K(n)-LOCAL SPHERES HANS-WERNER HENN Dedicated to the memory of Dieter Puppe

Abstract. Let p be an odd prime and Gn be the n-th (extended) Morava stabilizer group. We construct finite resolutions of the trivial Gn -module Zp by (direct summands of) permutation modules with respect to finite p-subgroups of Gn . Furthermore we discuss the problem of realizing these resolutions by finite resolutions of the K(n)-local sphere by spectra which are (direct summands of) wedges of homotopy fixed point spectra for the action of these finite p-subgroups on the Lubin-Tate spectrum En .

0. Introduction Let p be a prime and let K(n) be the n - th Morava K-theory at p. The category of K(n)-local spectra is a basic building block of the stable homotopy category of p-local spectra and, of course, the localization of the sphere, LK(n) S 0 , plays a central role in this category. The homotopy of LK(n) S 0 can be studied by the Adams-Novikov spectral sequence: if En denotes the periodic Landweber exact spectrum En whose coefficients in degree 0 classify deformations (in the sense of Lubin and Tate) of the Honda formal group law over Fpn then, up to a Galois extension, the E2 -page of the spectral sequence can be identified, by the Morava change of rings isomorphism, with the continuous cohomology of the Morava stabilizer group Sn with coefficients in (En )∗ . In this paper we discuss homological properties of the groups Sn , in particular resolutions of the trivial module Zp , and show how they can be used to construct finite resolutions of LK(n) S 0 in terms of spectra which are easier to understand. For example, if p − 1 does not divide n, then the mod-p cohomological dimension cdp (Sn ) of Sn is finite, equal to n2 , the trivial module Zp admits a projective resolution of length n2 and the E2 -term of the Adams-Novikov spectral sequence has a horizontal vanishing line. In homotopy theory this This paper is a sequel to the joint paper [GHMR1]. It is inspired by that paper and the author is happy to acknowledge the influence of numerous discussions with Paul Goerss, Mark Mahowald and Charles Rezk on this subject. Thanks are also due to the referee for his suggestions. 122

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allows us to construct a finite En -resolution of LK(n) S 0 in the sense of Miller [Mi] (at least up to a Galois extension in case n is still divisible by p). If p−1 divides n, then the mod-p cohomological dimension of Sn is infinite, so Zp does not admit a finite projective resolution, the E2 -term has no horizontal vanishing line and a finite En -resolution for LK(n) S 0 cannot exist. However, in analogy with discrete groups of finite virtual cohomological dimension one can hope to construct resolutions of the trivial module Zp by permutation modules on finite subgroups of Sn and then hope to realize those by finite resolutions (which will not be En -resolutions) of LK(n) S 0 via homotopy fixed point spectra of the form EnhF for suitable finite subgroups of Sn . This is in fact the main subject of this paper. There are various advantages of such resolutions. For instance, such a finite resolution gives rise to a spectral sequence with a horizontal vanishing line starting from the homotopy of the homotopy fixed point spectra and converging to π∗ (LK(n) S 0 ). This spectral sequence should be more manageable than the Adams-Novikov spectral sequence. For example, some of the delicate differentials in the latter might already be accounted for by more transparent phenomena in the homotopy of the homotopy fixed point spectra E2hF . In fact, this is essentially what happened in the calculation of the homotopy of the Toda-Smith complex V (1) at the prime p = 3 carried out in [GHM] (completing earlier work of Shimomura [Sh]). The resolutions can also be used to analyze the exotic part of Hopkins’ Picard groups (cf. [HMS]), i.e. the group of homotopy equivalence classes of invertible spectra in the category of K(n)-local spectra whose Morava module is isomorphic to that of the sphere S 0 . This will be pursued in a separate paper. On a more philosophical level, one can say that these resolutions capture to what extent the presence of finite p-subgroups in the stabilizer group influences the homotopy of π∗ (LK(n) S 0 ), very much in the same way as finite subgroups in a discrete group of finite virtual cohomological dimension influence the cohomology of the group. Here is an outline of the paper. In section 1 we recall background material on K(n)-localization, the stabilizer groups and homotopy fixed point spectra. Section 2 discusses algebraic and homotopy theoretic resolutions in the case n 6≡ 0 mod p − 1; there is a general existence result (Theorem 4) and beyond that we discuss the few cases n = 1 and p > 2 as well as n = 2 and p > 3 in which finite resolutions are known in an explicit form. The results in this section are mostly reformulations or reinterpretations of results which have been known for more than 25 years. They are included for completeness and in order to help develop our ideas on the interplay between homological properties of the groups Sn and homotopical properties of LK(n) S 0 . In section 3 we discuss the much more difficult case n ≡ 0 mod p − 1. We start by explaining how the

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well-known fibration Ψ3 −id

LK(1) S 0 → KOZ2 −→ KOZ2 (where KOZ2 is 2-adic real K-theory) can be regarded as the realization of a particular permutation resolution of the trivial S1 -module Z2 . Then we use this example as a role model which suggests possible generalizations for larger n. In section 3.3 we survey recent joint work with Goerss, Mahowald and Rezk in which the case n = 2, p = 3 was studied. In section 3.4 we comment on joint work in progress with the same coauthors which concerns the case n = 2, p = 2. In the final two sections we present new general results on the existence of finite resolutions of the trivial module Zp by permutation modules, at least for p > 2 (Proposition 17). If n = p − 1, we show how these algebraic resolutions can be realized by finite resolutions of LK(n) S 0 (Theorem 25 and Theorem 26). 1. Background 1.1 Localization with respect to Morava K-theory. 1.1.1. Let E be a spectrum and E∗ be the generalized homology theory determined by E. We recall that Bousfield localization with respect to E∗ is a functor LE from the homotopy category of spectra to itself together with natural maps λ : X → LE X which are terminal among all E∗ -equivalences. By [B] LE exists for each E. Classical examples are given by localization with respect to the Moore spectra MZ(p) , for the p-local integers Z(p) , resp. MQ, for the rationals, in which case LE is the homotopy theoretic version of arithmetic localization with respect to Z(p) resp. with respect to Q. 1.1.2. In this paper we will be concerned with the localization functors LK(n) with respect to Morava K-theory K(n). We refer to [HS] for a good general reference. Here we recall that K(0) = HQ is the rational Eilenberg-Mac Lane spectrum and is independent of p. Otherwise, for any fixed prime p we have K(n)∗ = Fp [vn±1 ] with generator vn in degree 2(pn − 1). Furthermore, K(n)∗ is a multiplicative periodic cohomology theory which admits a theory of characteristic classes, and the associated formal group law Γn is the Honda formal group law of height n. Localization with respect to K(n) plays a prominent role in stable homotopy theory because the functors LK(n) are elementary “building blocks” of the stable homotopy category of finite p-local complexes in the following sense. • The localization functor LK(n) is “simple” in the sense that the category of K(n)-local spectra contains no nontrivial localizing subcategory [HS, Theorem 7.5].

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• There is a tower of localization functors . . . → Ln → Ln−1 → . . . → L0 (with Ln = LK(0)∨...∨K(n) ) together with compatible natural maps X → Ln X such that X ' holimn Ln X for every finite p-local spectrum X [Ra2, Theorem 7.5.7]. Furthermore, for every X there is a homotopy pullback diagram (a “chromatic square”) LnX −→ LK(n) X y y Ln−1 X −→ Ln−1 LK(n) X

i.e. Ln is built from LK(n) and Ln−1 . (The diagram is easily established by using that LK(n) Ln−1 Z ' ∗ for any Z. Its existence is implicit in [Ho].) 1.2 The stabilizer groups.

1.2.1. The functors LK(n) are controlled by cohomological properties of the Morava stabilizer group Sn . We recall that Sn is the group of automorphisms of the p-typical formal group law Γn over the field Fq (with q = pn ) whose n [p]-series is given by [p](x) = xp . The group Sn can be extended to a slightly larger group Gn . In fact, because Γn is already defined over Fp the Galois group Gal(Fq /Fp ) of the finite field extension Fp ⊂ Fq acts on Aut(Γn ) = Sn and Gn can be identified with the semidirect product Sn o Gal(Fq /Fp ). In the sequel we will recall some of the basic properties of the group Sn resp. Gn . The reader is referred to [Ha] or [Ra1] for more details (see also [He] for a summary of what will be important in this paper). 1.2.2. The group Sn is equal to the group of units in the endomorphism ring of Γn , and this endomorphism ring can be identified with the maximal order On of the division algebra Dn over Qp of dimension n2 and Hasse invariant 1 . In more concrete terms, On can be described as follows: let WFq denote n the Witt vectors over Fq . Then On is the non-commutative ring extension of WFq generated by an element S which satisfies S n = p and Sw = w σ S, where w ∈ WFq and w σ is the image of w with respect to the lift of the Frobenius automorphism of Fq . The element S generates a two sided maximal ideal m in On with quotient On /m = Fq . Inverting p in On yields the division algebra Dn , and On is its maximal order. The action of the Galois group Gal(Fq /Fp ) on Sn is realized by conjugation by S inside D× n , the group of units of Dn , and the semidirect product Gn can

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therefore be described as the quotient of D× n by the central subgroup generated n × n ∼ by S , i.e. Gn = Dn /hS i. 1.2.3. Reduction mod m induces an epimorphism On× −→ F× q . Its kernel will be denoted by Sn and is also called the strict Morava stabilizer group. The group Sn is equipped with a canonical filtration by subgroups Fi Sn , i = nk , k = 1, 2, . . ., defined by Fi Sn := {g ∈ Sn |g ≡ 1 mod S in } . The intersection of all these subgroups contains only the element 1 and Sn is complete with respect to this filtration, i.e. we have Sn = limi Sn /Fi Sn . Furthermore, we have canonical isomorphisms Fi Sn /F 1 Sn ∼ = Fq i+ n

induced by x = 1 + aS in 7→ a ¯. Here a is an element in On , i.e. x ∈ Fi Sn and a ¯ is the residue class of a in On /m = Fq The associated graded object grSn with gri Sn = Fi Sn /Fi+ 1 Sn , i = n1 , n2 , . . . n becomes a graded Lie algebra with Lie bracket [¯ a, ¯b] induced by the commutator xyx−1 y −1 in Sn . Furthermore, if we define a function ϕ from the positive real numbers to itself by ϕ(i) := min{i + 1, pi} then the p - th power map on Sn induces maps P : gri Sn −→ grϕ(i) Sn which define on grSn the structure of a mixed Lie algebra in the sense of Lazard [La, Chap. II.1]. If we identify the filtration quotients with Fq as above then the Lie bracket and the map P are explicitly given as follows (cf. Lemma 3.1.4 in [He]). Lemma 1. Let a ¯ ∈ gri Sn , ¯b ∈ grj Sn . Then a) ni nj [¯ a, ¯b] = a ¯¯bp − ¯b¯ ap ∈ gri+j Sn .

b)

ni (p−1)ni ¯1+p +...+p a ni (p−1)ni Pa ¯= a ¯ + a¯1+p +...+p a ¯

if i < (p − 1)−1 if i = (p − 1)−1 if i > (p − 1)−1 .

1.2.4. Next we record some basic facts about finite p-subgroups of Sn . First of all, all finite abelian p-subgroups of Sn are cyclic. Sn is known to contain a cyclic subgroup of order pk if and only if pk−1 (p − 1) divides n, and then such a cyclic subgroup Cpk ⊂ Sn is unique up to conjugacy. Furthermore, if p > 2, or p = 2 and n is odd, then all finite p-subgroups are cyclic.

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The structure of centralizers of cyclic subgroups will be of importance for us. To get at it we note that the centralizer CDn (Cpk ) of Cpk in Dn is again a division algebra. It is central over the cyclotomic extension of Qp generated by Cpk and its dimension over its center is m2 , if n = mpk−1 (p − 1). Then the centralizer CSn (Cpk ) of Cpk in Sn can be identified with the group of units in the maximal order of CDn (Cpk ). 1.2.5. Recall that the cohomological p-dimension cdp (G) of a profinite group G is defined as cdp (G) = sup{n ∈ N|H n (G, M) 6= 0 for some finite continuous G−module M} where, here and elsewhere in this paper, H ∗ (G, M) is always continuous cohomology. Later on M may be a continuous profinite module over the completed group algebra Zp [[G]] := limU Zp [G/U] (with U running through all open normal subgroups of G). We refer to [SyW] for a discussion of the relevant homologial algebra. The cohomological p-dimension of the group Sn resp. Gn is n2 unless Sn resp. Gn contain non-trivial finite p-subgroups in which case it is infinite. By 1.2.4 this happens for Sn iff p − 1 divides n, and in the case of Gn this happens iff p or p − 1 divides n. However, even in these cases Sn and Gn are still virtually of finite cohomological p-dimension (i.e. they contain a finite index subgroup of finite cohomological p-dimension) and its virtual cohomological p-dimension vcdp (Sn ) remains n2 . The reader is referred to [La] or [SyW] for more details on these notions. 1.3 Homotopy fixed point spectra. 1.3.1. By Hopkins-Miller (cf. [Re]) the group Gn acts on the Lubin-Tate spectrum En ; we recall that En is the Landweber exact spectrum given by the 2-periodic theory with coefficients π∗ (En ) = π0 (En )[u±1 ] (with u ∈ π−2 (E)) whose associated formal group law over π0 (En ) is a universal deformation of Γn in the sense of Lubin and Tate [LT]. In particular there is a (non-canonical) isomorphism between π0 (En ) and WFq [[u1 , . . . , un−1 ]], the ring of formal power series over WFq in the variables u1 , . . . , un−1 . We can and will choose the fn to be p-typical with p-series universal deformation Γ [p]Γe n (x) = px +Γe n u1 xp +Γe n . . . +Γe n un−1 xp

n−1

n

+Γe n xp ,

in other words the classifying map BP∗ → π∗ (En ) sends the Araki generator i n vi to ui u1−p , if i < n, vn to u1−p , and vi to 0 if i > n. Let OGn be the orbit category of Gn , i.e. the objects of OGn are orbits Gn /K where K is a closed subgroup of Gn and morphisms are continuous Gn equivariant maps. By Devinatz-Hopkins [DH2] there is a contravariant functor

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from OGn to K(n)-local spectra which assigns to Gn /K the homotopy fixed point spectrum EnhK and this spectrum comes with an associated homotopy fixed point spectral sequence E2s,t = H s (K; πt (En )) =⇒ πt−s (EnhK ) . Furthermore, EnhGn can be identified with LK(n) S 0 and the Adams-Novikov spectral sequence for LK(n) S 0 can be identified with the associated homotopy fixed point spectral sequence E s,t ∼ = H s (Gn , (En )t ) =⇒ πt−s LK(n) S 0 . 2

Finally, EnhGn can be identified with the iterated homotopy fixed trum (EnhSn )hGal(Fq /Fp ) , the Galois group acts on the homotopy

point specfixed point

spectral sequence E2s,t ∼ = H s (Sn , (En )t ) =⇒ πt−s (EnhSn ) , and the action on the whole spectral sequence is coinduced. Thus we get isomorphisms π∗ LK(n) S 0 ∼ = π∗ (EnhSn )Gal(Fq /Fp ) , π∗ EnhSn ∼ = π∗ LK(n) S 0 ⊗Zp WFpn , and we may therefore say that EnhSn is equal to LK(n) S 0 , up to a Galois extension. 1.3.2. Hopkins and Devinatz also showed that for any closed subgroup K ⊂ Gn there is an isomorphism π∗ (LK(n) (En ∧ E hK )) ∼ = maps (Gn /K, (En )∗ ) n

cts

(where mapcts denotes continuous maps). The isomorphism is functorial on OGn . It is compatible with the obvious (En )∗ -module structures on both sides as well as with the actions of Gn on both sides, which is via the action on En on the left hand side and via the diagonal action on the space of continuous maps on the right hand side. In other words, the isomorphism is one of Morava modules where a Morava module M is a complete (En )∗ -module with a continuous action of Gn such that g(ax) = g(a)g(x) for g ∈ Gn , a ∈ (En )∗ , x ∈ M . By abuse of notation we will also say that M is a (twisted) (En )∗ [[Gn ]]-module. Typical examples of such modules are given by π∗ (LK(n) (En ∧ X)), at least under suitable conditions on X, e.g. if K(n)∗ X is evenly graded (see [HS], or [GHMR1] for a summary of what is important for us). In order to keep our notation compact we will write in the sequel (En )∗ X instead of π∗ (LK(n) (En ∧ X)). 1.3.3. We will need information about maps between various homotopy fixed point spectra. In the following F stands for function spectrum. We recall the following results from [GHMR1].

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1.3.3.1. Let U be an open subgroup of Gn . Functoriality of the homotopy fixed point spectra construction of [DH2] gives us a map EnhU ∧ Gn /U+ → En where as usual Gn /U+ denotes Gn /U with a disjoint base point added. Together with the product on En we obtain a map En ∧ EnhU ∧ Gn /U+ → En ∧ En → En whose adjoint induces an equivalence of En -module spectra Y LK(n) (En ∧ EnhU ) → En Gn /U

realizing the isomorphism of 1.3.2 above. Now let FEn be the function spectrum in the category of En -module spectra (see [EKMM] for details). If we apply FEn (−, En ) to this last equivalence we obtain another equivalence of En -module spectra Y FE n ( En , En ) → FEn (En ∧ EnhU , En ) Gn /U

which can be rewritten as an equivalence (still of En -module spectra) En ∧ Gn /U+ ' F (EnhU , En ) . The same reasoning shows that we can replace En in the target of the function spectrum by LK(n) (En ∧ I) where I is any spectrum and we obtain an equivalence LK(n) (En ∧ I) ∧ Gn /U+ ' F (EnhU , LK(n) (En ∧ I)) . 1.3.3.2. More generally, let K be any closed subgroup of GnT. Then there exists a decreasing sequence Ui of open subgroups Ui with K = i Ui and by [DH2] we have EnhK ' LK(n) hocolimi EnhUi . By passing to the limit we obtain an equivalence LK(n) (En ∧ I)[[Gn /K]] ' F (EnhK , LK(n) (En ∧ I)) where we have used the convention that if E is a spectrum and X = lim i Xi is an inverse limit of a sequence of finite sets with each Xi finite then Q E[[X]] is given as holimi E ∧(Xi )+ , i.e. as the fibre of the usual self map of i E ∧(Xi )+ . Note that if X is such a profinite set with continuous K-action and if E is a K-spectrum then E[[X]] is a K-spectrum via the diagonal action. If we concentrate (for simplicity) on the case I = S 0 and take homotopy fixed points in this equivalence with respect to another finite subgroup of Gn then we get the following result.

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Proposition 2 ([GHMR1, Prop. 2.6]). a) Let K1 be a closed subgroup and K2 a finite subgroup of Gn . Then there is a natural equivalence (where the homotopy fixed points on the left hand side are formed with respect to the diagonal action of K2 ) En [[Gn /K1 ]]hK2 ' F (EnhK1 , EnhK2 ) . b) If K1 is also an open subgroup then there is a natural decomposition Y En [[Gn /K1 ]]hK2 ' EnhKx K2 \Gn /K1

where Kx = K2 ∩ xK1 x−1 is the isotropy subgroup of the coset xK1 and K2 \Gn /K1 is the finite (!) set of double cosets. T c) If K1 is a closed subgroup and K1 = i Ui for a decreasing sequence of open subgroups Ui then Y F (EnhK1 , EnhK2 ) ' holimi En [[Gn /Ui ]]hK2 ' holimi EnhKx,i K2 \Gn /Ui

where Kx,i = K2 ∩ xUi x−1 is, as before, the isotropy subgroup of the coset xUi . Remark If Ui ⊂ Uj then the map Y EnhKx,i → K2 \Gn /Ui

Y

EnhKx,j

K2 \Gn /Uj

in the inverse system of part (c) of the proposition can be described as follows: if x ∈ Gn /Ui has isotropy group Kx,i and its image x0 ∈ Gn /Uj has isotropy group Kx0 ,j then the restriction of the map to the factor determined by x sends hK 0 hK En x,i via the transfer to the factor En x ,j determined by x0 . In particular, this implies that on homotopy groups the inverse system is Mittag-Leffler. In the next result Hom(En )∗ [[Gn ]] (−, −) denotes homomorphisms of Morava modules. Proposition 3 ([GHMR1, Prop. 2.7]). Let K1 and K2 be closed subgroups of Gn and suppose that K2 is finite. Then there is an isomorphism K ∼ = (En )∗ [[Gn /K1 ]] 2 −→ Hom(En )∗ [[Gn ]] ((En )∗ EnhK1 , (En )∗ EnhK2 )

such that the following diagram commutes

K2 hK2 π∗ En [[G −→ (En )∗ [[G n /K1 ]] n /K1 ]] ∼ ∼ y = y = hK1 hK2 π∗ F (En , En ) −→ Hom(En )∗ [[Gn ]] ((En )∗ EnhK1 , (En )∗ EnhK2 )

On finite resolutions of K(n)-local spheres

131

where the top horizontal map is the edge homomorphism in the homotopy fixed point spectral sequence, the left-hand vertical map is the isomorphism given by Proposition 2 and the bottom horizontal map is the En -Hurewicz homomorphism. 2. The case n 6≡ 0 mod p − 1 In this section we begin our discussion of LK(n) S 0 . The case n = 0 is both exceptional and trivial: K(0) = MQ and LK(0) S 0 is the rationalized sphere spectrum. From now on we will therefore assume n > 0. 2.1 Explicit examples I: the case n = 1 and p > 2. 2.1.1. We briefly review the case n = 1 which is well understood. In this case we have E1 = KZp (p-adic complex K-theory). The group G1 = S1 can be identified with Z× p , the group of units in the p-adic integers. If p is odd then ∼ C × Z , Z× = p−1 p where Cp−1 denotes the cyclic group of order p − 1 given by p the roots of unity in Zp . The homotopy fixed points E1hG1 can be formed in two steps, first with respect to Cp−1 and then with respect to Zp . Thus we obtain the following fibration (cf. [HMS]) in which ψ p+1 is the appropriate hC Adams operation and KZp p−1 can be identified with the Adams summand of p-adic complex K-theory ψ p+1 −id

p−1 p−1 . −→ KZhC LK(1) S 0 → KZhC p p

2.1.2. This fibration can also be considered as a suitable realization of a projective resolution of the trivial Zp [[G1 ]]-module Zp , and it is this point of view which turns out to be useful for finding generalizations of the above fibration for larger n. To get at this projective resolution we start with the following obvious short exact sequence of modules over the power series ring Zp [[t]] ×t

0 → Zp [[t]] −→ Zp [[t]] → Zp → 0 . Now recall that there is an isomorphism of complete algebras Zp [[t]] ∼ = Zp [[Zp ]] induced by sending t to g − e ∈ Zp [[Zp ]] if g is a topological generator of Zp , ∼ e.g. if g is the image of p + 1 ∈ Z× p = G1 in the quotient group G1 /Cp−1 = Zp . Therefore we can consider this sequence as an exact sequence of Zp [[Zp ]]modules, or even as an exact sequence of Zp [[G1 ]]-modules, and as such we can write it as g−e G1 1 0 → Zp ↑G Cp−1 −→ Zp ↑Cp−1 → Zp → 0

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1 where M ↑G Cp−1 denotes the induced module of a Zp [Cp−1 ]-module M, i.e. the b Zp [Cp−1 ] M. Because the trivial Zp [Cp−1 ] completed tensor product Zp [[G1 ]]⊗ 1 module Zp is projective we find that Zp ↑G Cp−1 is a projective Zp [[G1 ]]-module and the exact sequence is a projective resolution of the trivial Zp [[G1 ]]-module Zp which is even split as a sequence of continuous Zp -modules. So if we apply the functor Homcts (−, (KZp )∗ ) of continuous homomorphisms into (KZp )∗ to this sequence then we obtain another short exact sequence which, by 1.3.2, can be identified with the (`a priori long) exact sequence which is associated to our fibration: 0 → (KZp )∗ (LK(1) S 0 ) ∼ = (KZp )∗ → (KZp )∗ KZhCp−1 → (KZp )∗ KZhCp−1 → 0 .

p

p

2.2 The general case. 2.2.1. Following Miller [Mi] we say that a K(n)-local spectrum I is En -injective if the map I = S 0 ∧ I → LK(n) (En ∧ I) induced by the unit in En is split. Furthermore, a sequence X 0 → X → X 00 of K(n)-local spectra is En -exact if the composition X 0 → X 00 is trivial and if [X 0 , I] ←− [X, I] ←− [X 00 , I] is an exact sequence of abelian groups for each En -injective spectrum I. Finally an En -resolution of a K(n)-local spectrum X is an En -exact sequence of K(n)local spectra ∗ → X → I0 → I1 → . . . (i.e. every three term subsequence is En -exact) such that each I s , s ≥ 0, is En -injective. 2.2.2. The following result is a folk theorem whose roots can be traced back to the work of Morava [Mo]. Theorem 4. If n is neither divisible by p − 1 nor by p then LK(n) S 0 admits an En -resolution of length n2 . In fact, each of the En -injectives in the resolution can be chosen to be a direct summand of a finite wedge of En ’s. Proof. The idea of the proof is to start with information in homological algebra and use this to construct an En -resolution. If neither p − 1 nor p divides n then cdp (Gn ) = n2 . Therefore the trivial Zp [[Gn ]]-module Zp admits a projective resolution P• : 0 → P n 2 → . . . → P 0 → Z p → 0 of length n2 , and by [La] we may assume that each projective is finitely generated as Zp [[Gn ]]-module.

On finite resolutions of K(n)-local spheres

133

We want to construct an En -resolution X• of LK(n) S 0 X• : ∗ → X−1 → X0 → . . . → Xn2 → ∗ with X−1 = LK(n) S 0 such that the complex Homcts (P• , (En )∗ ) is isomorphic, as a complex of Morava modules, to the complex (En )∗ X• . For this we note that if F r is a free Zp [[Gn ]]-module of rank r then we have an isomorphism of Morava modules r _ r ∼ (1) Homcts (F , (En )∗ ) = (En )∗ ( En ) j=1

and the En -Hurewicz homomorphism (2)

[En , En ] → Hom(En )∗ [[Gn ]] ((En )∗ En , (En )∗ En )

is an isomorphism. In fact, (1) resp. (2) follow immediately from 1.3.2 resp. from Proposition 3. Property (2) allows us now to construct both the spectra Xs , s = 0, 1, . . . , n2 , (by lifting idempotents on finitely generated free Zp [[Gn ]]-modules to homotopy idempotents on corresponding wedges of En ’s) as well as the required maps between these spectra. Because En is K(n)-local and En is a ring spectrum, the spectra Xs will all be En -injective. It remains to show that the sequence is En -exact. For this it is enough to show that for any spectrum Z of the form Z := LK(n) (En ∧ I) with some spectrum I the complex [X• , Z] is exact. By our discussion in 1.3.3 we know that [En , Z] ∼ = limi (π0 (Z) ⊗ Zp [[Gn /Ui ]]) . = π0 (Z[[Gn ]]) ∼ Now assume first that π0 (Z) is p-complete, i.e. π0 (Z) ∼ = limj π0 (Z)/pj . Then we even have [En , Z] ∼ = π0 (Z[[Gn ]]) ∼ = limi,j (π0 (Z) ⊗ Z/pj [[Gn /Ui ]]) . e denote the This can be restated as follows. For a fixed abelian group A let A⊗− functor from profinite Zp -modules and continuous homomorphisms to abelian groups which sends the profinite Zp -module M ∼ = limα Mα , where each Mα is finite, to limα (A ⊗ Mα ). Therefore, as long as π0 (Z) is p-complete, we can write e p [[Gn ]] . [En , Z] ∼ = π0 (Z)⊗Z Lr More generally, if P is a direct summand in j=1 Zp [[Gn ]] and if X is the Wr corresponding direct summand in j=1 En then e [X, Z] ∼ = π0 (Z)⊗P

and we even obtain an isomorphism of complexes e •. [X• , Z] ∼ = π0 Z ⊗P

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Hans-Werner Henn

Because P• is split as a complex of continuous Zp -modules, [X• , Z] is exact provided π0 (Z) is p-complete. In addition, under this hypothesis on Z this complex is even naturally split in Z. In the general case we use that Z = LK(n) (En ∧I) can be written as homotopy inverse limit of a sequence Zn of spectra with p-complete homotopy groups, even bounded p-torsion homotopy groups (cf. [HS, Proposition 7.10]). Because [X• , Zn ] is naturally split in n, we see that limin [X• , Zn ] is split for i = 0, 1, in particular exact, and therefore [X• , Z] is exact, i.e. the sequence X• is En -exact. Remark 1 This result is a pure existence result. It says nothing about an explicit form of such a resolution. Remark 2 If n is divisible by p (but not by p−1) we can offer the two following substitutes of Theorem 4. Either we can use the existence of a finite projective resolution of the trivial n Zp [[Sn ]]-module Zp to get one for the induced Zp [[Gn ]]-module Zp ↑G Sn . This resolution can then be realized as in the proof of Theorem 4 to give an En resolution for EnhSn . which by 1.3.1 is, up to a Galois extension, equal to LK(n) S 0 . Or, if we insist on a resolution of LK(n)SS 0 , we can consider the formal group Γn over the algebraic extension K := r≥0 Fqpr of Fq with Galois group Gal(K, Fq ) ∼ = Zp . This will have the effect of replacing Gn = Sn o Gal(Fq /Fp ) by the group Gn (K) := Sn o Gal(K/Fp ). The advantage of doing this is that while Gn has elements of order p and therefore infinite mod-p cohomological dimension, the group Gn (K) has no elements of order p and its mod-p cohomological dimension is finite, equal to n2 + 1. As a consequence one gets a projective resolution of the trivial Gn (K)-module Zp of length n2 +1. If we also replace En by the corresponding Lubin-Tate spectrum En (K) whose homotopy groups in degree 0 classify deformations of Γn over K then our proof carries over verbatim: we only need to remark that 1.3.2 and the two properties (1) and (2) in the proof of Theorem 4 hold with En and Gn replaced by En (K) and Gn (K). 1 2.3 Explicit examples II: the case n = 2 and p > 3. As before we let q = pn . We start with an observation valid for all n > 1 (cf. section 1.3 of [GHMR1]). The reduced norm Sn → Z× p admits a canonical × extension Gn → Zp × Gal(Fq /Fp ) and by composing with the evident projection we obtain a homomorphism Gn → Z× p . Furthermore, if we identify the 1I

would like to thank Ethan Devinatz for a reassuring discussion of this point.

135

On finite resolutions of K(n)-local spheres

quotient of Z× p by its subgroup of elements of finite order by Zp we obtain a homomorphism Gn → Zp . The kernel of this homomorphism will be denoted G1n and the kernel of its restriction to Sn will be denoted by Sn1 . G1n contains a cyclic group Cq−1 of order q − 1 (the roots of unity in WFq ⊂ On ) and this subgroup is invariant with respect of the action of Gal(Fq /Fp ). Therefore G1n contains the semidirect product Cq−1 o Gal(Fq /Fp ). We will denote this finite subgroup by Fn(q−1) in the sequel. hFn(q−1)

For more information on π∗ (En

) we refer to the appendix.

Theorem 5. Assume n = 2 and p > 3. a) There exists a fibration hG12

LK(2) S 0 → E2

hG12

−→ E2

and an En -resolution hG12

∗ → E2

hF2(q−1)

where X ' Σ2(p−1) E2

hF2(q−1)

→ E2

∨ Σ2(p

hF2(q−1)

→ X → X → E2

2 −p)

hF2(q−1)

E2

→∗

.

b) There exists an En -resolution of the form ∗ → LK(2) S 0 ' E2hG2 →E2

hF2(q−1)

hF2(q−1)

→ E2

∨X →

hF2(q−1)

→ X ∨ X → X ∨ E2

hF2(q−1)

→ E2

→∗.

Given Theorem 4 this is a fairly straightforward consequence of the following purely algebraic result in which λp−1 denotes the Zp [F2(q−1) ]-module whose underlying Zp -module is WFq , on which Cq−1 ⊂ W× Fq acts via Cq−1 × WFq → WFq , (g, w) 7→ g p−1 w and on which the group Gal(Fq /Fp ) acts via the lift of Frobenius (cf. appendix). The algebraic result is in turn a consequence of the calculation of the cohomology of the relevant Morava stabilizer algebra in [Ra1, Theorem 6.3.22]. Proposition 7 below is the group theoretic version of Ravenel’s result. Theorem 6. Assume n = 2 and p > 3. a) There exists a short exact sequence of Zp [[G2 ]]-modules 2 2 0 → Z p ↑G → Z p ↑G → Zp → 0 G1 G1 2

2

and a projective resolution of the trivial Zp [[G12 ]]-module Zp G1

G1

G1

G1

2 2 2 2 0 → Zp ↑F2(q−1) → λp−1 ↑F2(q−1) → λp−1 ↑F2(q−1) → Zp ↑F2(q−1) → Zp → 0 .

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Hans-Werner Henn

b) There exists a projective resolution of the trivial Zp [[G2 ]]-module of the form G2 G2 2 0 → Z p ↑G F2(q−1) →(λp−1 ⊕ Zp ) ↑F2(q−1) → (λp−1 ⊕ λp−1 ) ↑F2(q−1) → G2 2 →(Zp ⊕ λp−1 ) ↑G F2(q−1) → Zp ↑F2(q−1) → Zp → 0 .

Proposition 7. Assume n = 2 and p > 3. a) H ∗ (S21 ; Fp ) is a 3-dimensional Poincar´e duality algebra and H 3 (S 1 ; Fp ) ∼ = Fp 2

is trivial as module over F2(q−1) . b) There is a canonical isomorphism of Zp [F2(q−1) ]-modules H 1 (S 1 ; Fp ) ∼ = λp−1 ⊗Z Fp . 2

p

c) The Bockstein homomorphism induces an isomorphism of Zp [F2(q−1) ]-modules H 1 (S21 ; Fp ) ∼ = H 2 (S21 ; Fp ) . Proof of the Proposition. We start by proving (b). S21 is a torsionfree p-adic Lie group of dimension 3; so by [La, V.2.5.8] H ∗ (S21 ; Fp ) is a 3-dimensional Poincar´e duality algebra which is therefore additively determined by H 1 (S21 ; Fp ) resp. its dual H1 (S21 ; Fp ). The filtration of S2 described in section 1.2.3 induces one of S21 . From the splitting S2 ∼ = S21 × Zp we deduce that gri S21 = gri S2 if i = k2 with k odd and that gri S21 is given by the kernel of the trace map Fq → Fp if i = k2 with k even. In particular, if i = k2 with k odd, there is an element b ∈ gr1 S21 with (b

p2i

− b) 6= 0, and then Lemma 1a shows that the commutator map gri S21 × p2i

gr1 S21 → gri+1 S21 given by [a, b] = a(b −b) is onto. Furthermore, if i = k2 with k odd and j = 2l with l odd then the commutator map gri S21 ×grj S21 → gri+j S21 p given by [a, b] = ab − bap is again onto. This implies that there is a canonical isomorphism H1 (S 1 ; Zp ) ∼ = F1/2 S 1 /F1 S 1 ∼ = Fq ∼ = H1 (S 1 ; Fp ) . 2

2

2

2

Furthermore, the conjugation action of ω ∈ Cq−1 is induced by (ω, 1 + aS) 7→ ω(1 + aS)ω −1 = 1 + ω 1−p aS while the action of Frobenius is induced by (S, 1 + aS) 7→ S(1 + aS)S −1 = 1 + aσ S and this implies that our isomorphism is an isomorphism of Zp [F2(q−1) ]-modules: H1 (S 1 ; Zp ) ∼ = H1 (S 1 ; Fp ) ∼ = λ1−p ⊗Z Fp . 2

2

p

By dualizing we get an isomorphism of Zp [F2(q−1) ]-modules H 1 (S 1 ; Fp ) ∼ = (λ1−p )∗ ⊗Z Fp . 2

p

137

On finite resolutions of K(n)-local spheres

Now λp−1 (and λ1−p ) are self-dual (the isomorphism is induced by the pairing (x, y) 7→ xy −1 + (xy −1 )σ ) and they are both isomorphic (the isomorphism is given by x 7→ xσ ) and therefore we obtain (b). Because of H1 (S21 ; Zp ) ∼ = H1 (S21 ; Fp ) ∼ = Fq we see that the Bockstein homo1 1 morphism β : H2 (S2 ; Fp ) → H1 (S2 ; Fp ) is onto, and hence β : H 1 (S21 ; Fp ) → H 2 (S21 ; Fp ) is mono. On the other hand Poincar´e duality gives an additive isomorphism H 1 (S21 ; Fp ) ∼ = H 2 (S21 ; Fp ) and hence the Bockstein gives an isomorphism H 1 (S21 , Fp ) ∼ = H 2 (S21 ; Fp ) which is clearly Zp [F2(q−1) ]-linear and thus (c) is proved. To prove (a) we consider the subgroup F1 S21 of S21 and note that this subgroup is invariant by the conjugation action of F2(q−1) . Using Lemma 1a as above shows that the closure of the commutator subgroup of F1 S21 is F 5 S21 2 and then Lemma 1b gives that the closure of the subgroup generated by pth powers and commutators is F2 S21 . It follows that H1 (F1 S21 ; Zp ) is torsion (∼ = Z/p2 ⊕ Z/p⊕ Z/p) and that we have an isomorphism of Zp [F2(q−1) ]-modules H1 (F1 S 1 ; Fp ) ∼ = gr1 S 1 ⊕ gr 3 S 1 . 2

2

2

2

Identifying the Zp [F2(q−1) ]-module structure on gr1 S21 ⊕ gr 3 S21 as before shows 2 that we have an isomorphism of Zp [F2(q−1) ]-modules Tr H1 (F1 S21 ; Fp ) ∼ = (Ker : Fq −→ Fp ) ⊕ λp−1

where Cq−1 acts trivially on the first summand and Frobenius acts by multiplication by −1. Again this module is self-dual so that we have an isomorphism of Zp [F2(q−1) ]-modules Tr H 1 (F1 S21 ; Fp ) ∼ = (Ker : Fq −→ Fp ) ⊕ λp−1 .

Now we use that F1 S21 is ´equi-p-valu´e in the sense of Lazard, hence its cohomology is the exterior algebra on H 1 (F1 S21 ; Fp ) [La, Proposition V.2.5.7.1)]. So if α is any endomorphism of H 1 (F1 S21 ; Fp ) then the induced homomorphism on H 3 (F1 S21 ; Fp ) ∼ = Fp is given by multiplication with the determinant of α. In particular, for the action of ω ∈ Cq−1 the determinant is 1; it is obviously Tr 1 on Ker : Fq −→ Fp and it is 1 on λp−1 because ω acts via multiplication by ω p−1 and hence its determinant is a (p − 1)-st power in F× p . For the action of Tr

Frobenius the determinant is again 1 because it is −1 on both Ker : Fq −→ Fp and on λp−1 . Therefore H 3 (F1 S21 ; Fp ) is trivial as Zp [F2(q−1) ]-module. Because H1 (F1 S21 ; Zp ) is a torsion group we deduce from the mod-p calculation and the universal coefficient theorem an isomorphism H 3 (F1 S21 ; Zp ) ∼ = Zp . 3 1 ∼ Likewise we find H (S2 ; Zp ) = Zp . Now a restriction-transfer argument shows that the Zp [F2(q−1) ]-module structure on H 3 (S21 ; Zp ) ∼ = Zp is trivial if and only 3 1 if it is trivial on H (F1 S2 ; Zp ). We have already seen that the latter is trivial after mod-p reduction and this implies that it was trivial before.

138 Proof of Theorem 6. modules (3)

Hans-Werner Henn a) The existence of the exact sequence of Zp [[G2 ]]2 2 0 → Z p ↑G → Z p ↑G → Zp → 0 G1 G1 2

2

∼ is an immediate consequence of the isomorphism G2 /G12 ∼ = Z× p /Cp−1 = Zp . (Note that this sequence is essentially the same exact sequence as that in section 2.1.2.) The projective resolution of Zp as Zp [[G12 ]]-module is now constructed by G12 using Proposition 7 as follows. The map Zp ↑F2(q−1) → Zp is just the G12 -linear extension of the identity of Zp (considered as an F2(q−1) -linear map). If N0 is its kernel then we can compute H0 (S21 ; N0 /(p)) from the long exact homology sequence associated to the short exact sequence G1

2 0 → N0 → Zp ↑F2(q−1) → Zp → 0

of Zp [F2(q−1) ]-modules and identify it with H1 (S21 ; Fp ) ∼ = λp−1 ⊗Zp Fp . Because λp−1 is projective as Zp [F2(q−1) ]-module (the order of F2(q−1) is prime to p!) we can lift the resulting map from λp−1 → H0 (S21 ; N0 /(p)) to an Zp [F2(q−1) ]-linear map λp−1 → N0 . G1

2 → N0 . By Let N1 be the kernel of the Zp [[G12 ]]-linear extension λp−1 ↑F2(q−1) a Nakayama Lemma type argument with H0 we see that this extension is onto (cf. Lemma 4.3 of [GHMR1]) and then we find an isomorphism of Zp [F2(q−1) ]modules H0 (S21 ; N1 /(p)) ∼ = H2 (S21 ; Fp ).

By iterating the procedure we construct a Zp [[G12 ]]-linear surjection G1

2 → N1 λp−1 ↑F2(q−1)

whose kernel N2 satisfies H0 (S21 ; N2 /(p)) ∼ = Fp as Zp [F2(q−1) ]= H3 (S21 ; Fp ) ∼ 1 module and Hi (S2 ; N2 /(p)) = 0 if i > 0. Finally by using the Nakayama Lemma once more we see that the G12 -linear G12 extension Zp ↑F2(q−1) → N2 of the F2(q−1) -linear projection Zp → H0 (S21 ; N2 /(p)) is an isomorphism. By splicing together the short exact sequences we obtain the projective resolution of Theorem 6a. b) We take the projective resolution obtained in (a) and induce it up to get 2 one of the Zp [[G12 ]]-module Zp ↑G . Then we use the exact sequence (3) and G12 construct the obvious double complex whose columns are these induced projective resolutions. The resulting double complex gives the projective resolution of (b). Proof of Theorem 5. a) For the fibration we can refer to Proposition 7.1 in [DH2]. In fact, the fibration “realizes” (in the same sense as before) the short

On finite resolutions of K(n)-local spheres

139 hG1

exact sequence of Zp [[G2 ]]-modules in Theorem 6a. The En -resolution of En 2 is now obtained as in the proof of Theorem 4 as the realization of the projective 2 resolution of Zp ↑G which is induced from the one given in Theorem 6a. To G12 finish the proof of (a) it remains to identify the spectrum which corresponds G12 to the module λp−1 ↑F2(q−1) . For this we refer to the appendix. b) The En -resolution of part (b) is nothing but the realization of the resolution of Theorem 6b obtained via (the proof of) Theorem 4. Remarks a) Ravenel [Ra1, Theorem 6.3.31] resp. Yamaguchi [Y] have also studied H ∗ (S3 , Fp ) for p ≥ 5 resp. p ≥ 3. In principle this can be used to obtain an explicit resolution for LK(3) S 0 if p ≥ 3. b) No other explicit resolutions seem to be known if n 6≡ 0 mod p − 1. 3. The case n ≡ 0 mod p − 1 3.1 Explicit examples III: the case n = 1 and p = 2. This case is again ∼ well understood. The isomorphism G1 = S1 ∼ = Z× 2 = C2 × Z2 allows, as before, to form the homotopy fixed points in two stages and we obtain the following fibration (cf. [HMS]) in which ψ 3 is again given by the appropriate Adams operation: (4)

ψ 3 −id

2 2 LK(1) S 0 → KZhC −→ KZhC . 2 2

2 The homotopy fixed points KZhC can be identified with 2-adic real K-theory 2 KOZ2 . Note, however, that this is not an example of Theorem 4. In fact, an En -resolution of finite length cannot exist in this case because the cohomological dimension cd2 (S1 ) is infinite. Nevertheless this is a very good substitute of such a resolution.

3.2 The general problem. The natural question arises whether there are generalizations of the fibre sequence (4) for higher n. What should they look like? In other words, can we explain the appearance of KOZ2 in (4) so that it fits into a more general framework? A good point of view is again provided by homological algebra as follows: the fibre sequence (4) is a homotopy theoretic analogue of the exact sequence of Z2 [[G1 ]]-modules G1 1 0 → Z 2 ↑G C2 → Z 2 ↑C2 → Z 2 → 0 .

This is not a free (neither a projective) resolution of the trivial module Z2 but rather a resolution by permutation modules.

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This suggests that we should look for a resolution of the trivial Gn -module Zp in terms of something like permutation modules on finite subgroups and try to realize those by appropriate homotopy fixed point spectra where realization is again in the sense of the isomorphism of Morava modules of 1.3.2 which gives us for each finite subgroup F of Gn a canonical isomorphism En∗ En hF ∼ = Homcts (Zp ↑FGn , En∗ ). This leads to the following questions. Questions. (Q1) Are there resolutions of finite length of the trivial Gn -module Zp by (finite) direct sums of permutation modules on finite (p)-subgroups of Gn ? (Q2) Can these algebraic resolutions be realized by “resolutions” of spectra? What do we mean by a “resolution” of a spectrum? (Q3) If the answers to (Q1) and (Q2) are yes, how unique are these resolutions? Remark The group Sn resp. Gn is of finite virtual cohomological p-dimension. If G is a discrete group which is virtually of finite cohomological dimension, then a permutation resolution of finite length can be obtained from the cellular chains of a contractible finite dimensional G-CW -complex on which G acts with finite stabilizers. Such spaces always exist and hence such resolutions always exist [Se]. In case G is profinite and vcdp (G) < ∞ then some sort of positive answer to (a) may be given by algebraically mimicking Serre’s construction: one considers a finite index open subgroup H with cdp (H) < ∞, then one takes a projective resolution of finite length of the trivial Zp [[H]]module Zp and finally one obtains the desired resolution by tensor induction from H to G. This ensures existence, but the drawback of this construction is that it tends to be not very efficient. In particular, the length of the resolution would be much larger than necessary (the vcd?) and the modules in the resolution would not be finitely generated. In the case of the stabilizer groups we will describe a construction which gives better qualitative (and in favorable cases quantitative) information on the form of such resolutions. However, before we turn to the general theory we will survey recent joint work with Goerss and Mahowald [GHM] resp. with Goerss, Mahowald and Rezk [GHMR1] in which we construct explicit and efficient resolutions in the case n = 2 and p = 3. 3.3 Explicit examples IV: the case n = 2 and p = 3. Throughout this section we assume n = 2 and p = 3. In this case there are two different explicit resolutions which we will call duality resolution resp. centralizer resolution. We will see in sections 3.5 and 3.6 below that the centralizer

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resolution can be generalized both algebraically and homotopy theoretically. The justification for its name will become clear in section 3.5. The duality resolution has the advantage of being more efficient and having an intriguing symmetry. In the sequel we describe both resolutions. For more details in the case of the duality resolution we refer to [GHMR1]. The centralizer resolution is implicit in [GHM] and is a special case of Theorem 26 below. 3.3.1. In the following results we use the phrase “resolution of spectra” in the following weak sense: we call a sequence of spectra ∗ → X−1 → X0 → X1 → . . . a resolution of X−1 if the composite of any two consecutive maps is nullhomotopic and if each of the maps Xi → Xi+1 , i ≥ 0 can be factored as Xi → Ci → Xi+1 such that Ci−1 → Xi → Ci is a cofibration for every i ≥ 0 (with C−1 := X−1 ). We say that the resolution is of length n if Cn ' Xn and Xi ' ∗ if i > n. 3.3.2. Before we can describe our resolutions we need to introduce certain finite subgroups of G12 and some of their representations (cf. [GHMR1] for more details). The group G12 contains a group G24 of order 24 which is isomorphic to the semidirect product C3 oQ8 such that the quaternion group Q8 acts non-trivially on C3 . If ω is a primitive 8-th root of unity in WFq then C3 is generated by the element s = − 12 (1 + ωS) while Q8 is generated by ω 2 and ωS. G12 contains also the semidirect product C8 o Gal(F9 /F3 ) generated by ω and S. This group which was denoted F2(32 −1) in section 2.3 can be identified with the semidihedral group SD16 of order 16. It has a unique non-trivial one dimensional representation χ over Z3 which is trivial on the subgroup < ω 2 , ωS >. (χ agrees with the representation λ4,− that is discussed in the appendix.) 3.3.3. The duality resolution. The following results are proved in [GHMR1]. Theorem 8 (Algebraic duality resolution). There exists a short exact sequence of Z3 [[G2 ]]-modules 2 2 0 → Z 3 ↑G → Z 3 ↑G → Z3 → 0 , G1 G1 2

an exact complex of

Z3 [[G12 ]]

G1

2

- modules G1

G1

G1

2 2 0 → Z3 ↑G224 → χ ↑SD → χ ↑SD → Z3 ↑G224 → Z3 → 0 , 16 16

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and an exact complex of Z3 [[G2 ]] - modules G1

G2 G2 2 2 0 → Z 3 ↑G G24 →χ ↑SD16 ⊕ Z3 ↑G24 → (χ ⊕ χ) ↑SD16 → G1

G2 2 2 →Z3 ↑G G24 ⊕ χ ↑SD16 → Z3 ↑G24 → Z3 → 0 .

Theorem 9 (Homotopy theoretic duality resolution). There exists a fibration hG12

LK(2) S 0 → E2

hG12

−→ E2

and resolutions of spectra of length 3 resp. 4 hG12

∗ → E2

→ E2hG24 → Σ8 E2hSD16 → Σ40 E2hSD16 → Σ48 E2hG24 → ∗

resp. ∗ → LK(2) S 0 → E2hG24 → E2hG24 ∨ Σ8 E2hSD16 → Σ8 E2hSD16 ∨ Σ40 E2hSD16 → → Σ40 E2hSD16 ∨ Σ48 E2hG24 → Σ48 E2hG24 → ∗ . Remarks a) The spectrum E2hG24 is a version of the Hopkins-Miller higher real K-theory spectrum EO2 at p = 3. Its coefficients are described in detail in [GHMR1]. The coefficients of E2hSD16 are given by the completion of Z3 [v1 ][v2±1 ] with respect to the ideal generated by v14 v2−1 (cf. the discussion in the appendix). b) The appearance of the 8-fold suspension is forced by the character χ, while the 40-fold suspension is there for purely aesthetic reasons (note that Σ40 E2hSD16 ' Σ8 E2hSD16 by periodicity), so that the homotopy theoretic resolution displays a similar kind of duality as the algebraic resolution. The appearance of the 48-fold suspension, however, is a genuinely homotopy theoretic phenomenon and cannot be avoided. c) The resolution appears to be self dual but there is no satisfactory explanation of this duality yet. And it is not at all clear whether there are generalizations, say to the case n = p − 1, p > 3, and what they could look like. 3.3.4. The centralizer resolution. For describing the centralizer resolution we will make use of the Z3 [SD16 ]-module λ2 (cf. 2.3 and appendix) and the unique non-trivial one dimensional representation χ e of G24 over Z3 which is trivial on s and on ωS. The following results are implicit in [GHM]. We will give a proof in section 3.6.6.

Theorem 10 (Algebraic centralizer resolution). There exists a short exact sequence of Z3 [[G2 ]]-modules 2 2 → Z3 → 0 , 0 → Z 3 ↑G → Z 3 ↑G G1 G1 2

2

143

On finite resolutions of K(n)-local spheres an exact complex of Z3 [[G12 ]] - modules G1

G1

G1

G1

G1

2 2 2 0 → Z3 ↑SD → λ2 ↑SD → χ ↑SD ⊕χ e ↑G224 → Z3 ↑G224 → Z3 → 0 16 16 16

and an exact complex of Z3 [[G2 ]] - modules G1

G1

G1

G1

2 2 2 0 → Z3 ↑SD → (Z3 ⊕ λ2 ) ↑SD → (λ2 ⊕ χ) ↑SD ⊕χ e ↑G224 → 16 16 16

G1

G1

G1

2 → χ ↑SD ⊕ (e χ ⊕ Z3 ) ↑G224 → Z3 ↑G224 → Z3 → 0 . 16

Theorem 11 (Homotopy theoretic centralizer resolution). There exists a fibration hG1 hG1 LK(2) S 0 → E2 2 −→ E2 2 and resolutions of spectra of length 3 resp. 4 hG12

∗ → E2

→ E2hG24 → Σ8 E2hSD16 ∨ Σ36 E2hG24 → → Σ4 E2hSD16 ∨ Σ12 E2hSD16 → E2hSD16 → ∗

resp. ∗ → LK(2) S 0 → E2hG24 → Σ8 E2hSD16 ∨ Σ36 E2hG24 ∨ E2hG24 → → Σ4 E2hSD16 ∨ Σ12 E2hSD16 ∨ Σ8 E2hSD16 ∨ Σ36 E2hG24 → → E2hSD16 ∨ Σ4 E2hSD16 ∨ Σ12 E2hSD16 → E2hSD16 → ∗ . 3.4 Work in progress (the case n = p = 2). In this case we do have an algebraic duality resolution but we have not yet completely succeeded in realizing it. However, there is an algebraic centralizer resolution which can be realized. To describe the homotopy theoretic resolutions we note that, for p = 2, S2 contains a subgroup of order 24, also denoted G24 (but not isomorphic to the group with the same label that we used in the last section). For p = 2 the group G24 is isomorphic to the semidirect product Q8 o C3 of the quaternion group Q8 with the cyclic group C3 or order 3 which cyclically permutes i, j and k. G24 contains cyclic subgroups of order 2, 4 and of order 6, We fix such subgroups and denote them by C2 resp. C4 resp. C6 . Then we have the following results which, up to Galois extension, give resolutions of LK(2) S 0 at p = 2. Details will appear in [GHMR2]. Theorem 12 (Centralizer resolution). There exists a fibration hS12

E2hS2 −→ E2

hS12

−→ E2

and a resolution of spectra of length 3 hS12

∗ → E2

→ E2hG24 ∨ E2hG24 → E2hC6 ∨ E2hC4 → E2hC2 → E2hC6 → ∗ .

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Theorem 13 (Duality resolution). There exists a fibration hS12

E2hS2 −→ E2

hS12

−→ E2

and a resolution of spectra of length 3 hS12

∗ → E2

→ E2hG24 → E2hC6 → E2hC6 → X3 → ∗ together with an isomorphism of Morava modules E∗ (X3 ) ∼ = E∗ (E2hG24 ).

Remarks a) The spectrum E2hG24 is a version of the higher real K-theory spectrum EO2 at p = 2. We have not been able yet to further identify the spectrum X3 . b) We expect that the resolutions described in this and the previous section will help to better understand the Shimomura-Wang calculation [ShW] of π∗ LK(2) S 0 at p = 3 and that they will be crucial for calculating π∗ LK(2) S 0 at p = 2. 3.5 Permutation resolutions in the case n = k(p − 1) for p odd. In this section we will give a positive answer to question (Q1) and the algebraic part of question (Q3) of section 3.2 above, at least if p is odd. 3.5.1. We start by introducing some relative homological algebra (cf. [EM]) in a form which parallels Miller’s discussion of En -injective spectra and En injective resolutions of spectra (cf. section 2.2.1). Let p be a fixed prime. If G is a profinite group we denote the collection of finite p-subgroups of G by Fp (G), or simply by F (G) or even F if G and p are clear from the context. Throughout this section we will make the following Assumption: G contains only finitely many conjugacy classes of finite psubgroups. We recall that all our modules will be profinite continuous modules for the completed group algebras and that induced modules are formed by using the completed tensor product. A Zp [[G]]-module P will be called F -projective if the canonical Zp [[G]]-linear map M P ↑G F→ P (F )∈F

is a split epimorphism (where the sum is taken over conjugacy classes of finite p-subgroups). It is clear that each Zp [[G]]-module which is induced from a Zp [F ]-module for some F ∈ F is F -projective, and that a Zp [[G]]-module

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P if and only if P is a retract of some module of the form L is F -projective G (F )∈F MF ↑F where each MF is a Zp [F ]-module.

The class of F –projectives determines in the usual way a class of F -exact sequences: a sequence of Zp [[G]]-modules M 0 → M → M 00 is called F -exact if the composition M 0 → M 00 is trivial and HomZp [[G]] (P, M 0 ) → HomZp [[G]](P, M) → HomZp [[G]](P, M 00 )

is an exact sequence of abelian groups for each F -projective Zp [[G]]-module P . It is obvious that the category of Zp [[G]]-modules has enough F -projectives. Finally an F -resolution of a Zp [[G]]-module M is a sequence of Zp [[G]]modules . . . → P1 → P0 → M → 0 where each Ps is F -projective and each 3-term subsequence is F -exact. Then it is clear that each module M admits an F -resolution and that any homomorphism of Zp [[G]]-modules M → N is covered by a map of F -resolutions which is unique up to chain homotopy. We will be interested in constructing F -resolutions of finite length of the trivial module Zp such that all modules in the resolution are finitely generated. 3.5.2. The construction of our F -resolutions will rely on the following three results. Lemma 14. Suppose G is a profinite group and H is a normal finite psubgroup. a) If P is F (G/H)-projective, then considered as a Zp [[G]]-module via the canonical projection π : G → G/H, P is F (G)-projective. b) If M 0 → M → M 00 is a sequence of Zp [[G/H]]-modules which is F (G/H)exact, then considered as a sequence of Zp [[G]]-modules via the canonical projection π : G → G/H, M 0 → M → M 00 is F (G)-exact. Proof. a) This follows immediately from the following observation. If F is a G/H finite p-subgroup of G/H and M is a Zp [F ]-module then M ↑F , considered as Zp [[G]]-module via π, is isomorphic to M ↑G π −1 F . b) If F is a finite p-subgroup of G and N is a Zp [F ]-module then there is a G/H ∼ natural isomorphism (N ↑G F )⊗Zp [H] Zp = (N ⊗Zp [F ∩H] Zp ) ↑F/F ∩H ) of Zp [[G/H]]modules. This implies that − ⊗Zp [H] Zp sends F (G)-projectives to F (G/H)projectives which in turn implies (b). Lemma 15. Suppose G is a profinite group and K is a closed subgroup of G which contains only a finite number of conjugacy classes of finite subgroups.

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a) If P is F (K)-projective, then P ↑G K is F (G)-projective. b) Assume that sequence of open normal subgroups U n of T there is a decreasing 0 G such that n Un = {1}. If M → M → M 00 is a sequence of Zp [[K]]-modules G 00 G which is F (K)-exact, then M 0 ↑G K → M ↑K → M ↑K is F (G)-exact. Proof. a) This is trivial. b) It is enough to show that for each F ∈ F (G) and each Zp [F ]-module L G 00 G the functor HomZp [F ](L, −) sends the sequence M 0 ↑G K → M ↑K → M ↑K to an exact sequence of abelian groups. This is in fact a consequence of the Mackey decomposition formula which describes the restriction of an induced module. In the case of profinite groups this requires some care so that it seems appropriate to give some details. To simplify notation we let Kn = K/K∩Un , Gn = G/Un and Fn = F/F ∩Un . Note that if F is finite then F = Fn if n is sufficiently large. Now let N be any profinite Zp [[K]]-module. Then we can write N = limi Ni (where i runs through some directed set I, not necessarily a countable sequence) with Ni finite and acted on trivially by K ∩ Uλ(i) for some increasing function λ : I → N. Then we have the classical Mackey decomposition formula for the Zp [Kλ(i) ]-modules Ni (where as usual (−) ↓G F denotes the restriction of a Zp [[G]]-module to a Zp [F ]-module) M G Gλ(i) ∼ gKλ(i) g −1 Fλ(i) g Ni ↑Kλ(i) ↓ ( N ) ↓ = i gKλ(i) g −1 ∩Fλ(i) ↑gKλ(i) g −1 ∩Fλ(i) λ(i) Fλ(i) g∈Fλ(i) \Gλ(i) /Kλ(i)

and by passing to the limit we obtain Gλ(i) Gλ(i) G ∼ G ∼ b N ↑G K ↓F = (Zp [[G]]⊗Zp [[K]] N ) ↓F = limi Ni ↑Kλ(i) ↓Fλ(i) M gKλ(i) g −1 Fλ(i) ∼ (g Ni ) ↓gKλ(i) = limi g −1 ∩Fλ(i) ↑gKλ(i) g −1 ∩Fλ(i) . g∈Fλ(i) \Gλ(i) /Kλ(i)

Now we consider our F (K)-split sequence M 0 → M → M 00 and factor the first homomorphism via the kernel M of M → M 00 as M 0 → M → M. It is enough G G G to show that M 0 ↑G K ↓F → M ↑K ↓F induces a surjection on HomZp [F ] (L, −). First we note that p : M 0 → M → 0 is F (K)-exact. This implies that for any finite p-subgroup H of K there exists an H-linear splitting s : M → M 0 , i.e. there exist increasing functions α : I → I, β : I → I and compatible families of Zp [H]-linear maps pi : (M 0 )α(i) → M i representing p and sα(i) : (M)βα(i) → (M 0 )α(i) representing s such that the composition (M )βα(i) → (M )i is the map in the given system for M . By explicitly choosing conjugations we may assume that we have such a splitting (with the same α and β) for all finite p-subgroups in the conjugacy class of H. And because we assume that there are only finitely

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many conjugacy classes of finite p-subgroups in K we may assume that the functions α and β work for all H ∈ F (K). 0 The Mackey decomposition formula for (M)βα(i) and Mα(i) gives us therefore, for any choice of double coset representatives in Gλβα(i) resp. Gλα(i) , well defined Zp [F ]-linear maps G

G

G

G

0 ↓ λβα(i) → Mα(i) ↑Kλα(i) ↓ λα(i) . seα(i) : (M )βα(i) ↑Kλβα(i) λβα(i) Fλβα(i) λα(i) Fλα(i)

Now the elementary theory of profinite sets (cf. Lemma 5.6.7 in [RZ]) tells us that the quotient map G → F \G/K admits a continuous section and any such section gives us a compatible choice of double coset representatives in Gn for all n. (We note that it is here that we use the assumption on the existence of the decreasing sequence of subgroups Un .) For any such choice the associated Zp [F ]-linear maps seα(i) are compatible with respect to i so that they patch together and define a Zp [F ]-linear map on the level of inverse limits. This G G G shows that M 0 ↑G K ↓F → M ↑K ↓F is split as a map of Zp [F ]-modules and hence we are done. Lemma 16. Suppose G is a profinite group and M is a Zp [[G]]-module which admits a finite projective resolution and which is projective as a Zp -module. Then M is projective as a Zp [F ]-module for every F ∈ F . Proof. By induction on the length of a finite projective resolution it is enough to show that for a finite p-group F a short exact sequence 0 → M1 → M2 → M3 → 0 of Zp [F ]-modules splits if M1 is projective as a Zp [F ]-module and if the sequence splits as a sequence of Zp -modules. The existence of a Zp -splitting of the inclusion M1 → M2 implies that any Zp [F ]-linear map ϕ from M1 to the coinduced module Hom(Zp [F ], M1 ) can be extended to an Zp [F ]-linear map ϕ e : M2 → Hom(Zp [F ], M1 ). Next, if M1 is projective as a Zp [F ]-module then it is a direct summand in the induced module M1 ↑F{1} , and because F is finite the induced module is isomorphic to the coinduced module. Now we take for ϕ any Zp [F ]-split inclusion of M1 into Hom(Zp [F ], M1 ). Then the composition of ϕ e with a Zp [F ]-linear splitting of ϕ provides the desired splitting. 3.5.3. Here is the promised answer to question (Q1) and the algebraic part of (Q3). Proposition 17. Suppose G is a virtually profinite p-group and S is a closed normal subgroup which is a profinite p-group. Furthermore assume that (1) H ∗ (S; Fp ) is a finitely generated Fp -algebra, (2) all finite p-subgroups of S are cyclic and there is a bound on their order,

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(3) the trivial module Zp admits a projective resolution of finite type and finite length over the p-completed group algebra of G/S and all its closed subgroups, T (4) there is a sequence of open subgroups Un of G such that n Un = {1}. Then the trivial Zp [[G]]-module Zp admits an F -resolution of finite length.

Furthermore, all F -projectives in this resolution can be chosen to be summands in finite direct sums of modules of the form Zp ↑G F with F ∈ F .

Remarks a) Assumption (1) implies by a recent result of Minh and Symonds [MS] that S has only a finite number of conjugacy classes of finite p-subgroups. In particular the bound in assumption (2) is already a consequence of assumption (1). Furthermore recent work of Symonds suggests that finite length F -resolutions should exist even if we skip condition (2). b) Assumption (3) implies that G/S has no elements of order p, in other words every finite p-subgroup of G is already contained in S. c) It is well-known that H ∗ (Sn ; Fp ) is a finitely generated algebra. As already mentioned in 1.2.4 and 1.2.5, Sn has finite p-subgroups if and only if n is divisible by p − 1. Furthermore, if p is odd, or p = 2 and n is odd, then all finite p-subgroups are cyclic and their conjugacy class is unique. The p-Sylow subgroup Sn of Sn is normal in Gn of index (pn − 1)n which is of order prime to p if n = k(p − 1) with k 6≡ 0 mod p. Therefore, for such n the assumptions of the proposition hold with G = Gn and S = Sn . If n = k(p − 1) with k divisible by p then we replace S Gn as in remark 2 of section 2.2 by Gn (K) := Sn o Gal(K/Fp ) where K := r Fqpr . This has the effect that all p-torsion elements of Gn (K) are already contained in Sn resp. its normal Sylow subgroup Sn . In this case it is clear that the assumptions hold with G = Gn (K) and S = Sn . In fact, G/S has a finite normal subgroup of order prime to p with quotient Zp . The somewhat artificially looking assumptions on the pair (G, S) in Proposition 17 have been introduced in order to cover this case. d) If n is even and p = 2 then Sn has finite 2-subgroups which are not cyclic. Thus for n even and p = 2 the proposition does not give any information. Nevertheless, if n = p = 2 the algebraic centralizer resolution mentioned in section 3.4 gives an explicit finite length F -resolution of the trivial Z2 [[S2 ]]module Z2 . Proof. The proof will be by induction over the order of the largest finite p-subgroup of G. We distinguish the following cases. Case 1: G has no non-trivial finite p-subgroups.

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In this case Quillen’s F -isomorphism theorem [Q] implies that H ∗ (S; Fp ) is a finite Fp -algebra. Because S is a profinite p-group this implies cdp (S) < ∞ and then assumption (3) implies cdp (G) < ∞. Finally, (3) and the spectral sequence of the group extension 1 → S → G → G/S → 1 show that H i (G, M) is a finite group for every finite discrete continuous Zp [[G]]-module M and this ensures the existence of a projective resolution of Zp of finite length in which all projectives are finitely generated (cf. Proposition 4.2.3 in [SyW]). Case 2: G has a normal finite p-subgroup F . By [MS] H ∗ (S/F, Fp ) is still a finitely generated Fp -algebra and hence G/F and S/F still satisfy our assumptions. Therefore, by induction hypothesis, the trivial G/F -module Zp admits an F (G/F )-resolution of finite length such that all F (G/F )-projectives are of the required form. Then Lemma 14 implies that the very same resolution is also an F (G)-resolution of finite length for the trivial G-module Zp , and all F (G)-projectives are as required. Case 3: The general case. We may suppose that G does not contain any finite normal p-subgroups. In this case we consider the short exact sequence of Zp [[G]]-modules M f ε (5) 0 → K −→ Zp ↑ G NG (E) −→ Zp → 0 (E)

where the sum is over conjugacy classes of non-trivial elementary abelian psubgroups of G (which by our assumption are all of order p), ε is the canonical augmentation and K is its kernel. First we note that the exact sequence (5) is F -exact. In fact, if F is a finite p-subgroup of G then it has a non-trivial central element of order p and F is contained in the normalizer of the elementary abelian p-subgroup E 0 generated by this element of order p. This means, that the action of F on G/NG (E 0 ) has a fixed point. Such a fixed point determines a Zp [F ]-linear splitting of the surjection in (5). In other words, the exact sequence splits upon restriction to every F ∈ F and this is equivalent to saying that the sequence is F -exact. L G It will therefore be enough to construct F -resolutions for (E) Zp ↑NG (E) and for K where the F -projectives have the required form. In fact, if we have two such resolutions we can lift f to a map of resolutions and then the resulting double complex will be an F -resolution for Zp with all F -projectives as required. The pair (NG (E), S ∩ NG (E)) satisfies the same assumptions as (G, S): assumptions (2), (3) and (4) are obvious. For (1) we note that S ∩ NG (E)) = NS (E) agrees with the centralizer CS (E) because E is of rank 1. In fact, because the p-rank of E is always 1, the p-group NS (E)/CS (E) injects into

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Aut(E) ∼ = Z/(p − 1) and hence it is trivial. Now CS (E) satisfies (1) because it is given by a component of Lannes’ T -functor (by Theorem 2.6 of [He]) and Lannes’ T -functor takes unstable finitely generated Fp -algebras to unstable finitely generated Fp -algebras ([DW, Theorem 1.4]). Furthermore, by [MS] the groups NS (E) and hence also NG (E) have only a finite number of conjugacy classes of finite p-subgroups. Therefore we can apply case 2 and deduce that for each E ∈ F (G) the trivial Zp [[NG (E)]]-module Zp admits an F (NG (E))resolution of finite length. Inducing this resolution gives, by Lemma 15, an F (G)-resolution of Zp ↑G NG (E) with all F -projectives as required. Finally consider K. The group G/S acts on the finite set of conjugacy classes of elementary abelian p-subgroups of S and the stabilizer of such a subgroup E is the image of NG (E) in G/S. In particular this image is of finite index in G/S and this implies that for each E ⊂ G we get an isomorphism of Zp [[S]]-modules M ∼ Zp ↑SNS (E 0 ) Zp ↑ G NG (E) = (E 0 )<S

where the direct sum is taken over conjugacy classes of E 0 ’s in S which become conjugate to E in G. Furthermore, as before the normalizer NS (E 0 ) can be identified with the centralizer CS (E 0 ). Therefore we end up with an isomorphism of Zp [[S]]-modules M

(E) 1. 3.6.1. Let us start by looking at the F -resolution of the trivial Zp [[Gn ]]-module constructed in the previous section. By 1.2.4 there is a unique conjugacy class of subgroups of Sn of order p. We pick a representative and denote it by E and its normalizer in Gn simply by N . So the exact sequence of the proof of Proposition 17 takes the following form (6)

f

n 0 → K −→ Zp ↑G N −→ Zp → 0 .

ε

By 1.3.2 this sequence can be realized by the cofibration LK(n) S 0 ' EnhGn −→ EnhN → C ι

where ι is induced by the inclusion of N into Gn and C is the cofibre of ι. Of course, as before realizing means that the map Homcts (ε, (En )∗ ) agrees with the map ι∗ Homcts (Zp , (En )∗ ) ∼ = Homcts (Zp ↑Gn , (En )∗ ) = (En )∗ LK(n) S 0 −→ (En )∗ E hN ∼ 2

N

induced by ι in (En )∗ . This map is injective and therefore E∗ C can be identified with Homcts (K, (En )∗ ). This suggests that we should start by constructing resolutions for EnhN and C. 3.6.2. We begin with C. Lemma 18. C admits an En -resolution of finite length in which each spectrum is a summand of a finite wedge of En ’s. Proof. We know from section 3.5 that K has a projective Zp [[Gn ]]-resolution P• : 0 → P d → . . . P 0 → K of finite length d in which each Ps is finitely generated. Let Q• be the following exact complex of Zp [[Gn ]]-modules n Q• : 0 → P d → . . . → P 0 → Z p ↑G N → Zp → 0

obtained by splicing the resolution of K and the exact sequence (6). By Proposition 3 there is a sequence of maps between spectra of the following form Y• : ∗ → LK(n) S 0 ' EnhGn −→ EnhN → X0 → . . . → Xd → ∗ ι

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where each Xi is a summand in a finite wedge of En ’s such that the complex Homcts (Q• , (En )∗ ) is isomorphic, as a complex of Morava modules to the complex (En )∗ (Y• ). Again by Proposition 3 the composite LK(n) S 0 → EnhN → X0 is null and we obtain a sequence of maps between spectra X• : ∗ → C → X0 → . . . → Xd → ∗ such that Homcts (P• , (En )∗ ) is isomorphic, as a complex of Morava modules to the complex (En )∗ X• . We claim that X• is, in fact, an En -resolution of C in the sense of 2.2.1. In fact, it is clear that all Xs for s ≥ 0 are En -injective and En -exactness of Y• is seen as in the proof of Theorem 4. En -exactness of X• follows immediately from that of Y• . 3.6.3. The case of EnhN is substantially more difficult and unlike in the case of C the resolution will not be an En -resolution although it will still have some good properties. In this subsection we will make the algebraic resolution of n Zp ↑ G N explicit. 3.6.3.1. First we investigate the structure of the group N = NGn (E). We choose a primitive (pn − 1)-st root of unity ω ∈ W× Fq ⊂ Sn and note that the element X := ω

p−1 2

S ∈ Dn satisfies X n = −p.

Lemma 19. The subfield Qp (X) of Dn is isomorphic to the cyclotomic extension Qp (ζp ) generated by a primitive p-th root of unity ζp and this isomorphism restricts to an isomorphism Zp [X]/(X n + p) ∼ = Zp [ζp ]. Proof (outline). In fact, this is a straightforward consequence of local class field theory but for the convenience of the reader we outline an elementary and direct proof. R := Zp [X]/(X n +p) is a local ring with maximal ideal generated by X. The powers of the maximal ideal induce, as in section 1.2.3, a decreasing complete filtration Fi with i = nk , k = 1, 2, . . ., of the group of units of R via Fi := {g ∈ R× |g ≡ 1 mod X in } and with filtration quotients gri := Fi /Fi+ 1 which are all canonically isomorn phic to Fp . Furthermore the p-th power map induces, as in the case of the groups Sn , maps gri → grϕ(i) which via these isomorphisms are given by the identity if i 6= n1 and by a 7→ ap − a if i = n1 (with notation as in section 1.2.3). The completeness of the filtration shows now that R contains a p-th root of unity, in fact for any a ∈ R with ap ≡ a mod (X) there is a p-th root of unity of the form 1 + aX mod(X 2 ). This shows that Qp (X) contains an isomorphic copy of the field Qp (ζp ) and because both are extensions of degree n = p − 1 of Q they have to be isomorphic. The final statement of the Lemma is now obvious.

On finite resolutions of K(n)-local spheres

153

We choose a primitive p-th root of unity in Qp (X) and we still denote it by ζp thus identifying Qp (X) with Qp (ζp ). We choose our subgroup E ⊂ Sn to be generated by ζp . If n = k(p − 1) then (cf. 1.2.4) the centralizer CDn (E) is a division algebra which is central over Qp (ζp ) of dimension k 2 . In particular, if n = p − 1, then CDn (E) = Qp (ζp ), and CD×n (E) = Qp (ζp )× . We need to know the structure of the group of units in Qp (ζp ) = Qp (X). This is well known in algebraic number theory. For the convenience of the reader we give some details. Consider the filtration by subgroups (7)

0 ⊂ F 2 ⊂ F 1 ⊂ Zp [ζp ]× ⊂ Qp (ζp )× n

n

where F 2 and F 1 are the filtration subgroups of Zp [ζp ]× introduced above. The n n valuation v, normalized by v(p) = 1, is a split epimorphism Qp (ζp )× → n1 Z ∼ =Z with kernel Zp [ζp ]× . Next, the description of the p-th power map given above shows that F 2 is a free Zp -module of rank n and the torsion-subgroup of Zp [ζp ]× n identifies with Zp [ζp ]× /F 2 ∼ = E × Z/n. In particular, we see that the filtration n (7) is split and we obtain an isomorphism (70 ) Qp (ζp )× ∼ = Z × Z/n × Z/p × Zn . p

Furthermore, the discussion above shows that we can choose generators as follows: • The element X satisfies v(X) = n1 and can therefore be chosen as a generator of the factor Z. pn −1 • The subgroup Z/n is the subgroup generated by ω p−1 . • Z/p ∼ = E is the subgroup generated by ζp . • The elements ηj := 1+X j ∈ Zp [ζp ]× ∼ = Zp [X]× , j = 2, . . . , n+1, qualify as a system of topological generators of Znp . We will need to understand the action of the Galois group Gal(Qp (ζp )/Qp ) on Zp [ζp ]× . It is clear that the three factors F 2 , E and Z/n are invariant with n respect to the action of the Galois group and that the action on Z/n ⊂ Z× p is trivial. Furthermore, the Galois group can be canonically identified with the group Aut(E) of automorphisms of E, in other words the action on E is the tautological action. To understand the action on F 2 we observe that the Galois automorphisms n can be realized by the conjugation action (in Dn ) of the quotient of the cyclic pn −1 (p−1)2

group generated by τ := ω elementary calculation shows

by the subgroup generated by τ n . Then an

τ∗ (ηj ) = 1 + ω

j(1−pn ) p−1

Xj ,

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i.e. ηj is an eigenvector in gr j with eigenvalue ω eigenvector with eigenvalue ω

n j(1−pn ) p−1

j(1−pn ) p−1

. To obtain a genuine

in Zp [ζp ]× we note that

n −1) 1 X ij(pp−1 τi ω n i=1

n

ej :=

is an idempotent in Zp [Z/n] which satisfies τ ej = ω by ηj0 := (ej )∗ ηj

j(1−pn ) p−1

ej . If we replace ηj

then the elements ηj0 , j = 2, . . . , n + 1, generate one-dimensional characters, say χ(j), and as a Galois-module F 2 is isomorphic to the direct sum of all the n different one-dimensional characters of Aut(E) over Zp . We note that if we rename χ(n + 1) as χ(1) then we get χ(j) = χ(1)⊗j for j = 2, . . . , n + 1. In the sequel we will interpret χ(j), for any integer j ∈ Z, as the tensor product χ(1)⊗j . After these preparations we will now describe NGn (E). By abuse of notation n ∼ × we continue to denote the image of elements of D× n in Gn = Dn /hS i by the same name. Proposition 20. Let n = p − 1 and p be odd. a) There is an exact sequence 1 → CGn (E) → NGn (E) → Aut(E) → 1 . b) There is an isomorphism CGn (E) ∼ = H × E × Znp where H ∼ = Z/2n × Z/ n2 is generated by X and τ n X 2 , E is generated by ζp , and the elements ηj = 1 + X j , j = 2, . . . , n + 1, form a system of topological generators of Znp . c) The action of Aut(E) leaves H, E and L Znp invariant: the action of Aut(E) n on E is the tautological action while Znp ∼ = j=1 χ(j) and χ(j) is generated by ηj 0 for j = 2, . . . , n and by ηn+1 0 for j = 1. d) The subgroup generated by ζp , H and τ is a finite subgroup F of N of order pn3 which contains H × E as a normal subgroup with quotient Aut(E). Proof. a) We have seen above that there is an exact sequence 1 → CD×n (E) → ND×n (E) → Aut(E) → 1

On finite resolutions of K(n)-local spheres

155

pn −1

where ω (p−1)2 projects to a generator of Aut(E). In particular all of the automorphisms of the group E are realized by conjugation in D× n and hence also × n ∼ in the central quotient Gn = Dn /hS i. This shows (a). b) We claim that the epimorphism from D× n to Gn induces an epimorphism from CD×n (E) to CGn (E) with kernel the central subgroup generated by S n . In fact, if we lift an element g ∈ CGn (E) to an element ge ∈ D× eζp ge−1 = ζp z n then g n with z ∈ hS i. But then we obtain 1 = geζpp ge−1 = (e gζp ge−1 )p = (ζp z)p = z p

and hence z = 1, in other words ge ∈ CD×n (E). Part (b) follows now from pn −1

the observation that S n = p = ω 2 X n , i.e. in terms of the decomposition (7’) of CD×n (E), the element S n has components (n, n2 ) in Z × Z/n and trivial components in Z/p × Znp . (c) and (d) are clear from the discussion above.

Remark The subgroup H is intrinsically defined as the subgroup of the abelian profinite group CGn (E) generated by elements of finite order prime to p. As such it is necessarily invariant with respect to the action of Aut(E). The splitting H ∼ = Z/2n × Z/ n2 , however, is visibly not invariant with respect to this action. 3.6.3.2. In the next step we will construct an explicit F (N )-resolution of the trivial Zp [[N ]]-module Zp . If we set N := N/(H × E) then we have N = Znp o Z/n where the semidirect product is taken with respect to the Z/nmodule structure on Znp given by the direct sum of the n different characters of Z/n over Zp . Proposition 21. a) The trivial Zp [[N ]]-module Zp admits a projective resolution P• : 0 → Pn → . . . → P1 → P0 → P−1 = Zp with Pr ∼ =

M

χ(i1 + . . . + ir ) ↑N Z/n

(i1 ,...,ir )

and where the direct sum is taken over all sequences (i1 , . . . , ir ) with n − 1 ≥ i1 > i2 > . . . > ir ≥ 0. (For r = 0 there is a unique summand 1 ↑N Z/n corresponding to the empty sequence.) b) Considered as a complex of Zp [[N ]]-modules, via the projection N → N , this complex is an F (N )-resolution in which all modules are summands in a finite direct sum of modules Zp ↑N E.

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Proof. a) We start by observing that H

∗

(Znp ; Zp )

∼ = Λ(

n−1 M

χ(i))

i=0

as a module over Zp [Z/n]. The resolution is now constructed step by step as in the proof of Theorem 6. The identity of Zp considered as a map of Zp [Z/n]-modules extends to a Zp [[N ]]-linear homomorphism P0 := Zp ↑N Z/n → Zp . Let N0 be its kernel. Then n we can compute H∗ (Zp , N0 /(p)) and in particular we find H0 (Znp ; N0 /(p))

∼ =

n−1 M

χ(i) ⊗Zp Fp ,

i=0

Ln−1 as a module over Zp [Z/n]. The Zp [Z/n]-module i=0 χ(i) is projective and Ln−1 n hence we can lift the resulting homomorphism i=0 χ(i) → H0 (Zp , N0 /(p)) to N0 and by Nakayama’s Lemma the Zp [[N ]]-linear extension P1 :=

n−1 M

χ(i) ↑N Z/n → N0

i=0

is onto. Then we repeat the game with the kernel N1 of this epimorphism, and so on. When we finally arrive at Nn−1 we see that H0 (Znp , Nn−1 /(p))

∼ = χ(

n−1 X

i) ⊗Zp Fp , Hi (Znp , Nn−1 /(p)) = 0 if i > 0 .

i=0

Pn−1 Then we can construct in the same manner a map Pn := χ( i=0 i) ↑N Z/n → Nn−1 which by Nakayama’s lemma is now even an isomorphism. b) This is an immediate consequence of Lemma 14.

3.6.4. Now we turn towards the problem of realizing the resolution P• conn structed in Proposition 21b, or rather the induced resolution P• ↑G N of the Gn induced module Zp ↑N , by a sequence of maps between spectra. Let F be the subgroup of N of order pn3 described in Proposition 20. The characters χ(i) of the last section can be considered as characters of F via the canonical projection F → F/(H × E) = Aut(E) ∼ = Z/n. We will also need to consider the group F1 := F ∩ Sn which is of order pn2 and is generated by ζp and τ , and the group F2 which is cyclic of order pn and is generated by ζp and τ n.

157

On finite resolutions of K(n)-local spheres Lemma 22. For any i ∈ Z there is an isomorphism of Morava modules (En )∗ (Σ2pni E hF ) ∼ = Homcts (χ(i) ↑Gn , (En )∗ ) . n

F

Proof. For i = 0 this is nothing but 1.3.2. More generally, for every k ∈ Z there is an isomorphism of Morava-modules k n (En )∗ (Σk EnhF ) → Homcts (Zp ↑G F , (En )∗ (S )) .

We will show that there is an invertible element ∆(i) in (En )∗ of degree 2pni on which F acts via χ(i). Then we will get the desired isomorphism by composing with the isomorphism of Morava modules n Homcts (Zp ↑FGn , (En )∗ (S 2pni )) → Homcts (χ(i) ↑G F , (En )∗ )

given by ϕ 7→ g 7→ σ −1 (ϕ(g))g∗ (∆(i))

∼ =

where σ is the suspension isomorphism E∗ = E∗ (S 0 ) −→ E∗+2pni (S 2pni ). p−1

We recall that F is generated by τ , ζp and X = ω 2 S. We need some information about the action of these elements on (En )∗ . For this we recall that the action of an element g ∈ Sn is determined as follows (cf. [DH1]): if we lift g to a power series ge(x) ∈ (En )0 [[x]] then there is a unique continuous ring homomorphism g∗ : (En )0 → (En )0 and a unique ∗ -isomorphism h ∈ fn ) to the formal group law H defined (En )0 [[x]] from the formal group law g∗ (Γ fn (e by H(x, y) = ge−1 Γ g(x), ge(y)) The action of g on u is then given by g∗ (u) = 0 0 ge (0)h (0)u.

In particular, if g(x) = ax with a ∈ F× uller lift of a will, q and if the Teichm¨ by abuse of notation, still be denoted by a then we can take as lift ge(x) = ax and the [p]-series of the formal group law H satisfies [p]H (x) = a−1 ([p]Γe n (ax)) = a−1 ([p]Γe n (ax))

= a−1 p(ax) +Γe n u1 (ax)p +Γe n . . . +Γe n un−1 (ax)p = px +H u1 ap−1 xp +H . . . +H un−1 ap

n−1 −1

xp

n−1

n−1

+Γe n (ax)p n

+H xp .

n

This shows that the ∗-isomorphism h is the identity, i.e. h(x) = x, and that g∗ is given by (8)

i

g∗ (ui ) = ap −1 ui , g∗ (u) = au .

In the case of τ we have a = ω (9)

pn −1 (p−1)2

so that

τ∗ (u) = ω

pn −1 (p−1)2

u.

The action of ζp is more difficult. However, we only need to know that ζp acts trivially modulo the maximal ideal m = (p, u1 , . . . , un−1 ) ⊂ (En )∗ , and in

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particular that (10)

ζp ∗ (u) ≡ u mod m .

In fact, this holds for any element g in the p-Sylow subgroup Sn of Sn . en is already defined over Zp [[u1 , . . . , un−1 ]] and The universal deformation Γ therefore the subgroup of Gn given by Galois automorphisms of Fq acts trivially on the ui and on u. Together with (8) this implies (11)

X∗ (u) = ω

p−1 2

u.

The action of τ and of ζp on (En )∗ is WFq -linear while the action of X is only Zp -linear and satisfies X∗ (wx) = w σ X∗ (x) if w ∈ WFq and x ∈ (En )∗ . Now consider the element ∆0 :=

Y

g∗ (u) .

g∈F2

This is clearly fixed by the subgroup F2 . Furthermore by (9) and (10) we have n n Y Y i(pn −1) p(p−1) pn −1 τ∗in (up ) = ω p p−1 up = ω p 2 p−1 upn = −upn mod m ∆0 ≡ i=1

i=1

so that ∆ is an invertible element of degree −2pn. Now F2 is normal in F and the quotient F/F2 is isomorphic to Z/n × Z/n with generators τ and X. This quotient acts on the F2 -fixed points and by (11) we find for any λ ∈ WFq Y Y p−1 p−1 X∗ (λ∆0 ) = λσ X∗ (∆0 ) = λσ g∗ X∗ (u) = λσ g∗ (ω 2 u) = λσ ω 2 pn ∆0 . 0

g∈F2

In particular we get X∗ (ω

− pn 2

g∈F2

∆0 ) = ω

− pn 2

∆0 , and thus pn

∆00 := ω − 2 ∆0 is still an invertible element in (En )−2pn which is fixed by τ n , ζp and X, and which satisfies pn pn Y τ∗ (∆00 ) = ω − 2 τ∗ (∆0 ) = ω − 2 g∗ τ∗ (u) pn

= ω− 2

Y

g∈F2

g∗ (ω

pn −1 (p−1)2

pn

u) = ω − 2 ω

pn −1 p p−1

pn −1

∆0 = ω (p−1) ∆00 .

g∈F2

Therefore, for any i ∈ Z, the class (∆00 )−i is an invertible element in (En )2pni on which F acts via χ(i) (with the convention adopted in the discussion before Proposition 20). We will need some partial information about the homotopy groups of the 2 spectra Σ2p ni EnhG , i ∈ Z, where G runs through various subgroups of F which contain the central subgroup Z ⊂ F generated by τ n .

On finite resolutions of K(n)-local spheres

159

Lemma 23. Suppose G is a subgroup of F which contains Z. a) If G is of order prime to p, then for any i ∈ Z the homotopy fixed point 2 spectral sequence converging to π∗ (Σ2p ni EnhG ) satisfies E2s,t = 0 if s > 0. The spectral sequence collapses at E2 and π∗ (EnhG ) ∼ = (En )G ∗ is concentrated in degrees divisible by 2n. b) If G contains an element of order p then for any i ∈ Z the homotopy fixed 2 point spectral sequence converging to π∗ (Σ2p ni EnhG ) satisfies 0 if q is even, 0 < q < 2n E 0,0 if q = 0 2 2 πq (Σ2p ni EnhG ) ∼ = 0 if q is odd, q 6≡ {1, 3, . . . , 2p − 3} mod 2pn s(q),s(q)+q E2 if q is odd, q ≡ {1, 3, . . . , 2p − 3} mod 2p2 n

where s(2q + 1) = 2(p − 2 − q 0 ) + 1 if 2q + 1 = 2q 0 + 1 + 2p2 nl and q 0 ∈ {0, 1, . . . , p − 2}. Proof. a) If G contains no elements of order p then it is clear that E2s,t = 0 if s > 0 and the spectral sequence collapses. Furthermore, G contains the central subgroup Z and this subgroup acts trivially on (En )0 (by (8)) and because of (τ n )∗ (u) = ω if t 6≡ 0 mod 2n.

pn −1 p−1

u (again by (8)) we see that E20,t = (En )G t is trivial

b) Because G always contains Z and because E is normal in F , the assumption implies that G contains F2 = Z × E. Because the index of F2 in G is prime to p the homotopy fixed point spectrum EnhG is a direct summand in EnhF2 and it is therefore enough to discuss the case G = F2 . In the case of G = F1 the homotopy fixed point spectral sequence has been analyzed by Hopkins and Miller. Their account remains unpublished. A summary of this analysis is given in section 2 of [N]. If p = 3 a rather detailed discussion which includes the case of F2 can be found in [GHM] and [GHMR1]. The approach used in these papers generalizes without much problems to the case of any p > 2. In the following we will describe the E2 -term and the differentials of this spectral sequence. First of all, let ρ be the (p−1)-dimensional WFq [F2 ]-module which restricted to E is the reduced regular representation and on which the central element pn −1 τ n acts by multiplication by ω p−1 . Then, as a graded Zp [F2 ]-algebra, (En )∗ is isomorphic to the completion of S∗ (ρ)[N −1 ] at its maximal ideal, where S∗ (ρ) is the graded Q symmetric algebra on ρ with ρ in degree −2, and we have inverted N := g∈F2 g∗ (e) ∈ S∗ (ρ) where e is a suitable generator of ρ (cf. [GHMR1, Lemma 3.2]). In fact, the isomorphism identifies N , up to a scalar, with the element ∆00 of the proof of Lemma 22. This isomorphism can be

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used to calculate the E2 -term of the homotopy fixed point spectral sequence as follows (cf. section 2 in [N], or Theorem 3.7 in [GHMR1] if p = 3): • The invariants E20,t are trivial unless t ≡ 0 mod 2n and periodic of period 2pn with periodicity generator ∆00 ∈ (En )−2pn . In the sequel we let ∆ := (∆00 )−1 ∈ (En )2pn . • Multiplication with p annihilates E2s,∗ if s > 0. Furthermore, there are 2 elements α ∈ E21,2p−2 and β ∈ E22,2p −2p such that, as a module over Fq [∆±1 ], • E22k,∗ , for k > 0, is free of rank 1 with generator β k , • E22k+1,∗ , for k ≥ 0, is free of rank 1 with generator αβ k . The elements α and β are infinite cycles and represent the images of the elements α1 ∈ π2p−3 (S 0 ), β1 ∈ π2(p2 −p−1) (S 0 ) with respect to the unit S 0 → EnhF2 of the ring spectrum EnhF2 . The only non-trivial differentials in this spectral sequence are d2p−1 and d2n2 +1 . They are forced by Toda’s relations α1 β1p = 0 and β1pn+1 = 0 and are determined by 2

d2p−1 (∆n ) = cαβ p−1 , d2n2 +1 (∆n α) = c0 β n

2 +1

, 2

i.e. d2p−1 (∆) = −c∆1−n αβ p−1 , and d2n2 +1 (∆α) = c0 ∆−p(p−2) β n +1 where c, c0 are suitable units in Fq . 2 Then we end up with the following result which is more precise than Lemma 23 above. Proposition 24 (cf. [N, Proposition 2.1, 2.2]). a) π∗ (EnhF2 ) is periodic of period 2p2 n and with periodicity generator ∆p . s,∗ b) E∞ is trivial if s is even and s > 2n2 , or if s is odd and s > 2n − 1. s,∗ c) If 2n2 ≥ s = 2k > 0 then E∞ is a free module over Fq [∆±p ] with generator β k , of total degree 2k(p2 − p − 1). s,∗ d) If 2n − 1 ≥ s = 2k + 1 > 0 then E∞ is a free module over Fq [∆±p ] with generators ∆l β k α, 2 ≤ l ≤ p of total degree 2pnl + 2k(p2 − p − 1) + 2p − 3. 0,t e) E∞ = 0 if t 6≡ 0 mod 2n.

After these preparations we can continue with the proof of part (b) of Lemma 23. First we investigate which of the generators in part (c) and (d) of Proposition 24 can contribute to πq for q as in Lemma 23b. We distinguish two cases according to the parity of q. 2Note

that the element ∆ in [N] corresponds to ∆p−1 = ∆n in this paper.

On finite resolutions of K(n)-local spheres

161

1) If 0 ≤ q = 2q 0 < 2n then it is enough to show that there is no k with 0 < k ≤ n2 such that 2k(p2 − p − 1) ≡ 2q 0 mod 2p2 n . Calculating modulo 2pn gives −2k ≡ 2q 0 mod 2pn and because of 0 < 2k ≤ 2n2 and 0 ≤ 2q 0 < 2n this is clearly impossible. In 0,0 particular, we see that πq = 0 if q is even and 0 < q < 2n, and π0 ∼ = E2 . 2) If q = 2q 0 + 1 then we have to consider the congruence (12)

2pnl + 2k(p2 − p − 1) + 2p − 3 ≡ 2q 0 + 1 mod 2p2 n .

Reducing mod 2pn gives −2k + 2p − 3 ≡ 2q 0 + 1 mod 2pn and thus k ≡ p − 2 − q 0 mod pn. In view of 0 ≤ k ≤ p − 2 this implies that for (12) to have a solution we must have q 0 ∈ {0, 1, . . . , p − 2} modulo pn, i.e. q ∈ {1, 3, . . . , 2p − 3} modulo 2pn. Furthermore, for such a q there is a unique k with 0 ≤ k ≤ p − 2 and a unique l such that ∆l β k α is of total degree q. It remains to check that the elements ∆l β k α with 0 ≤ k ≤ p − 2 which are not permanent cycles cannot be of total degree q ≡ 2q 0 + 1 mod 2p2 n with q 0 ∈ {0, 1, . . . , p − 2}. In fact, not being a permanent cycle is equivalent to l ≡ 1 mod p. Calculating modulo 2p2 n gives 2pnl+2k(p2 −p−1)+2p−3−(2q 0 +1) = 2pn(l+k)+2(−k+p−2−q 0) ≡ 2pn(l+k) and this cannot be 0 modulo 2p2 n if l ≡ 1 mod p and 0 ≤ k ≤ p − 2.

We can finally state and prove the following realization result. Theorem 25. There is a resolution of length n (in the sense of 3.3.1) X• : ∗ → EnhN := X−1 → X0 → . . . → Xn → ∗ such that the complex E∗ (X• ) is isomorphic, as a complex of Morava modules, n to the complex Homcts (P• ↑G N , (En )∗ ) where P• is the complex of Proposition 21. Furthermore, for r > 0 we have _ 2 Xr ' Σ2p n(i1 +...+ir ) EnhF (i1 ,...,ir )

where F is the finite subgroup of N of Proposition 20d and where the wedge is taken over all sequences (i1 , . . . , ir ) with n − 1 ≥ i1 > i2 > . . . > ir ≥ 0. (For r = 0 there is a unique summand EnhF corresponding to the empty sequence.) Proof. Because of χ(pi) ∼ = χ(i) Lemma 22 implies that the spectra Xr realize n the Morava modules Homcts (Pr ↑G N , (En )∗ ). So it remains to realize the maps

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and show that the resulting sequence of maps of spectra can be refined into a resolution in the sense of 3.3.1. 2

Because the index of F2 in F is prime to p, we have, for each i, that Σ2p ni EnhF 2 is a direct wedge summand in Σ2p ni EnhF2 . Furthermore, it follows easily from 2 Proposition 24 that EnhF is periodic of period 2p2 n and thus Σ2p ni EnhF is, for each i, a direct wedge summand in EnhF2 . Therefore, in order to realize the maps it is enough to show that the (En )∗ -Hurewicz homomorphism (13) π0 (F (EnhF2 , EnhF2 )) → Hom(En )∗ [[Gn ]] (En )∗ (EnhF2 ), (En )∗ (EnhF2 ) is an isomorphism (onto the group of degree preserving homomorphisms). By Proposition 3 this happens iff the canonical map F2 (14) π0 (En [[Gn /F2 ]]hF2 ) → (En )0 [[Gn /F2 ]]

is an isomorphism.

Now T we choose a decreasing sequence Uj of open subgroups of Gn with F2 = j Uj . Then En [[Gn /F2 ]] ' holimj En [[Gn /Uj ]] .

By Proposition 2 we have En [[Gn /Uj ]]hF2 '

Y

EnhFx,j

F2 \Gn /Uj

where Fx,j is the isotropy group of the coset xUj with respect to the action of F2 on Gn /Uj . Because Z ⊂ F2 is central each Fx,j contains Z. Then Lemma hF F 23b implies that the canonical maps π0 (En x,j ) → (En )0 x,j are isomorphisms for all x and j and therefore F π0 (En [[Gn /Uj ]]hF2 ) → (En )0 [[Gn /Uj ]] 2

is an isomorphism for each j. Furthermore, it is clear that (En )1 [[Gn /Uj ]] = 0 for all j, and by the remark on the Mittag-Leffler condition following Proposition 2 the relevant lim1 -terms for the homotopy groups of the homotopy limit holimj (En )[[Gn /Uj ]]hF2 are also trivial. Therefore we obtain the desired isomorphism (14) by passing to the limit. We have now proved that all maps Xr → Xr+1 can be (uniquely) realized and the compositions of two successive maps are trivial. It remains to construct the factorizations Xr → Cr → Xr+1 , 0 ≤ r ≤ n − 1, such that Cr−1 → Xr → Cr is a cofibration. This will be done inductively. We note that these factorizations will realize the splitting of the exact complex of Morava modules E∗ (X• ) into the usual short exact sequences. In particular, this will show that Cn ' Xn so that the resolution will be automatically of length n.

On finite resolutions of K(n)-local spheres

163

For r = 0 (where we take C−1 = X−1 ) this is just a consequence of the fact that the composition X−1 → X0 → X1 is null. Now suppose that we have already constructed the factorizations Xr → Cr → Xr+1 , 0 ≤ r ≤ k < n − 1 . We need to show that the composition Ck → Xk+1 → Xk+2 is null so that we can factor it through the cofibre Ck+1 of the map Ck → Xk+1 . For this it is enough to show that the induced map [Ck , Xk+2 ] → [Xk , Xk+2 ] is injective. Now the inductively already constructed part of the resolution ∗ → X−1 → X0 → . . . → Xk → Ck → ∗ can be viewed as a tower of (co)fibrations for Ck which we can use to compute π0 (F (Ck , Xk+2 )). In fact, there is an Adams type spectral sequence associated to this tower which has the form E1p,q =⇒ πq−p (F (Ck , Xk+2 )) with E1p,q

( πq (F (Xk−p, Xk+2 )) ∼ = 0

if 0 ≤ p ≤ k if p > k .

We will use this spectral sequence to show that [Ck , Xk+2 ] ∼ = Ker([Xk , Xk+2 ] → [Xk−1 , Xk+2 ]) thus finishing off the proof. We note that in terms of the spectral sequence this claim says that π0 (F (Ck , Xk+2 )) is isomorphic to the kernel of the differential d1 : E10,0 → E11,0 . It is therefore enough to show that E2q,q = 0 = E2q+1,q for q > 0. Now Proposition 2 (including the remark on the Mittag Leffler condition following it) together with Lemma 23 implies already E1q,q = 0 = E1q+1,q if q > 0 and q is even. W 2 Now let q > 0 be odd. We let Yk+2 := (i1 ,...,ik+2 ) Σ2p n(i1 +...+ik+2 ) En so that Xk+2 = (Yk+2 )hF . We claim that for p = q ≤ k and p = q + 1 ≤ k there are natural isomorphisms E1p,q ∼ = πq (F (Xk−p , (Yk+2 )hF )) ∼ = πq (F (Xk−p, Yk+2 )hF ) ∼ = H s(q),s(q)+q (F, Hom(En )∗ [[Gn ]] (En )∗ (Xk−p ), (En )∗ (Yk+2 ) ∼ = H s(q),s(q)+q F, Hom(E ) Homcts (Pk−p ↑Gn , (En )∗ ), π∗ (Yk+2 ) n ∗

N

where s(q) is as in Lemma 23 and the F -module structures in line 2 and 3 come from the action of F on Yk+2 . If we accept these isomorphisms for the

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n moment then we can finish off the proof because P• ↑G N is split exact as a complex of continuous Zp -modules, and this clearly implies that E2p,q = 0 if p > 0.

It remains to justify the chain of isomorphisms which identify E1p,q . The first two of the claimed isomorphisms are obvious and the last one holds because by 1.3.2 (En )∗ (Yr ) is a coinduced module, i.e. (En )∗ (Yr ) ∼ = Homcts (Zp [[Gn ]], π∗ (Yr )) . To get the third isomorphism it suffices to show that the homotopy fixed point spectral sequence gives, for q odd and 0 < q ≤ k < n, an isomorphism πq (F (EnhF , En )hF ) ∼ = H s(q),s(q)+q F, Hom(En )∗ [[Gn ]] (En )∗ (EnhF ), (En )∗ (En ) . In fact, by Proposition 3 we can identify Hom(En )∗ [[Gn ]] (En )∗ (EnhK ), (En )∗ (En ) with π∗ (F (EnhK , En )) whenever K is a closed subgroup of Gn . Now we replace first EnhF in the source by EnhU where U is an open subgroup of Gn . Then Proposition 2 together with Lemma 23 give the required identification. Finally we write EnhF ' LK(n) hocolimj (En )hUj and pass to the limit once more using the remark on the Mittag-Leffler condition following Proposition 2. 3.6.5. The resolution of LK(n) S 0 . It remains to construct a resolution of LK(n) S 0 . For this we start with an F (Gn )-resolution Q• : 0 → Qm → . . . → Q0 → Q−1 = Zp → 0 of finite length m of the trivial Zp [[Gn ]]-module Zp as given by Proposition 17 and Proposition 21. The modules Qr in this resolution are finitely generated 0 n projectives if r > n, and of the form Q0r ⊕ Pr ↑G N with Qr finitely generated projective and Pr as in Proposition 21, if r ≤ n. We can realize these modules by spectra Zr which are summands in a finite wedge of En ’s if r > n, and of the form Zr0 ∨ Xr with Zr0 a summand in a finite wedge of En ’s and Xr as in Theorem 25. Then the strategy of the proof of Theorem 25 can be applied to this situation. The Zp [[Gn ]]-linear maps Qr → Qr−1 can be uniquely realized by maps Zr−1 → Zr of spectra and the resulting sequence of spectra is a resolution of finite length in the sense of 3.3.1. In fact, the factorizations Zr → Cr → Zr+1 can be constructed just as in the proof of Theorem 25, if 0 ≤ r ≤ n, resp. as in the proof of Theorem 4, if r > n, by using that Q• is split exact as a complex of continuous Zp -modules. Theorem 26. Let p be odd and n = p − 1. Suppose Q• is an F (Gn )-resolution of length m of the trivial Zp [[Gn ]]-module Zp such that Qr is a finitely generated 0 n projective Zp [[Gn ]]-module if r > n while Qr ∼ = Q0r ⊕ Pr ↑G N with Qr finitely generated projective and Pr as in Theorem 25 resp. Proposition 21 if 0 ≤ r ≤ n.

On finite resolutions of K(n)-local spheres

165

Then there is a resolution of length m (in the sense of 3.3.1) Z• : ∗ → LK(n) S 0 := Z−1 → Z0 → . . . → Zn → . . . → Zm → ∗ such that the complex E∗ (Z• ) is isomorphic as a complex of Morava modules to the complex Homcts (Q• , (En )∗ ). Furthermore, Zr is a direct summand in a finite wedge of En ’s if r > n while for 0 ≤ r ≤ n _ 2 Zr ' Zr0 ∨ Σ2p n(i1 +...+ir ) EnhF (i1 ,...,ir )

where the wedge is taken over all sequences (i1 , . . . , ir ) with n − 1 ≥ i1 > i2 > . . . > ir ≥ 0 and Zr0 is a direct summand in a finite wedge of En ’s. 3.6.6. Proof of Theorem 10 and Theorem 11. We note that for p = 3 and n = 2 the group F of Theorem 26 (see also 3.6.4) is equal to the group G24 of Theorem 10. Likewise, the character χ(1) of Proposition 21 is equal to the character χ e of G24 . By Theorem 26 and the splitting on En discussed in the appendix below it is therefore enough to prove Theorem 10 and for this we 2 only need to show that the kernel K of the augmentation Z3 ↑G N → Z3 admits a projective resolution of the form G1

G1

G1

G1

2 2 2 2 →K→0. → χ ↑SD → (λ2 ⊕ χ) ↑SD → (Z3 ⊕ λ2 ) ↑SD 0 → Z3 ↑SD 16 16 16 16

In fact, because of the isomorphism G2 ∼ = G12 ×Z3 (cf. [GHMR1]) it is enough to G1 show that the kernel K 1 of the augmentation Z3 ↑N21 → Z3 with N 1 = NG12 (E) admits a projective resolution of the form G1

G1

G1

2 2 2 → K1 → 0 . → χ ↑SD → λ2 ↑SD 0 → Z3 ↑SD 16 16 16

From [He] we know that ExtiZ3 [[S 1 ]] (K 1 , F3 ) can be identified with the cokernel 2 of the map 1 Y ∗ 1 H (S2 ; F3 ) → H ∗ (ω i CS21 (E)ω −i ; F3 ) i=0

given by the inclusions ω CS21 (E)ω −i → S21 , i = 0, 1. Furthermore, this map is Z3 [SD16 ]-linear and from the explicit description of this map in Theorem 4.4 of [GHMR1] it is straightforward to see that, as Z3 [SD16 ]-modules, we obtain χ ⊗ Z 3 F3 if i = 0 λ ⊗ F if i = 1 2 Z3 3 ExtiZ3 [[S1 ]] (K 1 , F3 ) ∼ = 2 Z3 ⊗Z3 F3 if i = 2 0 if i > 2 . i

The resolution of K 1 is then constructed as in the proof of Theorem 6.

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Appendix: Splitting En with respect to the action of Fn(q−1) A.1. Let p be an odd prime, q = pn and F := Fn(q−1) = Cq−1 o Gal(Fq /Fp ) be as in section 2.3. The homotopy groups of En can be described as π∗ (En ) ∼ = WFq [v1 , . . . , vn−1 ][u±1 ]b, i

i

with vi := ui u1−p , i.e. ui = vi up −1 , and where b means completion with respect to the ideal (u1 , . . . , un−1 ). The elements vi are invariant with respect to the action of F and the action of a generator ω of the cyclic subgroup Cq−1 on u is given by multiplication with ω, i.e. ω∗ (u) = ωu (cf. formula (8) in the proof of Lemma 22), while the Galois group acts on the coefficients WFq only via Frobenius. In particular, the homotopy groups of the homotopy fixed points are given by the appropriate completion π∗ (EnhF ) ∼ = Zp [v1 , . . . , vn−1 ][vn±1 ]b, with vn = u−q . Furthermore we have an isomorphism of π∗ (EnhF )[F ]-modules (15)

π∗ (En ) ∼ =

q−1 M

π∗ (EnhF ) ⊗ WFq ui

i=0

and the action on the modules λi := WFq ui is as described above. We are interested in the decomposition of En obtained from a splitting of the group algebra Zp [F ] as a module over itself. The decomposition suggested by (15) is close but not quite equal to the decomposition obtained from such a splitting. A.2. From now on we will restrict attention to the case n = 2 and p odd which we have used in section 2.3, section 3.3 and 3.3.6. If i is divisible by p + 1 then ω i belongs to Zp and therefore the action of ω on λi commutes with the action of Frobenius. Therefore, for such i, λi splits into 2 one-dimensional pieces λi,+ and λi,− , with Frobenius acting trivially on λi,+ and by multiplication by −1 on λi,− . We claim that there is a direct sum decomposition of Zp [F ] into a direct sum of Zp [F ]-modules M M (16) Zp [F ] ∼ λi ⊕ λi,+ ⊕ λi,− . = i6≡0(p+1)

i≡0(p+1)

We leave it to the reader to check that the modules on the right hand side of this isomorphism are all indecomposable and that the only repetition in the decomposition comes from the isomorphisms λi ∼ = λpi , i 6≡ 0 mod p + 1, which are induced by Frobenius.

On finite resolutions of K(n)-local spheres

167

At the request of the referee we outline a direct construction of the decomposition given in (16): our identification of the roots of unity of WFq with Cq−1 specifies a character χ1 of Cq−1 defined over WFq . Let χi = χ1 ⊗i . Then the elements X 1 ei = χi (g −1 )g (q − 1) g∈C q−1

belong to WFq [Cq−1 ] and they are easily checked to be orthogonal idempotents which sum up to the element 1 ∈ WFq [Cq−1 ]. The elements ei form a basis of WFq [Cq−1 ] as a WFq -module and each ei generates a one-dimensional representation over WFq on which Cq−1 acts via χi . For i ≡ 0 mod (p + 1) the element ei lives already in Zp [Cq−1 ], while for p+1 i 6≡ 0 mod (p + 1) the elements ei + epi and ω 2 (ei − epi ) belong to Zp [Cq−1 ] and together they form a basis of Zp [Cq−1 ] as a Zp -module. Furthermore, the Zp -module generated by ei , i ≡ 0 mod (p + 1), is a Zp [Cq−1 ]-module on which Cq−1 acts via χi , and the Zp -submodule generated by δi := ei + epi p+1 and εi := ω 2 (ei − epi ) is also a Zp [Cq−1 ]-module which is isomorphic to λi restricted to Cq−1 . Now consider the isomorphism of Zp [Cq−1 ]-modules Zp [F ] ∼ = Zp [Cq−1 ] ⊕ Zp [Cq−1 ]σ where σ is Frobenius considered as an element of F . Then there is a Zp -basis of Zp [F ] given by ei ± σei , if i ≡ 0 mod(p + 1), and δi ± δi σ, εi ± εi σ if i 6≡ 0 mod(p + 1). In WFq [F ] we have σei = epi σ, in particular σei = ei σ if i ≡ 0 mod(p + 1), and therefore σ(ei ± ei σ) = ±(ei ± ei σ) if i ≡ 0 mod(p + 1) i.e. (ei ± ei σ) generates a direct summand isomorphic to λi,± . Furthermore we have σδi = δi σ, σεi = −εi σ which shows that the Zp -submodule generated by δi + δi σ and εi + εi σ is a Zp [F ]-module, and likewise the Zp -submodule generated by δi − δi σ and εi − εi σ is a Zp [F ]-module. Both of these modules are isomorphic to λi (or λpi ). A.3. Consequently, by the elementary theory of stable splittings we find that E2 splits, with respect to the F - action, into a direct sum of E2hF -module spectra whose homotopy groups are given by HomZp [F ] (λi,± ,

q−1 M

WFq uj ) ⊗ π∗ (E2hF ) ∼ = (WFq ui )± ⊗ π∗ (E2hF )

j=0

resp. HomZp [F ](λi ,

q−1 M j=0

WFq uj ) ⊗ π∗ (E2hF ) ∼ = Zp ui ⊕ Zp upi ⊗ π∗ (E2hF )

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where (WFq ui )± is the ±-eigenspace of the action of Frobenius on WFq ui . In the first case, the corresponding summand of E2 can be identified as E2hF module spectrum with Σ2i E2hF , in the second case with Σ2i E2hF ∨ Σ2pi E2hF and the multiplicity of this spectrum in a splitting of E2 (constructed via the action of F ) is 2.

References [B]

A.K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257-281 [DH1] E. Devinatz and M. Hopkins, The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts, Amer. J. Math. 117 (1995), 669–710 [DH2] E. Devinatz and M. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004), 1–48 [DW] W.G. Dwyer and C.W. Wilkerson Smith theory and the functor T , Comment. Math. Helvetici 66 (1991), 1–17 [EM] S. Eilenberg and J.C. Moore, Foundations of relative homological algebra, Memoirs of the Amer. Math. Soc. 55 (1965) [EKMM] A.D. Elmendorf, I. Kriz, M.A. Mandell and J.P. May, Rings, modules and algebras in stable homotopy theory, Amer. Math. Soc. Surveys and Monographs 47 (1996) [GHM] P. Goerss, H.-W. Henn and M. Mahowald, The homotopy of L2 V (1) for the prime 3, Categorical Decomposition Techniques in Algebraic Topology. Progress in Mathematics 215 (2003), 125–151 [GHMR1] P. Goerss, H.-W. Henn, M. Mahowald and C. Rezk, A resolution of the K(2)-local sphere, Annals of Mathematics 162 (2005), 777–822 [GHMR2] P. Goerss, H.-W. Henn, M. Mahowald and C. Rezk, in preparation [Ha] M. Hazewinkel, Formal groups and applications, Academic Press 1978 [He] H.-W. Henn, Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups, Duke Math. J. 91 (1998), 561–585 [HMS] M. Hopkins, M. Mahowald and H. Sadofsky, Constructions of elements in Picard groups, Topology and Representation Theory, Contemp. Math. 158 (1994), 89– 126 [Ho] M. Hovey, Bousfield localization functors and Hopkins’ chromatic splitting conˇ jecture, The Cech centennial (Boston, MA, 1993), Contemp. Math. 181 (1995), 225–250 [HS] M. Hovey and N. Strickland, Morava K-theories and Localisation, Memoirs of the American Mathematical Society 666 (1999) [La] M. Lazard, Groupes p-adiques analytiques, Inst. Hautes Etudes Sci. Publ. Math. 26 (1965), 389–603 [LT] J. Lubin and J. Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966), 49–60 [Mi] H. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, Journal of Pure and Applied Algebra 20, (1981), 287-312 [MS] Ph. A. Minh and P. Symonds, Cohomology and finite subgroups of profinite groups, Proc. Amer. Math. Soc. 132 (2004), 1581-1588 [Mo] J. Morava, Noetherian localizations of categories of cobordism comodules, Annals of Mathematics 121 (1985), 1-39

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L. Nave, On the nonexistence of Smith-Toda complexes, preprint, available at http://hopf.math.purdue.edu D. Quillen, The spectrum of an equivariant cohomology ring I,II, Annals of Mathematics 94 (1974), 549–572, 573–602 D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press 1986 D. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Studies, Princeton University Press 1992 L. Ribes and P. Zaleskii, Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 40, Springer Verlag 2000 C. Rezk, Notes on the Hopkins-Miller theorem, Homotopy theory via algebraic geometry and group representations. Contemp. Math. 220 (1998), 313–366 J.P. Serre, Cohomologie des groupes discrets, Annals of Math. Studies 70 (1971), 77–169 K. Shimomura, The homotopy groups of the L2 -localized Toda-Smith complex V (1) at the prime 3, Trans. Amer. Math. Soc. 349 (1997), 1821–1850 K. Shimomura and X. Wang, The homotopy groups π∗ (L2 S 0 ) at the prime 3, Topology 41 (2002), 1183–1198 P. Symonds and T. Weigel, Cohomology of p-adic analytic groups, New Horizons in pro-p groups, Progress in Math. 184 (2000), 349–410 A. Yamaguchi, The structure of the cohomology of Morava stabilizer algebra S(3), Osaka Journal Math. 29 (1992), 347–359

´matique Avanc´ Institut de Recherche Mathe ee, C.N.R.S. - Universit´ e Louis Pasteur, 7 rue Ren´ e Descartes, F-67084 Strasbourg, France

CHROMATIC PHENOMENA IN THE ALGEBRA OF BP∗ BP -COMODULES MARK HOVEY Abstract. We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗ BP comodules. We show, following [HS03a], that the category of E(n)∗ E(n)comodules is a localization, in the abelian sense, of the category of BP∗ BP comodules. This gives analogues of the usual structure theorems, such as the Landweber filtration theorem, for E(n)∗ E(n)-comodules. We recall the work of [Hov02a], where an improved version Stable(Γ) of the derived category of comodules over a well-behaved Hopf algebroid (A, Γ) is constructed. The main new result of the paper is that Stable(E(n)∗ E(n)) is a Bousfield localization of Stable(BP∗ BP ), in analogy to the abelian case.

Introduction The object of this paper is to describe some of the author’s recent work, much of it joint with Neil Strickland, on comodules over BP∗ BP and related Hopf algebroids. The basic idea of this work is to realize the chromatic approach to stable homotopy theory in the algebraic world of comodules. This means, in particular, constructing and understanding the localization functor Ln , or the associated finite localization functor Lfn , in the abelian category BP∗ BP -comod of graded BP∗ BP -comodules. It also means constructing Ln and Lfn in some kind of associated derived category Stable(BP∗ BP ) of chain complexes of BP∗ BP -comodules. Ultimately, we would like to understand LK(n) in the algebraic setting as well, and this should involve the Morava stabilizer groups. In the present paper, we confine ourselves to Ln and Lfn . Topologically, Lfn is localization away from a finite spectrum of type n + 1, and Ln is localization with respect to the homology theory E(n). These functors are probably different in the ordinary stable homotopy category (because Ravenel’s telescope conjecture [Rav84] is widely expected to be false), but they turn out to agree on the abelian category of BP∗ BP -comodules. We then have the following theorem. Theorem A. Let Ln denote localization away from BP∗ /In+1 in the category of BP∗ BP -comodules. Then there is an equivalence of categories between E(n)∗ E(n)-comodules and Ln -local BP∗ BP -comodules. Date: October 26, 2006.

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Chromatic phenomena in the algebra of BP∗ BP -comodules

171

Note that this is localization in an abelian sense. The localization Ln turns out to be the localization functor that inverts all maps whose kernel and cokernel are vn -torsion. This theorem is really a special case of a more general theorem proved in [HS03a]. In general, if (A, Γ) is a flat Hopf algebroid and B is a Landweber exact A-algebra, there is an induced flat Hopf algebroid (B, ΓB ). We prove in [HS03a] that, in this situation, the category of ΓB -comodules is always equivalent to some localization of the category of Γ-comodules. In the case of BP∗ BP , we give a partial classification of such localizations. In particular, E(n)∗ E(n) can be replaced by E∗ E for any Landweber exact commutative ring spectrum E with E∗ /In 6= 0 and E∗ /In+1 = 0. This tells us that all such theories E have equivalent categories of E∗ E-comodules, even though the categories of E∗ -modules can be drastically different. It also leads to structural results about E∗ E-comodules analogous to those of Landweber [Lan76] for BP∗ BP -comodules. To extend this result to the derived category setting, we first must decide what we mean by the derived category. The ordinary derived category, obtained by inverting homology isomorphisms, is usually badly behaved. For example, the analogue of S 0 in the derived category of BP∗ BP -comodules is BP∗ thought of as a complex concentrated in degree 0, but this is not a small object (see the introduction to Section 3). The following is a corollary of the main result of [Hov02a]. Theorem B. Suppose E is a commutative ring spectrum that is Landweber exact over BP . Then there is a bigraded monogenic stable homotopy category Stable(E∗ E) such that π∗∗ S 0 ∼ = Ext∗∗ E∗ E (E∗ , E∗ ). In particular, the sphere, which is E∗ concentrated in degree 0, is a small object of Stable(E∗ E). This stable homotopy category is the homotopy category of a model structure on chain complexes of E∗ E-comodules. The weak equivalences are homotopy isomorphisms, where homotopy is suitably defined. Every cofibrant object is dimensionwise projective over E∗ and every complex of relatively injective comodules is fibrant. Just as in the ordinary stable homotopy category, there exist interesting nontrivial complexes with no homology. On Stable(BP∗ BP ), we define Lfn to be finite localization away from BP∗ /In+1 , which turns out to be a small object in Stable(BP∗ BP ). We define Ln to be Bousfield localization with respect to the homology theory corresponding to E(n)∗ . In Stable(BP∗ BP ), the homology functor H is represented by BP∗ BP , so is somewhat analogous to the BP -homology of a spectrum. Then the homology theory corresponding to E(n)∗ is in fact HE(n), ordinary homology with coefficients in E(n)∗ . Because of this, some of the things one would expect to be true about Ln are false. For example, the two localizations

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Lfn and Ln on Stable(BP∗ BP ) are definitely different in general, so the most naive version of the telescope conjecture is false in Stable(BP∗ BP ). Also, Ln is not a smashing localization, though Lfn is. Thus Ln is less important in Stable(BP∗ BP ) then Lfn . These statements are also true in the category Stable(E(n)∗ E(n)), where f Ln is the identity functor (as it is localization away from E(n)∗ /In+1 = 0), and Ln is localization with respect to ordinary homology (which already has coefficients in E(n)∗ ). Thus Ln Stable(E(n)∗ E(n)) is the classical unbounded derived category of E(n)∗ E(n)-comodules. The main new result of this paper is then the following theorem. Theorem C. There is an equivalence of stable homotopy categories between the localization Lfn Stable(BP∗ BP ) and Stable(E(n)∗ E(n)). As above in Theorem A, we can replace E(n) in Theorem C by any Landweber exact commutative ring spectrum E with E∗ /In 6= 0 and E∗ /In+1 = 0. Theorem C also implies a similar equivalence between Ln Stable(BP∗ BP ) and Ln Stable(E(n)∗ E(n)). From a computational point of view, Theorem C gives rise to the following change of rings theorem. Theorem D. Suppose that M is a finitely presented BP∗ BP -comodule and N is a BP∗ BP -comodule such that N = Ln N and the right derived functors Lin N are 0 for i > 0. Then there is a change of rings isomorphism Ext∗∗ (M, N ) ∼ = ExtE(n) E(n) (E(n)∗ ⊗BP M, E(n)∗ ⊗BP N ). BP∗ BP

∗

∗

∗

This change of rings theorem includes the Miller-Ravenel change of rings theorem [MR77] and the change of rings theorem of the author and Sadofsky [HS99] as special cases. Also, E(n) can be replaced by any Landweber exact commutative ring spectrum E with E∗ /In 6= 0 and E∗ /In+1 = 0. Because the basic structure of the abelian category of comodules over a flat Hopf algebroid is not as well known as it should be, we first summarize this in Section 1. The results in this section were mostly proved in [Hov02a]. We then describe our proof of Theorem A and related results about E(n)∗ E(n)-comodules in Section 2. Further details can be found in [HS03a]. We introduce the stable homotopy category of comodules in Section 3, describing the proof of Theorem B and looking at some particular features of Stable(BP∗ BP ) and Stable(E(n)∗ E(n)). Some of the material in this section can be found in [Hov02a], but some of it is new. We discuss the relation between Stable(BP∗ BP ) and Stable(E(n)∗ E(n)) in Section 4, where we prove Theorems C and D. All of the results in this section are new. It is a pleasure to thank my coauthor Neil Strickland. Many of the theorems in this paper are joint work with him, much of it done at the Isaac Newton Institute for Mathematical Sciences in Fall 2002. I would like to thank the Universitat de Barcelona, the Universitat Aut`onoma de Barcelona, the Centre de Recerca Matematica, and the Isaac Newton Institute for Mathematical

Chromatic phenomena in the algebra of BP∗ BP -comodules

173

Sciences for their support during our collaboration. I would also like to thank John Greenlees for many discussions about the material in this paper. 1. Comodules The object of this section is to give an overview of the structural properties of the category Γ-comod. We begin by recalling the structure maps of a Hopf algebroid (A, Γ). A Hopf algebroid (A, Γ) has the following structure maps, which are all maps of commutative rings. • • • •

The counit : Γ → − A, corepresenting the identity map of an object. The left unit ηL : A → − Γ, corepresenting the source of a morphism. The right unit ηR : A → − Γ, corepresenting the target of a morphism. The diagonal ∆ : Γ → − Γ ⊗A Γ, corepresenting the composite of two composable morphisms. Note that this is a tensor product of Abimodules, with the left A-module structure given by ηL and the right A-module structure given by ηR . • The conjugation χ : Γ → − Γ, corepresenting the inverse of a morphism.

There are many relations between these structure maps, but they are all easily obtained from corresponding facts about groupoids. For example, the source and target of the identity morphism at x are both x, so ηL = ηR = 1. e where Note that the conjugation is best thought of as a map χ : Γ → − Γ, e denotes Γ with the opposite A-bimodule structure, so that A acts on the Γ e by ηR . Similarly, the multiplication map is best thought of as left on Γ e→ e ⊗A Γ. µ : Γ ⊗A Γ − Γ, although we could also think of it as having domain Γ With this convention, the fact that composition of a map with its inverse is the identity gives the following commutative diagram, 1⊗χ

∆

e Γ −−−→ Γ ⊗A Γ −−−→ Γ ⊗A Γ

µ

y

Γ −−−→

A

−−−→

Γ

ηL

and a similar diagram involving χ ⊗ 1 and ηR . This is a great deal simpler than the corresponding diagram in [Rav86, Definition A1.1.1(f)]. We recall that a Γ-comodule is a left A-module M together with a counital and coassociative coaction map ψ : M → − Γ ⊗A M of left A-modules, where again Γ is an A-bimodule. A map of comodules is a map of A-modules that preserves the coaction, so we get a category Γ-comod of Γ-comodules. We then have the following proposition [Rav86, Proposition 2.2.8], which explains the importance of Hopf algebroids and comodules in algebraic topology.

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Proposition 1.1. Suppose E is a ring spectrum such that E∗ E is a commutative ring that is flat over E∗ . Then (E∗ , E∗ E) is a Hopf algebroid, and E∗ X is naturally an E∗ E-comodule for X a spectrum. Other good examples of Hopf algebroids include Hopf algebras, which are just Hopf algebroids where ηL = ηR . In particular, a commutative ring A can be thought of as the discrete Hopf algebroid (A, A); a comodule over a discrete Hopf algebroid is just an A-module. Also, if G acts on a commutative ring R by ring automorphisms, the ring of R-valued functions on G is a Hopf algebroid. The right unit is defined by ηR (r)(g) = g(r). This is dual to the twisted group ring R[G]. We will summarize the properties of Γ-comod in the following theorem, but for it to make sense we need to recall some definitions. Definition 1.2. (a) A Hopf algebroid (A, Γ) is flat if ηR makes Γ into a flat A-module. The conjugation shows that it is equivalent to assume ηL is flat. Since is a right inverse for both ηL and ηR , both of these maps are then faithfully flat. (b) A category is complete if it has all small limits, and cocomplete if it has all small colimits. It is bicomplete if it is both complete and cocomplete. (c) Given a regular cardinal λ, a category I is said to be λ-filtered if every subcategory J of I with fewer than λ morphisms has an upper bound in I; that is, there is an object C in I and a natural transformation from the inclusion functor J → − I to the constant J -diagram at C. An object M of a cocomplete category C is said to be λ-presented if C(M.−) commutes with λ-filtered colimits. (d) Suppose C is a closed symmetric monoidal category with monoidal structure X∧Y , unit A, and closed structure F (X, Y ). An object M in C is said to be dualizable if the natural map F (M, A)∧X → − F (M, X) is an isomorphism for all X ∈ C. An object M is said to be invertible if there is an N and an isomorphism M ∧ N ∼ = A. Theorem 1.3. Suppose (A, Γ) is a flat Hopf algebroid. Then Γ-comod is a bicomplete, closed symmetric monoidal abelian category. We also have: (a) Filtered colimits are exact. (b) Given a regular cardinal λ and a comodule M, M is λ-presented if and only if M is λ-presented as an A-module. (c) For any comodule M, there is a cardinal λ such that M is λ-presented. (d) A comodule M is dualizable if and only if it is projective and finitely generated over A. (e) A comodule M is invertible under the symmetric monoidal product if and only if it is invertible as an A-module. We denote the symmetric monoidal structure by M ∧ N with unit A and the closed structure by F (M, N ).

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This theorem is a summary of the results of [Hov02a, Section 1]. We will just discuss some of the issues that arise. First of all, left adjoints are generally easy to construct, since the forgetful functor from Γ-comod to A-mod is itself a left adjoint (its right adjoint is the extended comodule functor discussed in the following paragraph). Thus, one generally forms the left adjoint in A-mod and notices that it has a natural comodule structure. This is true for colimits and for the symmetric monoidal structure M ∧ N . This is defined to be M ⊗A N , the tensor product of left A-modules, with the coaction given as the composite ψ⊗ψ

g

M ⊗A N −−→ (Γ ⊗A M) ⊗A (Γ ⊗A N ) → − Γ ⊗ A M ⊗A N where g(x ⊗ m ⊗ y ⊗ n) = xy ⊗ m ⊗ n. The key to constructing right adjoints is the extended comodule functor from A-mod to Γ-comod that takes M to Γ ⊗A M, with coaction ∆ ⊗ 1. This is the right adjoint to the forgetful functor. As such, it is generally easy to define a desired right adjoint R on extended comodules. For example, one can easily see that we must define the product of extended comodules by Γ Y

(Γ ⊗A Mi ) ∼ = Γ ⊗A

Y

Mi ,

and the closed structure with target an extended comodule by F (M, Γ ⊗A N ) ∼ = Γ ⊗A HomA (M, N ). It is less easy to see how one defines these right adjoints on maps between extended comodules that are not necessarily extended maps, but this can generally be done. Having done this, we use the exact sequence of comodules ψ

ψg

0→ − M− → Γ ⊗A M −→ Γ ⊗A N, where g : Γ⊗A M → − N is the cokernel of ψ, to define RM = ker R(ψg). Since R is supposed to be a right adjoint, it must be left exact, so we must define R in this way. Note that if M is an A-module and N is a Γ-comodule, we have the two tensor products Γ ⊗A (M ⊗A N ) and (Γ ⊗A M) ∧ N . It is useful to know that these are the same [Hov02a]. Lemma 1.4. Suppose (A, Γ) is a flat Hopf algebroid, M is an A-module and N is a Γ-comodule. Then there is a natural isomorphism of comodules (Γ ⊗A M) ∧ N → − Γ ⊗A (M ⊗A N ). Although Theorem 1.3 indicates that the category of Γ-comodules is a very well-behaved abelian category, one obvious property is missing, and that is the existence of a set of generators. Recall that a set of objects G is said to generate an abelian category C if, whenever f is a nonzero map in C, there exists an object G ∈ G such that C(G, f ) is also nonzero. For example, A is a generator of A-mod. This issue of generators is already complicated for Hopf

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algebras; for a finite group G and a field k, the natural generators for the category of k[G]-modules (which is isomorphic to the category of comodules over the ring of k-valued functions on G) are the simple k[G]-modules. There is no canonical description of these in general. However, any simple k[G]module is finitely generated, and of course projective, over k. Referring to part (d) of Theorem 1.3, one might then expect that the set of isomorphism classes of dualizable comodules forms a set of generators for Γ-comod. Sadly, this appears to be false in general, so we need a hypothesis. Definition 1.5. A Hopf algebroid (A, Γ) is called an Adams Hopf algebroid when Γ is a filtered colimit of dualizable comodules. This hypothesis is really due to Adams [Ada74, Section III.13], who used it for the Hopf algebroid (E∗ , E∗ E) to set up universal coefficient spectral sequences. We learned it from [GH00], as well as the following lemma. Lemma 1.6. Suppose (A, Γ) is an Adams Hopf algebroid. Then it is flat, and the dualizable comodules generate the category of Γ-comodules. In categorical language, the category of comodules over an Adams Hopf algebroid is a locally finitely presentable Grothendieck category. All of the Hopf algebroids that commonly arise in algebraic topology, as well as all Hopf algebras over fields, are known to be Adams [Hov02a, Section 1.4]. Note that it may be that one can take smaller sets of generators than all of the dualizable comodules. For example, the set {BP∗ Xn } will serve as a set of generators for BP∗ BP -comodules, where Xn is the 2n-skeleton of BP . However, BP∗ by itself is definitely not a generator for the category of BP∗ BP -comodules. To see this, let P M be the set of primitives in a BP∗ BP comodule M. Then one can easily check that if BP∗ is a generator of the category of BP∗ BP -comodules then any comodule map that is surjective after applying P is in fact surjective. In particular, the map M f Σ|x| BP∗ → − M x∈P M

would be surjective. This is easily seen to be false for M = BP∗ (CP 2 ), for example. Remark 1.7. Theorem 1.3 lists the good points of the category Γ-comod; we now list some of the bad points of Γ-comod. (a) The forgetful functor from Γ-comod to A-modules, or even down to abelian groups, does NOT have a left adjoint; there is no free comodule functor. (b) Γ-comod does not, in general, have enough projectives. If we take (A, Γ) = (Fp , A), where A denotes the dual Steenrod algebra, it is generally believed that there are no nonzero projective comodules.

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(c) If (A, Γ) is not a Hopf algebra, that is if ηL and ηR are not equal, then the forgetful functor from Γ-comod to A-mod is not in general surjective on objects. For example, there is no BP∗ BP -comodule structure on vn−1 BP∗ for n > 0 [JY80, Proposition 2.9]. (d) Products are not in general exact. Hence the inverse limit functor limΓ on sequences ··· → − Mn → − ··· → − M1 → − M0 may have nonzero derived functors limiΓ for all i > 0. Because there are not enough projective comodules, the homological algebra of comodules always involves injectives, or, better, relative injectives. Because the forgetful functor is exact, if I is an injective A-module, then Γ ⊗A I is an injective Γ-comodule. From this it is easy to check that there are enough injectives. However, injective A-modules are complicated, whereas relative injectives have much better properties. A comodule I is defined to be relatively injective if Γ-comod(−, I) takes A-split short exact sequences of comodules to short exact sequences. The following proposition sums up the properties of relative injectives and is wellknown; details can be found in [Hov02a, Section 3.1]. Proposition 1.8. Suppose (A, Γ) is a flat Hopf algebroid. (a) The relatively injective Γ-comodules are the retracts of extended comodules. (b) The coaction ψ : M → − Γ ⊗A M defines an A-split embedding of M into a relatively injective comodule. (c) Relatively injective comodules are closed under coproducts and products. (d) If I is relatively injective, so is I ∧ M and F (M, I) for all comodules I. (e) If P is a comodule that is projective over A, and I is relatively injective, then ExtnΓ (P, I) = 0 for all n > 0. We take this proposition to mean that, to understand Ext∗Γ (M, N ), we must simultaneously resolve M by comodules that are projective over A and N by relative injectives. We return to this point in Section 3. We close this section with a brief description of naturality. There is, of course, a natural notion of a map Φ : (A, Γ) → − (B, Σ) of Hopf algebroids. The map Φ corepresents a natural functor of groupoids, so consists of ring maps Φ0 : A → − B and Φ1 : Γ → − Σ satisfying certain conditions. Such a map induces a symmetric monoidal functor Φ∗ : Γ-comod → − Σ-comod that takes M to B ⊗A M, with comodule structure given by the composite 1⊗ψ g⊗1 B ⊗A M −−→ B ⊗A Γ ⊗A M −−→ Σ ⊗A M ∼ = Σ ⊗B (B ⊗A M)

where g(b ⊗ x) = bΦ1 (x). It is clear that Φ∗ preserves colimits, so should have a right adjoint Φ∗ . It does, but, as usual, Φ∗ is hard to define. We define

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Φ∗ (Σ ⊗B M) = Γ ⊗A M, and then extend this definition to all Σ-comodules in the same way we did for the product of comodules. An important new feature that arises in the study of Hopf algebroids is the notion of weak equivalence. Definition 1.9. A map Φ of Hopf algebroids is defined to be a weak equivalence if Φ∗ is an equivalence of categories. If Φ is a weak equivalence between discrete Hopf algebroids, then Φ is an isomorphism, but a central point of the author’s work on Hopf algebroids is that there are important non-trivial weak equivalences of Hopf algebroids that are not discrete. In general, we have the following characterization of weak equivalences. Theorem 1.10. The map Φ of Hopf algebroids is a weak equivalence if and only if the composite ηR Φ ⊗1 A −→ Γ −−0−→ B ⊗A Γ is a faithfully flat ring extension and the map B ⊗ A Γ ⊗A B → − Σ that takes b ⊗ x ⊗ b0 to ηL (b)Φ1 (x)ηR (b0 ) is a ring isomorphism. The “if”half of this theorem is the main result of [Hov02b]. The “only if” half is much easier and was proven in [HS03a]. This theorem has a better formulation. Hollander [Hol01] has constructed a model structure on presheaves of groupoids on a Grothendieck topology C; the fibrant objects are stacks; see also [Jar01]. In particular, we can take our Grothendieck site to be the flat topology on Aff, the opposite category of commutative rings (with a cardinality bound so we get a small category). In this topology, a cover of R is a finite collection of flat extensions Si of R Q such that Si is faithfully flat over R. Any Hopf algebroid (A, Γ) defines a presheaf of groupoids Spec(A, Γ) by definition; this presheaf is in fact a sheaf in the flat topology by faithfully flat descent [Hov02b]. We can then rephrase Theorem 1.10 as follows. Corollary 1.11. A map Φ of Hopf algebroids is a weak equivalence if and only if Spec Φ is a weak equivalence of sheaves of groupoids in the flat topology. This point of view suggests that one should reconsider the results of this section for quasi-coherent sheaves over a sheaf of groupoids, since a quasicoherent sheaf over Spec(A, Γ) is the same thing as a Γ-comodule [Hov02b]. We have not carried out this program. One reason for this is that we don’t see any clear applications. But another reason is that we do not know whether the Adams condition is invariant under weak equivalence. The difficulty is that, while dualizable comodules and filtered colimits are preserved by weak equivalences, Γ is not. One could simply demand that dualizable sheaves generate the category of quasi-coherent sheaves, but again we do not know

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if this is sufficient to provide a useful theory, or even whether it holds for interesting non-affine sheaves of groupoids. It would be interesting to know if there are equivalences of categories of comodules that are not given by maps of Hopf algebroids, as occurs in Morita theory. Since Hopf algebroids are a generalization of commutative rings, and there are no non-trivial Morita equivalences of commutative rings, it is reasonable to guess that every equivalence of categories of comodules is a zig-zag of weak equivalences. 2. Landweber exact algebras The object of this section is to study the relation between Γ-comodules and ΓB -comodules, where B is a Landweber exact A-algebra. The main application is to the relation between BP∗ BP -comodules and E(n)∗ E(n)comodules. In particular, we sketch the proof of Theorem A and its corollaries in this section. More details can be found in [HS03a]. Given a Hopf algebroid (A, Γ), an A-algebra B is said to be Landweber exact over A if B ⊗A (−) takes exact sequences of Γ-comodules to exact sequences of B-modules. This is called Landweber exactness because Landweber gave a characterization of Landweber exact BP∗ -algebras in his famous Landweber exact functor theorem [Lan76]. One can check that B is Landweber exact over A if and only if the composite ηR

A −→ Γ → − B ⊗A Γ is a flat ring extension. This condition is reminiscent of the characterization of weak equivalences given in Theorem 1.10. We can make it even more so by defining ΓB = B ⊗A Γ ⊗A B. We then have the following lemma, which is easy to prove but can also be found in [HS03a]. Lemma 2.1. Suppose (A, Γ) is a Hopf algebroid and B is an A-algebra. Then (B, ΓB ) is a Hopf algebroid, and the evident map (A, Γ) → − (B, ΓB ) is a map of Hopf algebroids. If (A, Γ) is flat and B is Landweber exact, then (B, ΓB ) is a flat Hopf algebroid. Thus, if B is Landweber exact over A, the map Φ : (A, Γ) → − (B, ΓB ) is almost a weak equivalence, in that ηR

A −→ Γ → − B ⊗A Γ is flat, and B ⊗ A Γ ⊗A B → − ΓB is an isomorphism. The only thing stopping Φ from being a weak equivalence is that B ⊗A (−) may not be faithful on the category of Γ-comodules. The

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idea of the following theorem, proved in [HS03a], is that we can force Φ to be faithful by localizing the category of Γ-comodules. Theorem 2.2. Suppose (A, Γ) is a flat Hopf algebroid, and B is Landweber exact over A. Then the map Φ : (A, Γ) → − (B, ΓB ) yields an equivalence Φ∗ : LT (Γ-comod) → − ΓB -comod where LT (Γ-comod) is the localization of Γ-comod with respect to the hereditary torsion theory T consisting of all Γ-comodules M such that B ⊗ A M = 0. A hereditary torsion theory is just a full subcategory closed under subobjects, quotient objects, extensions, and arbitrary direct sums. The localization LT is obtained by inverting all maps f of Γ-comodules whose kernel and cokernel are in T . Note that the Hopf algebroids that arise in algebraic topology are graded, so B will be a graded A-algebra, and our hereditary torsion theories will also be graded, in the sense that M is in T if and only if all shifts of M are in T . Because it is so surprisingly easy, we will give the proof of Theorem 2.2. Proof. Consider the natural transformation M : Φ∗ Φ∗ M → − M. We claim that this map is a natural isomorphism. One can check this by calculation for extended Σ-comodules M. Since is a natural transformation of left exact functors (because B is Landweber exact), and every Σ-comodule is the kernel of a map of extended comodules, M is an isomorphism for all M. After this, the rest of the proof of Theorem 2.2 is purely formal. A priori, the category LT (Γ-comod) may not be an actual category, since it may not have small Hom sets, always a danger with localization. However, it does exist in a higher universe. The natural transformation ηM : M → − Φ ∗ Φ∗ M becomes an isomorphism upon applying Φ∗ , and therefore, since Φ∗ is exact, the kernel and cokernel of ηM are in T . This gives us the desired equivalence, and incidentally shows that LT (Γ-comod) actually does exist as an honest category, since it is equivalent to ΓB -comod. Theorem 2.2 gives the following corollary, which it is difficult to imagine proving directly. Corollary 2.3. Suppose (A, Γ) is a flat Hopf algebroid, B is Landweber exact over A, and every nonzero Γ-comodule has a primitive. Then every nonzero ΓB -comodule has a primitive. In particular, every E(n)∗ E(n)-comodule has a primitive.

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This corollary is immediate, as LT (Γ-comod) is the full subcategory of Γ-comod consisting of the local objects. To get further information, we need to identify the hereditary torsion theories that can arise. Let Tn denote the collection of all vn -torsion BP∗ BP comodules, so that T0 is the collection of all p-torsion comodules, and T−1 is the collection of all comodules. One can easily check that Tn is a hereditary torsion theory. It is less obvious, but true, that Tn is the smallest graded hereditary torsion theory containing BP∗ /In+1 . Theorem 2.4. Let T be a graded hereditary torsion theory of BP∗ BP -comodules. If T contains a nonzero finitely presented comodule, then T = Tn for some n. This theorem is proved in [HS03a], using the ideas behind the Landweber filtration theorem. Note that this theorem explains why Ln and Lfn agree on the category of BP∗ BP -comodules. The only reasonable definition of Ln is localization with respect to the hereditary torsion theory of all comodules M such that E(n)∗ ⊗BP∗ M = 0, and Lfn is localization with respect to the hereditary torsion theory generated by BP∗ /In+1 . Both of these torsion theories are Tn . Because of this theorem, it is natural to make the following definition. Definition 2.5. Suppose B is a BP∗ -algebra. Define the height of B to be the largest integer n such that B/In is nonzero. If there is no such n, define the height of B to be infinite. From a formal group law point of view, the height of B is the largest possible height of any specialization of the formal group law of B. So the height of E(n) is n, and the height of BP itself is ∞. Here is the main theorem of [HS03a]. Theorem 2.6. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B is a graded Landweber exact A-algebra of height n ≤ ∞. Then the functor M 7→ B ⊗ A M defines an equivalence of categories Ln (Γ-comod) → − ΓB -comod, where Ln is localization with respect to Tn for n < ∞ and L∞ is the identity localization. This theorem is almost a corollary of Theorem 2.2 and Theorem 2.4, except for the infinite height case. We do not have a classification of graded hereditary torsion theories of BP∗ BP -comodules that do not contain a nonzero finitely presented comodule. There are probably uncountably many such torsion theories. However, if B is Landweber exact and B ⊗A M = 0 for some nonzero M, then B/I∞ B = 0. Indeed, M must have a nonzero primitive, and so we conclude by Landweber exactness that B/IB = 0 for some proper invariant ideal I. Since I ⊆ I∞ , it follows that B/I∞ B = 0. But this means

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that 1 ∈ I∞ B, so 1 ∈ In B for some n. Thus B ⊗A A/In = 0. This proves that if B has infinite height, then B ⊗A (−) does not kill any nonzero BP∗ BP comodules. The following corollary is immediate. Corollary 2.7. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B and B 0 are both Landweber exact A-algebras of the same height. Then the category of ΓB -comodules is equivalent to the category of ΓB 0 -comodules. In particular, the category of E(n)∗ E(n)-comodules is equivalent to the category of vn−1 BP∗ (vn−1 BP )-comodules, even though the category of E(n)∗ modules is very different from the category of vn−1 BP∗ -modules. One way to think of the Miller-Ravenel change of rings theorem [MR77, Theorem 3.10] as an isomorphism of certain Ext groups in these two categories. This is now obvious; the Ext groups are isomorphic because the categories they are taken in are equivalent. We point out that the equivalence of categories of comodules in Corollary 2.7 is in fact induced by a zig-zag of weak equivalences of Hopf algebroids. Any map B → − B 0 of Landweber exact BP∗ -algebras of the same height induces a weak equivalence of Hopf algebroids (B, ΓB ) → − (B 0 , ΓB 0 ), where we are still denoting BP∗ BP by Γ. If B and B 0 are Landweber exact BP∗ -algebras of the same height, there may not be a map of BP∗ -algebras between them. However, if we let C = B ⊗BP∗ Γ ⊗BP∗ B 0 , then C has a left and right BP∗ -algebra structure, which we denote by CL and CR . There are maps of BP∗ -algebras B → − CL and B 0 → − CR , and conjugation induces an isomorphism CL → − CR . Since CL is also Landweber exact, of the same height as B and B 0 , this yields the desired zig-zag of weak equivalences. To further understand the structure of the category of E(n)∗ E(n)-comodules, we would like to understand the localization functor Ln better. The following theorem is a summary of the results of [HS03b], and is joint work of the author and Strickland. Theorem 2.8.

(a) A comodule M is Ln -local if and only if

Hom∗BP∗ BP (BP∗ /In+1 , M) = Ext1,∗ BP∗ BP (BP∗ /In+1 , M) = 0, which is true if and only if Hom∗BP∗ (BP∗ /In+1 , M) = Ext1,∗ BP∗ (BP∗ /In+1 , M) = 0, (b) Ln , thought of as an endofunctor of the category of BP∗ BP -comodules, is left exact and preserves finite limits, filtered colimits, and arbitrary direct sums. It has right derived functors which we denote Lin . (c) For i > 0, Lin (M) is isomorphic to the i + 1st local cohomology group of the BP∗ -module M with respect to In+1 . In particular, Lin (M) = 0 for i > n.

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(d) Suppose m < n and M is a vm−1 -torsion comodule on which (vm , vm+1 ) is a regular sequence. Then M is Ln -local. (e) If vm acts invertibly on a comodule M for some m ≤ n, then M is Ln -local and Lin M = 0 for all i > 0. (f) If a comodule M is vn−1 -torsion, then Ln M = vn−1 M. (g) We have k < n, BP∗ /Ik −1 Ln (BP∗ /Ik ) = vn BP∗ /In k = n, 0 k > n. and, for i, n > 0, ( BP∗ /(p, v1 , . . . , vk−1 , vk∞ , . . . , vn∞ ) i = n − k > 0, i Ln (BP∗ /Ik ) = 0 otherwise.

These derived functors Lin can be used to compute BP∗ (Ln X) from BP∗ X by means of a spectral sequence. Theorem 2.9. Let X be a spectrum. There is a natural spectral sequence E∗∗∗ (X) with dr : Ers,t → − Ers+t,t+r−1 and E2 -term E2s,t (X) ∼ = (Lsn BP∗ X)t , converging to BPt−s (Ln X). This is a spectral sequence of BP∗ BP -comodules, in the sense that Ers,∗ is a graded BP∗ BP -comodule for all r ≥ 2 and dr : Ers,∗ → − s+r,∗ Er is a BP∗ BP -comodule map of degree r−1. Furthermore, every element in E20,∗ that comes from BP∗ X is a permanent cycle. This theorem is proved in [HS03b]. It is very closely related to the local cohomology spectral sequence of Greenlees [Gre93] and Greenlees and May [GM95]. One way of putting it is that we show that the Greenlees spectral sequence is a spectral sequence of comodules in this case. When X = S 0 , this implies that the spectral sequence collapses with no extensions, and we recover Ravenel’s computation [Rav84] of BP∗ Ln S 0 . We now derive some corollaries of Theorem B, proved in [HS03a]. Corollary 2.10. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B is a Landweber exact A-algebra of height n. Then Z(p) n>m=0 Q n=m=0 ΓB -comod(B, B/Im ) = Fp [vm ] n>m>0 −1 Fp [vm , vm ] n = m > 0. Proof. Let Φ : (A, Γ) → − (B, ΓB ) be the evident map of Hopf algebroids, so that Ln = Φ∗ Φ∗ by Theorem 2.6. Then we have ΓB -comod(B, B/Im ) ∼ = Γ-comod(A, Φ∗ (B/Im )) ∼ = Γ-comod(A, Ln (A/Im )),

so the result follows from Theorem 2.8 and the analogous calculation for B = BP∗ [Rav86, Theorem 4.3.2].

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This corollary in turn gives rise to the expected structural results about ΓB -comodules, proved in [HS03a]. Theorem 2.11. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B is a Landweber exact A-algebra of height n. (a) If I is an invariant radical ideal in B, then I = Im for some m ≤ n. (b) If M is a finitely presented ΓB -comodule, then there is a filtration 0 = M 0 ⊆ M1 ⊆ · · · ⊆ M t = M of M by subcomodules such that, for each s ≤ t, there is a k ≤ n and an r such that Ms /Ms−1 ∼ = sr B/Ik B. Proof. Note that the theorem is invariant under the equivalences of categories of Theorem 2.6. Thus we might as well assume that B = E(n)∗ or BP∗ , and in case B = BP∗ the result is due to Landweber [Lan76]. So we assume B = E(n)∗ . Now, for part (a), suppose I is an invariant radical ideal, and choose the largest k such that Ik ⊆ I. If Ik 6= I, the comodule I/Ik must have a nonzero primitive y. This primitive must also be a primitive in E(n)∗ /Ik . By Corollary 2.10, it must be a power of vk . Since I is radical, this means that Ik+1 ⊆ I. This contradication implies that Ik = I. For part (b), we construct the filtration Mi by induction on i, taking M0 = 0. Having built Mi , if Mi 6= M, we choose a nonzero primitive y in M/Mi . We claim that some multiple z of y is a primitive whose annihilator is Ik for some k ≤ n. Indeed, if y is p-torsion, then we can multiply y by a power of p to obtain a nonzero primitive y1 with py1 = 0. Since v1 is a primitive mod p, if y1 is v1 -torsion we can multiply y1 by power of v1 to obtain a nonzero primitive y2 with py2 = v1 y2 = 0. Continuing in this fashion, we end up with a primitive z such that Ik z = 0 and z is not vk -torsion. Corollary 2.10 implies that Ann z = Ik , as required. We now choose an element w in M whose image in M/Mi is z, and let Mi+1 denote the subcomodule generated by Mi and w. Then Mi+1 /Mi ∼ = sr E(n)∗ /Ik , where r is the degree of z. Since M is finitely presented over the Noetherian ring E(n)∗ , this process must stop, and so Mt = M for some t. 3. The stable homotopy category Stable(Γ) The object of this section is to discuss the construction and basic properties of the stable homotopy category Stable(Γ) associated to an Adams Hopf algebroid (A, Γ). In practice, we are most interested in Stable(E∗ E) for E a commutative Landweber exact ring spectrum. This means our Hopf algebroids should be graded, and the stable homotopy category Stable(Γ) should be bigraded. However, the grading just adds unnecessary complexity to the notation, so we will forget about it for most of this section.

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Construction of Stable(Γ). The usual derived category D(Γ) of Γ-comodules is obtained from Ch(Γ), the category of unbounded chain complexes of Γcomodules, by inverting the homology isomorphisms. This is not the right thing to do to form Stable(Γ). To see this, note that there is a model structure on Ch(Γ) whose homotopy category is the derived category, as there is on Ch(A) for any Grothendieck category A [Bek00]. The cofibrations in this model structure are the monomorphisms, the fibrations are the epimorphisms with DG-injective kernel, and the weak equivalences are the homology isomorphisms. Here a complex X is DG-injective if each Xn is injective and every map from an exact complex to X is chain homotopic to 0. In particular, bounded above complexes of injectives are DG-injective, from which it follows that D(Γ)(A, A)∗ = Ext∗Γ (A, A). Here we are thinking of A as a complex concentrated in degree 0. It is the analog of the sphere S, since it is the unit of ∧. Now, there are often nonnilpotent elements in ExtiΓ (A, A) for i > 0. For example, the well-known element β1 is non-nilpotent in Ext∗BP∗ BP (BP∗ , BP∗ ) for p > 2. Since β1 corresponds to a self-map of BP∗ in D(BP∗ BP ), we can form β1−1 BP∗ . But this complex has trivial homology, so is 0 in D(BP∗ BP ) even though β1 is not nilpotent. This is not good for several reasons; we should be able to see β1−1 BP∗ because it is an important object, and the fact that we can’t also implies that A is not a small object of D(BP∗ BP ). We should be inverting homotopy isomorphisms, not homology isomorphisms. To do this, we need to define the homotopy groups. Definition 3.1. Suppose (A, Γ) is an Adams Hopf algebroid, X ∈ Ch(Γ), and P is a dualizable Γ-comodule. We define the homotopy groups of X with coefficients in P by πnP (X) = H−n (Γ-comod(P, LA ∧ X)), where LA is the cobar resolution of A. We define a map f of complexes to be a homotopy isomorphism if πnP (f ) is an isomorphism for all n and all dualizable comodules P . The stable homotopy category of Γ, Stable(Γ) is defined to be the category obtained from Ch(Γ) by inverting the homotopy isomorphisms. The reason for the sign in the definition of πnP is so that πnP (M) ∼ = ExtnΓ (P, M) for a comodule M. Homotopy groups and homotopy isomorphisms satisfy the expected properties [Hov02a]. Proposition 3.2. Suppose (A, Γ) is an Adams Hopf algebroid, and P is a dualizable comodule.

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(a) A short exact sequence of complexes induces a long exact sequence in the homotopy groups π∗P (−). Note that the boundary map raises dimension by one, because of the sign in the definition of πnP (−). (b) Homotopy groups commute with filtered colimits, so homotopy isomorphisms are closed under filtered colimits. (c) Every chain homotopy equivalence is a homotopy isomorphism, and every homotopy isomorphism is a homology isomorphism. (d) The natural map X → − LA ∧ X is a homotopy isomorphism for all X. To understand Stable(Γ), and for that matter to even see that it is a category at all, we need a model structure on Ch(Γ) in which the weak equivalences are the homotopy isomorphisms. This is the main goal of [Hov02a], where the following theorem is proved. Theorem 3.3. Suppose (A, Γ) is an Adams Hopf algebroid. Then there is a proper symmetric monoidal model structure on Ch(Γ) in which the weak equivalences are the homotopy isomorphisms. Furthermore, if X is cofibrant, then X ∧ (−) preserves homotopy isomorphisms. We call this model structure the homotopy model structure. The cofibrations in the homotopy model structure are dimensionwise split monomorphisms with cofibrant cokernel. If X is cofibrant, then each Xn is projective over A. More precisely, if X is cofibrant, then X is a retract of a complex Y that admits a filtration Y i such that each map Y i → − Y i+1 is a dimensionwise split monomorphism and the quotient Y i+1 /Y i is a complex of relatively projective comodules with trivial differential. Here a comodule is relatively projective if it is a retract of a direct sum of dualizable comodules. A characterization of the fibrations is given in [Hov02a]. Fibrations are of course surjective. Every complex of relative injectives is fibrant, and every fibrant complex is equivalent in a precise sense to a complex of relative injectives. A fibrant replacement of X is given by LB ∧ X. The following theorem is also proved in [Hov02a]. Theorem 3.4. The homotopy model structure is natural, in the sense that a map Φ : (A, Γ) → − (B, Σ) induces a left Quillen functor Φ∗ : Ch(Γ) → − Ch(Σ) of the homotopy model structures. Furthermore, if Φ is a weak equivalence, then Φ∗ is a strong Quillen equivalence, in the sense that both Φ∗ and Φ∗ preserve and reflect homotopy isomorphisms. Global properties of Stable(Γ). We now establish some of the essential properties of Stable(Γ). At this point, we begin to use some of the standard notational conventions of ordinary stable homotopy. Thus, we will begin using S for the image of A in Stable(Γ), thinking of it as analogous to the usual zero-sphere. Similarly, we will sometimes use [X, Y ]∗ for graded maps in Stable(Γ). In practice, Γ is usually graded and so Stable(Γ) is bigraded. However, the internal suspension in the category of Γ-comodules is usually not relevant, so we tend to omit it from the notation.

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Theorem 3.5. Suppose (A, Γ) is an Adams Hopf algebroid. The category Stable(Γ) is a closed symmetric monoidal triangulated category. The dualizable comodules form a set of small, dualizable, weak generators for Stable(Γ). This theorem is really a corollary of Theorem 3.3 and general facts about model categories. It is proved in [Hov02a]; another way to say it is that Stable(Γ) is a unital algebraic stable homotopy category in the sense of [HPS97]. One drawback of Stable(Γ) is that it is not in general monogenic. That is, A and its suspensions are not generally enough to generate the whole category. This is unavoidable even for Hopf algebras. Indeed, if G is a finite group and k is a field, the stable homotopy category of the Hopf algebra of functions from G to k is closely related to the stable module category much studied in modular representation theory [Ben98], as explained in [HPS97]. If G is a p-group, the stable module category is monogenic, but not in general. However, one certainly expects Stable(BP∗ BP ) and Stable(E(n)∗ E(n)) to be monogenic, so we need a condition on (A, Γ) that will ensure that Stable(Γ) is monogenic. Recall that a full subcategory D of an abelian category is called thick if it is closed under retracts and, whenever two out of three terms in a short exact sequence are in D, so is the third. Proposition 3.6. Suppose (A, Γ) is an Adams Hopf algebroid, and every dualizable comodule is in the thick subcategory generated by A. Then Stable(Γ) is monogenic. This proposition is proved in [Hov02a]. To apply it, we note that the filtration theorem for E∗ E-comodules, part (b) of Theorem 2.11, implies that every finitely presented E∗ E-comodule is in the thick subcategory generated by E∗ when E is a commutative ring spectrum that is Landweber exact over BP∗ . Thus we get the following corollary. Corollary 3.7. Suppose E is a commutative ring spectrum that is Landweber exact over BP . Then Stable(E∗ E) is monogenic in the bigraded sense. In particular, a map f in Ch(E∗ E) is a homotopy isomorphism if and only if k πn,k (f ) = πns E∗ (f ) is an isomorphism for all n and k. The bigrading arises because we can suspend a complex X either internally, by suspending each graded comodule Xn , or externally by suspending the complex X. We would like to understand the relation between a comodule M and its image in Stable(Γ). The following proposition is proved in [Hov02a]. Proposition 3.8. Suppose (A, Γ) is an Adams Hopf algebroid. (a) A short exact sequence of comodules, or even complexes, gives rise to a cofiber sequence in Stable(Γ). (b) If M is in the thick subcategory generated by A, then M is a small object of Stable(Γ).

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Mark Hovey (c) If M and N are comodules, then there is a natural map ExtkΓ (M, N ) → − Stable(Γ)(M, N )k that is an isomorphism for M in the thick subcategory generated by A.

Part (b) is not actually proved in [Hov02a], but follows immediately from part (a). In particular, of course, we have Stable(Γ)(A, A)∗ ∼ = π∗A (A) ∼ = Ext∗Γ (A, A). This is the stable homotopy of the sphere in Stable(Γ). The following point is also valuable. Proposition 3.9. Suppose E is a commutative ring spectrum that is Landweber exact over BP . Then Stable(E∗ E) is a Brown category, so that every homology functor is representable. This proposition follows from Theorem 4.1.5 of [HPS97]. Indeed, we can assume E = E(n) or BP , and then one can easily check using the cobar resolution that Ext∗∗ E∗ E (E∗ , E∗ ) is countable. Ordinary homology. We now describe ordinary homology in Stable(Γ). Since homotopy isomorphisms are in particular homology isomorphisms, the ordinary homology of a chain complex X is a homology theory on Stable(Γ). Proposition 3.10. Let (A, Γ) be an Adams Hopf algebroid. Ordinary homology is represented on Stable(Γ) by Γ itself, as usual thought of as a complex concentrated in degree 0. Because of this proposition, we will sometimes denote Γ by H. Proof. For a complex X, we have Γ∗ (X) ∼ = π∗ (QΓ ∧ QX), where Q denotes cofibrant replacement. Since QX ∧ (−) preserves homotopy isomorphisms by Theorem 3.3, we have π∗ (QΓ ∧ QX) ∼ = π∗ (Γ ∧ QX). Since Γ ∧ QX is already fibrant, as it is a complex of relative injectives, we have π∗ (Γ ∧ QX) ∼ = Ch(Γ)(A, Γ ∧ QX)/ ∼, where ∼ denotes the chain homotopy relation. This is in turn isomorphic to Ch(A)(A, QX)/ ∼∼ = H∗ (QX) ∼ = H∗ X by adjointness.

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Note that the Hopf algebroid (H∗ , H∗ H) associated to homology is isomorphic to (A, Γ) itself, concentrated in degree 0. Thus H∗ X is naturally a graded (A, Γ)-comodule, which is bigraded in case (A, Γ) is graded. We get an Adams-Novikov spectral sequence based on H whose E2 -term is E s,t ∼ = Exts (A, Ht X), 2

Γ

which in good cases will converge to π∗ X. In particular, if X = S, this spectral sequence is concentrated in degrees (s, 0), and so collapses and converges to π∗ S ∼ = Ext∗Γ (A, A). Thus, if we take (A, Γ) = (BP∗ , BP∗ BP ), we have built a stable homotopy category in which the usual Adams-Novikov spectral sequence collapses. Note that ordinary cohomology is somewhat complicated. This is actually already true in the derived category D(A). Indeed, in D(A) we have H ∗ (X) ∼ = Ch(A)(QX, S 0 A)∗ and there is no really convenient interpretation of these groups. Similarly, in Ch(Γ), we have H ∗ (X) ∼ = Ch(Γ)(QX, S 0 Γ)∗ ∼ = Ch(A)(QX, S 0 A)∗ . The ordinary derived category of Γ, obtained by inverting the homology isomorphisms, is the Bousfield localization of Stable(Γ) with respect to H. As we have said before, this is a non-trivial localization. Indeed, suppose x is a non-nilpotent class in Exts (A, A) with s > 0. Then x corresponds to a self-map S −s → − S, which is necessarily 0 on homology. Hence the −1 telescope x S will have no homology, but will be nonzero. In particular, if (A, Γ) = (BP∗ , BP∗ BP ), there are non-nilpotent classes in Ext. For example, α1 is non-nilpotent when p = 2 by [Rav86, Theorem 4.4.37]. Homology with coefficients. We now consider ordinary homology with coefficients in an A-module B. Proposition 3.11. Let (A, Γ) be an Adams Hopf algebroid, and let B be an A-module. Then Γ ⊗A B represents the homology theory HB on Stable(Γ) defined by (HB)∗ (X) ∼ = H∗ (B ⊗A QX). If B is Landweber exact over A, then (HB)∗ (X) ∼ = H∗ (B ⊗A X) ∼ = B ⊗A H∗ X. In particular, in this case the Hopf algebroid (HB∗ , HB∗ HB) is isomorphic to (B, ΓB ) concentrated in degree 0. Proof. For an object X of Stable(Γ), we have (HB)∗ (X) ∼ = π∗ (Γ ⊗A B) ∧ QX) ∼ = H∗ (B ⊗A QX),

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where we have used Lemma 1.4 to manipulate the tensor product. In particular, HB∗ (S) ∼ = B concentrated in degree 0. If B is Landweber exact over A, then B ⊗A (−) will preserve homology isomorphisms of complexes of comodules, so (HB)∗ (X) ∼ = H∗ (B ⊗A X) ∼ = B ⊗A H∗ X. In particular,

(HB)∗ (HB) ∼ = B ⊗ A Γ ⊗A B ∼ = ΓB . Thus (HB)∗ X is naturally a graded comodule over (B, ΓB ), when B is Landweber exact over A. Thus we get theories HE(n) when (A, Γ) = (BP∗ , BP∗ BP ) and B = E(n)∗ . The Adams-Novikov spectral sequence based on HB when B is Landweber exact will then have E2 -term E s,t ∼ = Exts (B ⊗A Ht X, B ⊗A Ht Y ). 2

ΓB

In particular, when X = Y = S, this spectral sequence must collapse, since the E2 -term is concentrated where t = 0. However, it is not entirely clear to what it converges. The obvious guess is π∗ LHB S, where LHB denotes Bousfield localization with respect to HB. Bousfield’s convergence results [Bou79] should be re-examined to see if they apply in a more general setting to answer this question. Note that if B is an A-algebra that is also a field, then HB will be a field object of Stable(Γ). In particular, if p is a prime ideal in A with residue field kp , then we can form Hkp . If we apply this to the case (A, Γ) = (BP∗ , BP∗ BP ), we get field spectra HK(n) corresponding to the Morava K-theories, but we also get many other field spectra, including HFp corresponding to the prime ideal I∞ . Note that the objects HK(n) do not detect nilpotence in Stable(BP∗ BP ), since there are non-nilpotent self-maps of S that are zero on homology with any coefficients. 4. Landweber exactness and the stable homotopy category Recall that in Section 2 we showed that the abelian category of E(n)∗ E(n)comodules is a localization of the abelian category of BP∗ BP -comodules. In Section 3, we introduced stable homotopy categories of E(n)∗ E(n) and BP∗ BP -comodules. It is therefore natural to conjecture that Stable(E(n)∗ E(n)) is a Bousfield localization of Stable(BP∗ BP ). The goal of this section is to prove this conjecture, thereby proving Theorem C. The functor Φ∗ . Throughout this section, we let (A, Γ) = (BP∗ , BP∗ BP ), B = E(n)∗ , and we let Φ : (A, Γ) → − (B, ΓB ) be the induced map of Hopf algebroids. The map of Hopf algebroids Φ induces a functor Φ∗ : Γ-comod → − ΓB -comod

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and a left Quillen functor Φ∗ : Ch(Γ) → − Ch(ΓB ) by Theorem 3.4. To prove Theorem C, we must show that Φ∗ induces a Quillen equivalence upon suitably localizing Ch(Γ). The object of the present section is to prove the following theorem. Theorem 4.1. The functor Φ∗ : Ch(Γ) → − Ch(ΓB ) preserves weak equiva∗ lences. Its right adjoint Φ reflects weak equivalences. We prove this theorem in a series of propositions. Proposition 4.2. Let D denote the class of all X ∈ Ch(Γ) such that the map Φ∗ QX → − Φ∗ X is a weak equivalence in Ch(ΓB ), where Q is a cofibrant replacement functor in Ch(Γ). Then D is a thick subcategory. Proof. Note that D is obviously closed under retracts. To see that D is thick, suppose we have a short exact sequence X0 → − X→ − X 00 in Ch(Γ) such that two out of three terms are in D. By Proposition 3.8(a), this is a cofiber sequence in Stable(Γ). Since Φ∗ Q is the total left derived functor of the left Quillen functor Φ∗ , we conclude that Φ∗ QX 0 → − Φ∗ QX → − Φ∗ QX 00 is a cofiber sequence in Stable(ΓB ). On the other hand, because Φ∗ is exact, the sequence Φ∗ X 0 → − Φ∗ X → − Φ∗ X 00 is a short exact sequence in Ch(ΓB ), and hence, applying Proposition 3.8(a) again, is also a cofiber sequence in Stable(ΓB ). There is a map from the first of these cofiber sequences to the second, and by assumption it is an isomorphism on two out of three terms. Since Stable(ΓB ) is a triangulated category, we conclude that it is also an isomorphism on the third term, and so D is thick. Our next goal is to show that D is closed under filtered colimits. For this we need to recall some standard model category theory. Suppose I is a small category, and M is a cofibrantly generated model category, such as Ch(Γ). Then there is a cofibrantly generated model category structure on the diagram category MI [Hir03, Theorem 12.7.1] in which the weak equivalences and fibrations are taken objectwise. Furthermore, the cofibrations in MI are in particular objectwise cofibrations [Hir03, Proposition 12.7.3]. Proposition 4.3. The class D of Proposition 4.2 is closed under filtered colimits.

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Proof. Suppose F : I → − Ch(Γ) is a functor from a filtered small category I such that F (i) ∈ D for all i ∈ I. We must show that colim F (i) ∈ D. Let QF be a cofibrant replacement of F in the model category on Ch(Γ)I discussed prior to this proposition. Because the constant diagram functor obviously preserves fibrations and trivial fibrations, the colimit is a left Quillen functor [Hir03, Theorem 12.7.9]. Hence colim QF is cofibrant. Furthermore, each map QF (i) → − F (i) is a homotopy isomorphism, and so, since homotopy commutes with filtered colimits, we conclude that the map colim QF → − colim F is a weak equivalence. Therefore, colim QF is a cofibrant replacement of colim F . To show that colim F ∈ D, then, we need only show that the map Φ∗ (colim QF ) → − Φ∗ (colim F ) is a homotopy isomorphism. Since Φ∗ itself commutes with colimits, this is equivalent to showing that the map colim Φ∗ QF → − colim Φ∗ F is a homotopy isomorphism. Since QF is cofibrant, and cofibrations of diagrams are in particular objectwise cofibrations, we conclude that QF (i) is a cofibrant replacement for F (i) for all i ∈ I. Since F (i) ∈ D, then, each map Φ∗ QF (i) → − Φ∗ F (i) is a homotopy isomorphism. Hence, again using the fact that homotopy commutes with filtered colimits, colim Φ∗ QF → − colim Φ∗ F is a homotopy isomorphism, so colim F ∈ D. We now know that D is a thick subcategory that is closed under filtered colimits and (obviously) contains all the cofibrant objects of Ch(Γ). This should mean that it has to be all of Ch(Γ), and that is what we now prove. Proposition 4.4. If X ∈ Ch(Γ), then the map Φ∗ QX → − Φ∗ X is a weak equivalence. Proof. The proposition is just saying that the class D of Propositions 4.2 and 4.3 is all of Ch(Γ). We prove this in three steps. We first show that the complexes S n M are in D, where M is a finitely presented Γ-comodule and S n M denotes the complex whose only non-zero entry is M in degree n. We then show that all finitely presented complexes are in D, and finally, we show that every complex is a filtered colimit of finitely presented complexes, so is in D by Proposition 4.3. For the first step, it is clear that S n A is in D since it is cofibrant. The collection of all M such that S n M is in D is a thick subcategory by Proposition 4.2; by induction, therefore, it contains A/Ik for all k. The Landweber filtration theorem then implies that it contains all finitely presented M. Now suppose X is a finitely presented complex. For the purposes of the present proof, we take this to mean that Xn is finitely presented for all n and 0 for almost all n; this is in fact equivalent to X being a finitely presented object of Ch(Γ) in the categorical sense. We easily prove by induction on the nunber of non-zero entries in X that X ∈ D. Indeed, the base case of one non-zero entry is handled in the preceding paragraph. For the induction step, let X 0 be the subcomplex of X obtained by removing the non-zero entry in

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the largest possible degree. Then X 0 ∈ D by the induction hypothesis, and the quotient X/X 0 ∈ D by the preceding paragraph. Since D is thick, X ∈ D. Now suppose X is an arbitrary complex. Let F /X denote the category of all maps F → − X, where F is a finitely presented complex. This is easily seen to have a small skeleton and to be a filtered category. There is an obvious inclusion functor i : F /X → − Ch(Γ), and an obvious map f : colim i → − X. We claim that f is an isomorphism. To see that f is surjective, choose x ∈ Xn . Since the comodule Xn is a filtered colimit of finitely presented comodules, there is a finitely presented comodule F and a map F → − Xn whose image n contains x. This gives a map of complexes D F → − X whose image contains x, and so f is surjective. To see that f is injective, suppose j : F → − X is an object of F /X and x ∈ Fn has jx = 0. Let K denote the kernel of the map j, so that x ∈ Kn . Now K may not be finitely presented, but at least there is a map F 0 → − Kn from a finitely presented comodule whose image contains x. This corresponds to a map D n F 0 → − K of complexes, which induces an object F/D n F 0 → − X of F /X and a map F → − F/D n F 0 in F /X that sends x to 0. Thus x is 0 in colim i and so f is injective. We can now give the proof of Theorem 4.1. Proof of Theorem 4.1. Suppose f : X → − Y is a weak equivalence in Ch(Γ). Then Qf is a weak equivalence between cofibrant objects, so Φ∗ Qf is a weak equivalence since Φ∗ is a left Quillen functor. But we have a commutative square Φ∗ Qf

Φ∗ QX −−−→ Φ∗ QY y y Φ∗ X −−−→ Φ∗ Y Φ∗ f

where the vertical maps are weak equivalences, by Proposition 4.4. Hence Φ∗ f is a weak equivalence as well. To prove the second part of Theorem 4.1, suppose f is a map in Ch(ΓB ) such that Φ∗ f is a weak equivalence in Ch(Γ). Then Φ∗ Φ∗ f is a weak equivalence in Ch(Γ) by what we have just proved. But f is naturally isomorphic to Φ∗ Φ∗ f by Theorem 2.2. Localization. We have just seen that Φ∗ : Ch(Γ) → − Ch(ΓB ) preserves weak equivalences. But of course it does not reflect weak equivalences, since Φ∗ (A/In+1 ) = 0. We dealt with this problem already in the abelian category world by localizing Γ-comod so as to force 0 → − A/In+1 to be an isomorphism. We now want to do the same thing for chain complexes. More precisely, we define Lfn Ch(Γ) to be the category Ch(Γ) equipped with the model structure that is the Bousfield localization of the homotopy model

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structure with respect to the maps 0 → − sk A/In+1 for all k, where sk A/In+1 denotes the complex which is A/In+1 in degree k and 0 elsewhere. Recall from [Hir03] that this means that a left Quillen functor F : Ch(Γ) → − M defines a left Quillen functor F : Lfn Ch(Γ) → − M if and only if 0 → − (LF )(sk A/In+1 ) is an isomorphism in ho M, where LF denotes the total left derived functor of F . The homotopy category of Lfn Ch(Γ) is the finite localization Lfn Stable(Γ) in the sense of Miller [Mil92] of Stable(Γ) away from A/In+1 . The total left derived functor of the identity, thought of as a functor from Ch(Γ) to Lfn Ch(Γ), is the finite localization functor Lfn on Stable(Γ). Proposition 4.5. The Quillen functor Φ∗ : Ch(Γ) → − Ch(ΓB ) induces a left Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ). Proof. We need to show that (LΦ∗ )(A/In+1 ) = 0. In light of Proposition 4.4, (LΦ∗ )X is naturally isomorphic to Φ∗ X for any X ∈ Ch(Γ). Thus (LΦ∗ )(A/In+1 ) ∼ = Φ∗ (A/In+1 ) = 0. Note that Φ∗ still preserves weak equivalences when thought as a functor from Lfn Ch(Γ), Proposition 4.6. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) preserves weak equivalences, and its right adjoint Φ∗ reflects weak equivalences. Proof. Suppose f : X → − Y is a weak equivalence. Factor f = pi, where i is a trivial cofibration and p is a trivial fibration. Then Φ∗ i is a weak equivalence since Φ∗ is a left Quillen functor. On the other hand, since the trivial fibrations do not change under Bousfield localization, p is a weak equivalence in Ch(Γ). Thus Theorem 4.1 implies that Φ∗ p is a weak equivalence. Hence Φ∗ f = (Φ∗ p)(Φ∗ i) is a weak equivalence, as required. The proof that Φ∗ reflects weak equivalences is the same as the proof of the corresponding part of Theorem 4.1. To prove Theorem C, we will show that Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence. To do so, we will use the following lemma, which is proved in [Hov99, Corollary 1.3.16].

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Lemma 4.7. Suppose F : C → − D is a left Quillen functor of model categories, with right adjoint U . Then F is a Quillen equivalence if and only if the following two conditions hold. (a) U reflects weak equivalences between fibrant objects. (b) The map X → − U RF X is a weak equivalence for all cofibrant X in C, where R denotes fibrant replacement and the map is induced by the unit of the adjunction. We have already seen in Proposition 4.6 that Φ∗ reflects all weak equivalences. We point out that there is a much simpler proof that Φ∗ reflects weak equivalences between fibrant objects; if X is fibrant in Ch(ΓB ), then adjointness implies that π∗ (Φ∗ X) ∼ = π∗ X. The other condition of Lemma 4.7 is harder to check. Here are the main points of the argument. (a) We first show, using the fact that Φ∗ preserves filtered colimits, that it suffices to show that A → − Φ∗ (LB) is an Lfn -equivalence, where LB denotes the cobar resolution of B. (b) A Bousfield class argument that shows that it suffices to prove that − vk−1 A/Ik ∧ QΦ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism for all 0 ≤ k ≤ n, where Q denotes cofibrant replacement. (c) We show that, although Φ∗ (LB) is not cofibrant, it is still nice enough that it suffices to check that − vk−1 A/Ik ∧ Φ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism. (d) It was proved in [Hov02b] that the Hopf algebroid (vk−1 A/Ik , vk−1 Γ/Ik ) is weakly equivalent to (vk−1 B/Ik , vk−1 ΓB /Ik ). Hence it suffices to show that v −1 B/Ik → − v −1 B/Ik ∧ Φ∗ Φ∗ (LB) ∼ = v −1 B/Ik ∧ LB k

k

k

is a homotopy isomorphism, and this is clear. We now fill in the details of this argument, beginning with Step (a). Proposition 4.8. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence if and only if the map A→ − Φ∗ (LB) is an Lfn -equivalence. Recall that LB denotes the cobar resolution of B as a ΓB -comodule. Before proving this proposition, we need a lemma.

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Lemma 4.9. The total right derived functor RΦ∗ : Stable(ΓB ) → − Lfn Stable(Γ) preserves coproducts. Proof. First note that because Lfn is a finite localization, it is in particular smashing [Mil92]. Thus the coproduct in Lfn Stable(Γ) is the same as the coproduct in Stable(Γ). Also note that a fibrant replacement in Stable(ΓB ) is given by LB ∧ (−), which clearly preserves coproducts. Hence, it suffices to show that Φ∗ : ΓB -comod → − Γ-comod ∗ preserves coproducts. Since Φ certainly preserves finite coproducts, and any coproduct is a filtered colimit of finite coproducts, it suffices to show that Φ∗ preserves all filtered colimits. This follows from the fact that Ln = Φ∗ Φ∗ preserves filtered colimits (Theorem 2.8); more details can be found in [HS03b]. Proof of Proposition 4.8. Lemma 4.7 tells us that if Φ∗ is a Quillen equivalence, then A → − Φ∗ RΦ∗ A must be a weak equivalence in Lfn Ch(Γ). Since Φ∗ A = B, and since LB is a fibrant replacement for B in Ch(ΓB ), we conclude that A → − Φ∗ (LB) is an Lfn -equivalence. Conversely, suppose A → − Φ∗ (LB) is an Lfn -equivalence. By Lemma 4.7 and Proposition 4.6, it suffices to show that X → − Φ∗ RΦ∗ X is an Lfn -equivalence for all cofibrant X. This is equivalent to proving that η

X X −→ (RΦ∗ )(LΦ∗ )X

is an isomorphism in Lfn Stable(Γ) for all X, where RΦ∗ denotes the total right derived functor of Φ∗ and LΦ∗ denotes the total left derived functor of Φ∗ . Let D denote the full subcategory of Lfn Stable(Γ) of those X such that ηX is an isomorphism. By hypothesis, D contains Lfn A. Since LΦ∗ and RΦ∗ both preserve exact triangles, D is a thick subcategory. As a left adjoint, LΦ∗ preserves coproducts, and Lemma 4.9 assures us that RΦ∗ also preserves coproducts. Thus D is a localizing subcategory. In any monogenic stable homotopy category, the only localizing subcategory that contains the unit is the whole category. We are now reduced to showing that A → − Φ∗ (LB) is an Lfn -equivalence. The theory of Bousfield classes gives us the following lemma. Lemma 4.10. A map f of cofibrant objects in Ch(Γ) is an Lfn -equivalence if and only if vk−1 A/Ik ∧ f is a weak equivalence for all k with 0 ≤ k ≤ n. Proof. As usual, let < X > denote the Bousfield class of X in Stable(Γ), thought of as the collection of all Y in Stable(Γ) such that X ∧ Y = 0. The cofiber sequences vk A/Ik −→ A/Ik → − A/Ik+1

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197

imply that < A/Ik >=< vk−1 A/Ik > ∨ < A/Ik+1 >, by [Rav84, Lemma 1.34]. Thus, we find n _ < A >= < vk−1 A/Ik > ∨ < A/In+1 > . k=0

As in the usual stable category, this implies that Lfn is localization Ln homotopy −1 with respect to k=0 vk A/Ik . It follows that f is an Lfn -equivalence if and only if vk−1 A/Ik ∧L f is a weak equivalence for all k with 0 ≤ k ≤ n, where ∧L denotes the total left derived functor of ∧. Recall from Theorem 3.3 that if X is cofibrant, then X ∧ (−) preserves homotopy isomorphisms. It follows that, if X is cofibrant, (−) ∧L X ∼ = (−) ∧ X in Stable(Γ). Hence, if f is a map of cofibrant objects, then f is an Lfn -equivalence if and only if vk−1 A/Ik ∧ f is a weak equivalence for all k with 0 ≤ k ≤ n. By combining Proposition 4.8 with Lemma 4.10, we get the following corollary, which is Step (b) of the argument on page 195. Corollary 4.11. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence if and only if the map − vk−1 A/Ik ∧ QΦ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism for all 0 ≤ k ≤ n, where Q denotes a cofibrant replacement functor in Ch(Γ). To accomplish Step (c) of the argument on page 195, we need to know something about Φ∗ (LB). Lemma 4.12. We have TorjA (A/Ik , Φ∗ (LB)m ) = 0 for all j > 0, k ≥ 0, and m ∈ Z. Proof. Recall that ⊗B m

(LB)−m = ΓB ⊗B ΓB for m ≥ 0, and is 0 otherwise. Here ΓB is the cokernel of the left unit ηL : B → − ΓB . Since this map is split as a map of B-modules, ΓB is a flat ⊗B m B-module. Therefore ΓB is also a flat B-module, and hence a filtered colimit of projective B-modules. Now, we have ⊗B m Φ∗ (LB)−m = Γ ⊗A ΓB . j Since TorA (A/Ik , −) commutes with filtered colimits, it suffices to show that TorjA (A/Ik , Γ ⊗A M) = 0

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for all j > 0, all k, and all projective B-modules M. But then we can easily reduce to the case M = B, so we must show that TorjA (A/Ik , Γ ⊗A B) = 0 for all j > 0 and all k. We prove this by induction on k, using the exact sequences vk 0→ − A/Ik −→ A/Ik → − A/Ik+1 → − 0. We are reduced to showing that vk is not a zero-divisor on (Γ ⊗A B)/Ik . Since Ik is invariant, this is the same as showing that vk is not a zero-divisor on Γ ⊗A (B/Ik ). Since vk is itself invariant modulo Ik and Γ is flat over A, this is in turn equivalent to showing that vk is not a zero-divisor on B/Ik . This follows because B is Landweber exact. With this lemma in hand, we can carry out Step (c) of our argument. Proposition 4.13. The map vk−1 A/Ik ∧ QΦ∗ (LB) → − vk−1 A/Ik ∧ Φ∗ (LB) is a homotopy isomorphism in Ch(Γ). In particular, Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence if and only if − vk−1 A/Ik ∧ Φ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism for all 0 ≤ k ≤ n. Proof. Let q denote the map q : QΦ∗ (LB) → − Φ∗ (LB). Because homotopy commutes with filtered colimits, it suffices to show that A/Ik ∧ q is a homotopy isomorphism for all k. Now, q is a trivial fibration in the homotopy model structure, so we have a short exact sequence of complexes q

0→ − K→ − QΦ∗ (LB) → − Φ∗ (LB) → − 0. The long exact sequence in homotopy implies that πtA (K) = 0 for all n. Lemma 4.12 implies that we have a short exact sequence 0→ − A/Ik ∧ K → − A/Ik ∧ QΦ∗ (LB) → − A/Ik ∧ Φ∗ (LB) → − 0 for all k. The long exact sequence in homotopy implies that we need only check that πtA (A/Ik ∧ K) = 0 for all n. To se this, note that the short exact sequence defining K realizes Km as the first syzygy of Φ∗ (LB)m , since (QΦ∗ (LB))m is projective over A. Hence Lemma 4.12 implies that TorjA (A/Ik , Km ) = 0

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for all j > 0. Hence we have short exact sequences 0→ − A/Ik ∧ K → − A/Ik ∧ K → − A/Ik+1 ∧ K → − 0 for all k. The long exact sequence in homotopy and induction on k now complete the proof. The final step of the argument on page 195 requires us to know more about the Hopf algebroids (vk−1 A/Ik , vk−1 Γ/Ik ) and (vk−1 B/Ik , vk−1 ΓB /Ik ). Lemma 4.14. Let (C, Σ) denote (vk−1 A/Ik , vk−1 Γ/Ik ), and let (CB , ΣB ) denote (vk−1 B/Ik , vk−1 Γ/Ik ). Then: (a) Both (C, Σ) and (CB , ΣB ) are Adams Hopf algebroids; (b) The stable homotopy categories Stable(Σ) and Stable(ΣB ) are monogenic; (c) If 0 ≤ k ≤ n, Φ induces a weak equivalence of Hopf algebroids Φ : (C, Σ) → − (CB , ΣB ). Proof. For part (a), we know that Γ is a filtered colimit of comodules that are finitely generated projective A-modules. By tensoring with vk−1 A/Ik , we se that Σ is a filtered colimit of comodules that are finitely generated projective C-modules, and so (C, Σ) is an Adams Hopf algebroid. Similarly, (CB , ΣB ) is an Adams Hopf algebroid. For part (b), the proof is again the same for (C, Σ) and (CB , ΣB ), so we concentrate on (C, Σ). We will use Proposition 3.6, so we need to show that every dualizable Σ-comodule is in the thick subcategory generated by C. We will do this by showing that every finitely presented Σ-comodule has a Landweber filtration. To do so, we will use the Hopf algebroid (vk−1 A, vk−1 Γvk−1 ), obtained by inverting vk and ηR vk . A Σ-comodule M is just a (vk−1 Γvk−1 )-comodule on which Ik acts trivially. Since Ik is finitely generated, M is finitely presented if and only if it is finitely presented as a vk−1 Γvk−1 -comodule. Since vk−1 A is Landweber exact of height k, Theorem 2.11 gives us a Landweber filtration of M as a vk−1 Γvk−1 -comodule in which each filtration quotient is isomorphic to vk−1 A/Ij for some j ≤ k. Since M is killed by Ik , in fact each filtration quotient must be isomorphic to vk−1 A/Ik , giving us our Landweber filtration of M as a Σ-comodule. Part (c) is a special case of Theorem E of [Hov02b]. Lemma 4.14 allows us to carry out the final step of our argument. Proposition 4.15. Suppose f is a map in Ch(Γ), and 0 ≤ k ≤ n. Then vk−1 A/Ik ∧f is a homotopy isomorphism in Ch(Γ) if and only if vk−1 B/Ik ∧Φ∗ f is a homotopy isomorphism in Ch(ΓB ). Proof. Let C = vk−1 A/Ik and let Σ = vk−1 Γ/Ik , as in Lemma 4.14. The category of Σ-comodules is just the full subcategory of Γ-comodules on which Ik acts trivially and vk acts invertibly. This follows from the fact that Ik is

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invariant and vk is primitive modulo Ik . Thus, if X is a complex in Ch(Σ), we can also think of X as a complex in Ch(Γ). As such, X has homotopy groups π∗C (X) and π∗A (X). We claim that these are naturally isomorphic. Indeed, one can easily check that LC, the cobar complex of (C, Σ), is just vk−1 LA/Ik . Hence LC ∧ X is naturally isomorphic to LA ∧ X. From this, one can easily check the desired isomorphism. Therefore, using parts (a) and (b) of Lemma 4.14, we conclude that vk−1 A/Ik ∧ f is a homotopy isomorphism in Ch(Γ) if and only if it is a weak equivalence in Ch(Σ). Similarly, let CB = vk−1 B/Ik and ΣB = vk−1 ΓB /Ik . We find that vk−1 B/Ik ∧ Φ∗ f is a homotopy isomorphism in Ch(ΓB ) if and only if it is a weak equivalence in Ch(ΣB ). Now, use part (c) of Lemma 4.14 and Theorem 3.4 to conclude that vk−1 A/Ik ∧ f is a weak equivalence in Ch(Σ) if and only if v −1 B/Ik ⊗ −1 (v −1 A/Ik ∧ f ) ∼ = v −1 B/Ik ∧ Φ∗ f k

vk A/Ik

k

k

is a weak equivalence in Ch(ΣB ).

We can now complete the proof of Theorem C, which we first restate in stronger form. Theorem 4.16. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence. Furthermore, Φ∗ and its right adjoint Φ∗ preserve and reflect weak equivalences. Proof. Proposition 4.13 and Proposition 4.15 imply that, to check that Φ∗ is a Quillen equivalence, we only need to check that the map − vk−1 B/Ik ∧ Φ∗ Φ∗ (LB) vk−1 B/Ik → is a homotopy isomorphism for 0 ≤ k ≤ n. But Φ∗ Φ∗ is natually isomorphic to the identity functor by Theorem 2.2. Hence we need only check that the map − vk−1 B/Ik ∧ LB vk−1 B/Ik → is a homotopy isomorphism, which follows from Proposition 3.2(d). We have already seen that Φ∗ preserves weak equivalences and that Φ∗ reflects them in Proposition 4.6. Suppose f is a map in Ch(Γ) such that Φ∗ f is a weak equivalence. Since Φ∗ preserves weak equivalences, it follows that Φ∗ Qf is a weak equivalence. But, since Φ∗ is a Quillen equivalence, it must reflect weak equivalences between cofibrant objects by [Hov99, Corollary 1.3.16]. Hence Qf is a weak equivalence, and so f is a weak equivalence. This proves that Φ∗ reflects weak equivalences. Now suppose g is a weak equivalence in Ch(ΓB ). By Theorem 2.2, g is naturally isomorphic to Φ∗ Φ∗ g. Since Φ∗ reflects weak equivalences, we conclude that Φ∗ g is a weak equivalence.

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We point out that it is possible to further localize the Quillen equivalence in Theorem 4.16 to obtain a Quillen equivalence Φ∗ : Ln Ch(Γ) → − Ln Ch(ΓB ) where Ln is Bousfield localization with respect to HE(n) is the first case, and ordinary homology H in the second. The change of rings theorem. In this final part, we show how our work implies the Miller-Ravenel change of rings theorem. To begin with, here is our generic change of rings theorem. Recall our notational conventions: (A, Γ) = (BP∗ , BP∗ BP ), B = E(n)∗ , and ΓB = E(n)∗ E(n). Theorem 4.17. Suppose X ∈ Ch(Γ) and Y is an Lfn -local object of Stable(Γ). Then Stable(Γ)(X, Y )∗ ∼ = Stable(ΓB )(Φ∗ X, Φ∗ Y )∗ . Proof. First of all, since Φ∗ preserves weak equivalences, we have Stable(ΓB )(Φ∗ X, Φ∗ Y ) ∼ = Stable(ΓB )(Φ∗ QX, Φ∗ QY ). Also, by Theorem 4.16, we have Stable(ΓB )(Φ∗ QX, Φ∗ QY ) ∼ = (Lfn Stable(Γ))(X, Y ). Since Y is already Lfn -local, we have (Lfn Stable(Γ))(X, Y ) ∼ = Stable(Γ)(X, Y ), as required.

We claim that this corollary captures the Miller-Ravenel change of rings theorem. To see this, we need the following lemma. Lemma 4.18. Suppose N is a Γ-comodule with Ln N = N and Lin N = 0 for i > 0. Then N is an Lfn -local object of Stable(Γ). Given this lemma, we immediately get the following corollary, which is Theorem D and more like the usual change of rings theorems. Corollary 4.19. Suppose M is a finitely presented BP∗ BP -comodule, and N is a BP∗ BP -comodule such that Ln N = N and Lin N = 0 for all i > 0. Then Ext∗∗ (M, N ) ∼ = ExtE(n) E(n) (E(n)∗ ⊗BP M, E(n)∗ ⊗BP N ). BP∗ BP

∗

∗

∗

This corollary includes both the Miller-Ravenel change of rings theorem [MR77], by taking N with N = vn−1 N , and the change of rings theorem of the author and Sadofsky [HS99], by taking N with vj−1 N = N for some j ≤ n. Here we are using Theorem 2.8(e) to verify the hypothesis of Corollary 4.19. Proof. Simply apply Theorem 4.17, using Lemma 4.18 to see that N is Lfn local, and Proposition 3.8(c) to identify the groups in question as Ext groups.

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We still owe the reader a proof of Lemma 4.18, which, incidentally, is presumably a special case of a spectral sequence that computes H∗ (Lfn X) for X ∈ Stable(Γ) from the derived functors Lin H∗ X, in analogy to the spectral sequence of Theorem 2.9. The converse of Lemma 4.18 is true as well, though we do not need it. Proof of Lemma 4.18. Note that, by definition, N is Lfn -local if and only if Stable(Γ)(A/In+1 , N )∗ = 0. This is equivalent to ExtiΓ (A/In+1 , N ) = 0 for all i, by Proposition 3.8(c). Now ExtiΓ (A/In+1 , N ) = 0 for i = 0, 1 if and only if Ln N = N , by Theorem 2.8(a). Suppose in addition that Lin N = 0 for i > 0. We claim that we can find cosyzygies Cj of N in the category of Γ-comodules such that Ln Cj = Cj and Lin Cj = 0 for i > 0. If we assume this, then for i > 0 we have Exti (A/In+1 , N ) ∼ = Ext1 (A/In+1 , Ci−1 ) = 0. Γ

Γ

We now construct the cosyzygies Cj by induction on j, taking C0 = N . Suppose we have constructed Cj . Since Ln Cj = Cj , Cj has no vn -torsion. It follows that the injective hull, or indeed any essential extension of Cj , has no vn -torsion. Therefore, we can find an short exact sequence 0→ − Cj → − Ij → − Cj+1 → − 0 of Γ-comodules where Ij is an injective comodule with no vn -torsion. In particular, Ln Ij = Ij by Theorem 2.8(a). If we apply Ln to this sequence, we find that Ln Cj+1 = Cj+1 and Lin Cj+1 = 0 for i > 0, completing the induction step. References [Ada74] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill., 1974, Chicago Lectures in Mathematics. MR 53 #6534 [Bek00] Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475. MR 1 780 498 [Ben98] D. J. Benson, Representations and cohomology. I, second ed., Cambridge University Press, Cambridge, 1998, Basic representation theory of finite groups and associative algebras. MR 99f:20001a [Bou79] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281. [GH00] Paul G. Goerss and Michael J. Hopkins, Andr´e-Quillen (co)-homology for simplicial algebras over simplicial operads, Une d´egustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), Amer. Math. Soc., Providence, RI, 2000, pp. 41–85. MR 2001m:18012 [GM95] J. P. C. Greenlees and J. P. May, Completions in algebra and topology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 255–276. MR 96j:55011

Chromatic phenomena in the algebra of BP∗ BP -comodules [Gre93] [Hir03]

[Hol01] [Hov99] [Hov02a] [Hov02b] [HPS97] [HS99]

[HS03a] [HS03b] [Jar01]

[JY80]

[Lan76] [Mil92]

[MR77]

[Rav84] [Rav86]

203

J. P. C. Greenlees, K-homology of universal spaces and local cohomology of the representation ring, Topology 32 (1993), no. 2, 295–308. MR 94c:19007 Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1 944 041 Sharon Hollander, Homotopy theory for stacks, Ph.D. thesis, MIT, 2001. Mark Hovey, Model categories, American Mathematical Society, Providence, RI, 1999. MR 99h:55031 Mark Hovey, Homotopy theory of comodules over a Hopf algebroid, preprint, 2002. Mark Hovey, Morita theory for Hopf algebroids and presheaves of groupoids, Amer. J. Math. 124 (2002), 1289–1318. Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 98a:55017 Mark Hovey and Hal Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, J. London Math. Soc. (2) 60 (1999), no. 1, 284–302. MR 2000h:55017 Mark Hovey and Neil P. Strickland, Comodules and Landweber exact homology theories, preprint, 2003. , Local cohomology of BP ∗ BP -comodules, preprint, 2003. J. F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology Homotopy Appl. 3 (2001), no. 2, 361–384 (electronic), Equivariant stable homotopy theory and related areas (Stanford, CA, 2000). MR 1 856 032 David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), no. 1, 117– 136. MR 81b:55010 Peter S. Landweber, Homological properties of comodules over M U ∗ (M U) and BP∗ (BP), Amer. J. Math. 98 (1976), no. 3, 591–610. MR 54 #11311 H. R. Miller, Finite localizations, Boletin de la Sociedad Matematica Mexicana 37 (1992), 383–390, special volume in memory of Jos´e Adem, in book form, edited by Enrique Ram´ırez de Arellano. Haynes R. Miller and Douglas C. Ravenel, Morava stabilizer algebras and the localization of Novikov’s E2 -term, Duke Math. J. 44 (1977), no. 2, 433–447. MR 56 #16613 Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR 85k:55009 , Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press Inc., Orlando, FL, 1986. MR 87j:55003

Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: [email protected]

NUMERICAL POLYNOMIALS AND ENDOMORPHISMS OF FORMAL GROUP LAWS KEITH JOHNSON

Abstract. Endomorphism rings of formal group laws are used to construct families of polynomials which take integer values at the integers. Applications are made to the study of Hopf algebroids of stable cooperations for generalized homology theories and to the study of generalized Bernoulli numbers.

Primary 13B25, 14L05; Secondary 55N20, 11B68

A numerical polynomial is a polynomial with coefficients in Q which takes integer values at the integers, or more generally, with coefficients in a number field K taking values in A, the ring of integers of K, when evaluated at elements of A. The study of such polynomials has a long history and a substantial literature (for example [5], [17], [21], [22]). The connection with formal group laws and topology stems from the results of [1] where the notion of a stably numerical Laurent polynomial was introduced: a Laurent polynomial f (x) ∈ Q[x, x−1 ] is stably numerical if xk f (x) is numerical for some integer k. The main result of [1] is that the Hopf algebra of degree 0 stable cooperations for complex K-theory, K0 K, is isomorphic to the algebra of stably numerical Laurent polynomials for Z. This is essentially a result connecting stably numerical Laurent polynomials with the multiplicative formal group law, and this connection was described explicitly in [9]. The point of the current paper is that this method is applicable to a large class of formal group laws, that its use provides a large supply of stably numerical Laurent polynomials, both for Z and for the integers in other number fields, and that when it is applied to the formal group laws associated to elliptic curves new and useful information about the Hopf algebroid of stable cooperations for elliptic homology results. It also gives information about the generalized Bernoulli numbers introduced in [20]. The paper is organized as follows: In §1 we recall the basic properties of formal group laws that we require and, for a given formal group law, F , construct a sequence of numerical polynomials, {βi }. In §2 we relate these polynomials for certain formal group laws to the set of generators for the Hopf algebroid of stable cooperations for elliptic homology constructed in [12] and so deduce an integrality condition for elements of that algebroid. In §3 we relate generalized Bernoulli numbers to the polynomials constructed 204

Numerical Polynomials and Endomorphisms of Formal Group Laws 205 in §1 and deduce results related to the classical Von Staudt theorem. In the special case of formal group laws associated to elliptic curves we recover some number-theoretic results due to Carlitz ([6], [7], [8]). 1. Formal Group Laws Let R denote a commutative ring with unit. We will make use of the following definitions and terminology concerning formal group laws: Definition 1.1. A formal group law over R is a power series F (x, y) ∈ R[[x, y]] with the properties F (0, x) = F (x, 0) = x, F (x, F (y, z)) = F (F (x, y), z) and F (x, y) = F (y, x) Definition 1.2. If R is torsion-free then the logarithm of a formal group law, F , over R is the power series logF (x) ∈ R ⊗ Q[[x]] given by Z x dt logF (x) = . ∂F 0 ∂y (t, 0) The inverse (with respect to composition) of log F (x) is the exponential function of F , denoted expF (x). These series have the property that F (x, y) = expF (logF (x) + logF (y)). Definition 1.3. A homomorphism between formal group laws, F and G, over R is a power series α(x) ∈ R[[x]] such that α(0) = 0 and α(F (x, y)) = G(α(x), α(y)) Let J(α) denote the coefficient of x in α(x). α is an isomorphism if J(α) is a unit in R and it is a strict isomorphism if J(α) = 1. Definition 1.4. An endomorphism of a formal group law, F , is a homomorphism of F with itself, i.e. a power series α(x) ∈ R[[x]] with the property α(F (x, y)) = F (α(x), α(y)). The set of endomorphisms forms a ring, using F for addition and composition for multiplication, which we denoted EndR (F ). Any formal group law, F , has a family of endomorphisms defined inductively by [1]x = x, [2]x = F (x, x), [n]x = F ([n − 1]x, x) and F ([−n]x, [n]x) = 0. These define a homomorphism Z → EndR (F ). If R is torsion-free then all endomorphisms of F are of the form α(x) = expF (a logF (x))

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where a = J(α). In particular if R is torsion-free then [n]x = expF (n logF (x)) Definition 1.5. Let R be an algebra over a ring A. A formal A-module is a formal group law, F , over R together with a homomorphism A → EndR (F ) extending the homomorphism Z → EndR (F ) and whose composition with J gives the A-algebra structure map for R. The most common example of a formal A-module occurs when R is an algebra over Z(p) , the integers localized at p. In this case F is always a formal Z(p) -module. (Since [k](x) ≡ kx mod x2 the series [k](x) has an inverse in Z(p) [[x]] if k is an integer prime to p. Thus F has endomorphisms [1/k](x) and so, for any integer l, endomorphisms [l/k](x) = [l]([1/k](x))). In general if F is a formal A-module we will use [a]x to denote the endomorphism of F associated to a. Two examples of formal group laws which we will be particularly interested in are: (a) The multiplicative formal group law F (x, y) = x + y + xy over R = Z. (b) The Euler formal group law associated to the Jacobi elliptic curve y 2 = 1 − 2δx2 + x4 = Sδ, (x): q q F (x, y) = (x Sδ, (y) + y Sδ, (x))/(1 + x2 y 2 ) over R = Z[1/2][δ, ].

Definition 1.6. Let R be a torsion free A-algebra. The polynomial p(w) ∈ (R ⊗ Q)[w] is (A, R)-numerical if p(a) ∈ R whenever a ∈ A. The Laurent polynomial p(w) ∈ (R⊗Q)[w, w −1 ] is stably (A, R)-numerical if there is an integer k such that w k p(w) is (A, R)-numerical. Similarly p(w1 , . . . , wn ) ∈ (R⊗ Q)[w1 , . . . , wn ] is (A, R)-numerical if p(a1 , . . . , an ) ∈ R whenever a1 , . . . an ∈ A and is stably (A, R)-numerical if there are integers k1 , . . . , kn such that (w1k1 . . . wnkn )p(w1 , . . . , wn ) is (A, R)-numerical. If R = A then we will contract these terms to A-numerical and stably Anumerical respectively, and to simply numerical if A can be inferred from the context. Given a formal group law or A-module, F , defined over a torsion free ring R, we may produce a sequence of (Z, R)-numerical or (A, R)-numerical polynomials by expanding the power series expF (w logF (x)) in powers of x: expF (w logF (x)) =

∞ X i=1

βi xi

Numerical Polynomials and Endomorphisms of Formal Group Laws 207 The βi ’s are easily seen to be polynomials in (R ⊗ Q)[w] (with βi of degree ≤ i) and, in view of the fact that for a ∈ Z (or for a ∈ A if F is a formal A-module) ∞ X βi (a)xi = [a](x) ∈ R[x] i=1

we see that the βi ’s are (Z, R) or (A, R)-numerical. In the case of Lazard’s universal formal group law these polynomials were defined, using the umbral calculus, in [25]. In a similar way we may produce stably (Z, R)-numerical or stably (A, R)numerical polynomials by expanding the series (1/v) expF ((v/u) logF (ux)) : (1/v) exp ((v/u) log (ux)) = F

F

∞ X

αi (u, v)xi

i=1

Since the αi ’s are homogeneous and αi (1, w) = (1/w)βi (w) the αi ’s are clearly stably (Z, R)-numerical or (A, R)-numerical. This may appear to be only a minor variation but it has a feature of particular interest in that the αi ’s are the coefficients of a strict isomorphism i.e. α1 (u, v) = 1). Given a formal group law F over a ring R we may, for any unit u ∈ R ∗ , define a related formal group law Fu by Fu (x, y) =

1 F (ux, uy) u

This formal group law is, of course, isomorphic to F (via α(x) = ux) and so any two such formal group laws Fu and Fv are isomorphic. If the invertible power series ux and vx lie in the endomorphism ring of F (for example if R is a Z(p) -module and u, v are integers prime to p) then among the isomorphisms from Fu to Fv is v v −1 [ ](ux) u which is the composition of isomorphisms from Fu to F and F to Fv with an automorphism of F . This particular isomorphism is the unique strict isomorphism from Fu to Fv and X v v [ ](ux) = αi (u, v)xi+1 u i=1 ∞

−1

A particular case of this (originally appearing in [9](see also [12])), in which the coefficient polynomials, αi , are familiar is that of the multiplicative group law x + y + xy. The logarithm in this case is the analytic log logF (x) = ln(1 + x) =

∞ X (−1)i−1 xi i=1

i

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and so expF (x) = ex − 1. Thus v −1 [v/u](ux) = v −1 expF ((v/u) logF (ux)) = v −1 (e(v/u)ln(1+ux) − 1) = v −1 ((1 + ux)v/u − 1) ∞ X v/u −1 =v (ux)i i i=1 and so αi (u, v) = (v − u)(v − 2u) . . . (v − (i − 1)u)/i! is the i-th homogeneous binomial polynomial. An alternative approach to the integrality and divisibility questions concerning formal group laws considered here is by way of the umbral calculus. A development of this material for similar applications in algebraic topology can be found in [24] and [11]. 2. Stable Cooperations and Elliptic Homology If E∗ ( ) is a complex oriented homology theory then there is associated to it a formal group law, FE over the coefficient ring E∗ . A basic problem is to calculate E∗ E, the Hopf algebroid of stable cooperations for E∗ ( ) which in some cases may be described in terms of FE . If we restrict our attention to those theories possessing a Conner-Floyd isomorphism E∗ ( ) ∼ = E∗ ⊗M U∗ MU∗ ( ) then E∗ E may be identified with the subalgebra of E∗ ⊗ E∗ ⊗ Q generated L over E∗ by the coefficients of the series expLFE (logR FE (x)). Here expFE and logR FE are the exponential and logarithm series of the two formal group laws L FE = ηL∗ FE and FER = ηR∗ FE induced from FE by the left and right counit maps ηL , ηR : E∗ → E∗ E. The Hopf algebroid structure of E∗ ⊗ E∗ ⊗ Q over E∗ is determined by ηL = id ⊗ 1 and ηR = 1 ⊗ id. A more detailed account of this approach to E∗ E can be found in section 1 of [12]. If the homology theory is complex K-theory, K∗ ( ) then K∗ = Z[t, t−1 ], the associated formal group law is the graded multiplicative law FK = x+y +txy, FKL and FKR are Fu and Fv respectively where F is the ungraded multiplicative law and u = ηL (t) and v = ηR (t). It follows that K∗ K is the subalgebra of K∗ ⊗ K∗ ⊗ Q generated over K∗ by the homogeneous binomial polynomials. For elliptic homology, Ell∗ ( ), as defined in [18], Ell∗ = Z[1/2][δ, , ∆−1 ] were ∆ is the discriminant of the associated Jacobi elliptic curve, the associated formal group law is the Euler formal group law of section 1 which we will denote FEll , the logarithm is Z x dt p logEll (x) = Sδ, (t) 0

Numerical Polynomials and Endomorphisms of Formal Group Laws 209 −1 and Ell∗ Ell is a subalgebra of Ell∗ ⊗ Ell∗ ⊗ Q = Q[δL , L , δR , R , ∆−1 L , ∆R ] where δL = ηL (δ) etc. If we form the series

m(x) =

expLEll (logR Ell (x))

=

∞ X

mi (δL , δR , L , R )xi

i=1

then the coefficients, which are polynomials in δL , δR , L , and R , generate Ell∗ Ell over Ell∗ . The Hopf algebroid Ell∗ Ell has been studied in [12], [19] and [3]. To obtain integrality conditions that these polynomials must satisfy we relate them to formal group laws of the form Fu considered in section 1 by defining, for integers k, l, homomorphisms φ : Ell∗ → Z[1/2] and Φ : Ell∗ → Z[1/2][t] by φ(δ) = k, φ() = l, Φ(δ) = kt2 , Φ() = lt4 . These maps have the property Φ∗ FEll = (φ∗ FEll )t Proposition 2.1. For any integers k, l the polynomials mi (ku2 , kv 2 , lu4 , lv 4 ) ∈ Q[u, v] are stably Z[1/2]-numerical. Proof. If we make Q[u, v] into a Hopf algebroid over Q[t] by taking u = ηL (t), v = ηR (t) then Φ∗ extends to a map of Hopf algebroids Φ∗ ⊗ Φ∗ : Ell∗ ⊗ Ell∗ ⊗ Q → Q[u, v] such that mi (ku2 , kv 2 , lu4 , lv 4 ) = (Φ∗ ⊗ Φ∗ )(mi ) We also have ∞ X (Φ∗ ⊗ Φ∗ )(mi )xi = expΦ∗ FL (logΦ∗ FR (x)) i=1

= exp(φ∗ F )v (log(φ∗ F )u (x)) 1 v = expφ∗ F ( logφ∗ F (ux)) v u and so the coefficient polynomials of this series are stably Z[1/2]-numerical by the results of section 1. For special values of k, l we can say more. An elliptic curve having the property that its endomorphism ring, A, is not a subring of Q is said to have complex multiplication. For curves of this sort the associated formal group law will, after extension of its ring of definition, be a formal A-module. There is an extensive literature devoted to this subject including [14], in which the possibilities for the endomorphism ring are enumerated. It is known that in characteristic 0 such curves have endomorphism rings which are orders in imaginary quadratic field extensions of Q, and also that there are only 13 isomorphism classes of such curves whose j-invariants are rational. These are catalogued in [13], p. 376. Since the j-invariant of the curve associated to the Euler formal group law is a rational function of δ and , we may

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enumerate some values of k, l for which we obtain an extension of the previous proposition. Proposition 2.2. For the following values of δ, the endomorphism ring of the Jacobi elliptic curve y 2 = 1 − 2δx2 + x4 is the order Z + τ Z in the field Q(τ ). δ τ j(τ ) 0 3 −3 −21 2 3 6 21 18 7 33 21 513

1 1 −3 −7 2 12 32 448 √ 162 − 90i √ 7 57 − 40 2√ 417 + 240√3 249 − 176 2 √ 129537 + 48960i 7

i i √ (1 + i√3)/2 (1 √+ i 7)/2 i√2 i 3 2i√ i 7√ 1√+ i 7/2 i√2 i 3 2i√ i 7

1728 1728 0 −3375 8000 54000 287496 16581375 −3375 8000 54000 287496 16581375

Corollary 2.3. For k, l taking the values of δ, listed in the previous proposition the polynomials mi (ku2 , kv 2 , lu4 , lv 4 ) ∈ Q[1/2, τ ][u, v] are stably numerical for the ring Z + τ Z. The values of k, l given in corollary 2.3 are not the only ones for which the polynomials mi are stably numerical for a ring larger than Z. There are irrational values of the j-invariant for which the associated Euler curve has complex multiplication and so the same phenomena occurs however these are not so easy to classify. A consequence of this is that the conditions on the polynomials mi we have described cannot completely characterize the Hopf algebroid Ell∗ Ell. There are two other approaches to Ell∗ Ell which do give complete characterizations, however. One, due to Baker ([3]) uses generalized modular forms while the other, due to Laures ([19]) uses 2 variable q-series. This raises the possibility of using these essentially topological results to obtain number theoretic information about elliptic curves by way of further integrality conditions that the polynomials mi might satisfy. 3. Bernoulli Numbers Definition 3.1 ([20]). The Bernoulli numbers of a formal group law F over R are the elements BnF of R defined by ∞ X x xn = BnF expF (x) n=0 n!

Numerical Polynomials and Endomorphisms of Formal Group Laws 211 It follows easily from this definition that these numbers can be computed inductively in terms of the coefficients of the exponential series exp F (x) = P n xn /n! using the recurrence ( n X n+1 1 if n = 0 n−j BjF = j 0 if n > 0 j=0

Arithmetic properties and topological applications of these numbers have been studied in [10], [18], [24]. For the multiplicative formal group law they specialize to the classical Bernoulli numbers which we will denote simply by Bn . In this case a connection between these numbers and numerical polynomials is given by: Proposition 3.2.

Bn (w n 2n

− 1) is stably numerical and not divisible.

One interpretation of this result is as equating the denominator of Bn /2n and the largest integer m(n) (ordered by divisibility) for which (w n −1)/m(n) is stably numerical. The first of these is given by Von Staudt’s theorem while the second can be calculated one prime factor at a time using the structure of (Z/pZ)∗ . The natural generalization of this result requires a preliminary definition: Definition 3.3. If F is a formal group law over the torsion-free ring R, then the algebra of F -numerical polynomials, which we will denote A F , is the subalgebra of (R ⊗ Q)[w] generated by the polynomials {βi (w), i = 1, 2, . . . } associated to F by the construction given in §1. The algebra of stably F numerical polynomials is the subalgebra AF [w −1 ]. ¿From §1 it is clear that elements of AF are numerical for Z or for A if F is a formal A-module. AF need not, however, equal the full algebra of polynomials numerical for Z or A. For example if F is the additive formal group law, then AF contains only constant polynomials. Based on this definition we have the following generalization of Proposition 3.2. In the case of formal group laws of singular elliptic curves an equivalent result may be found in [8]. Proposition 3.4.

F Bn n

(w n − 1) is stably F-numerical.

Proof. First note that, from the definition of the BnF ’s we have: ∞ ∞ w 1 1 X BnF w n xn 1 X BnF xn − = − expF (wx) expF (x) x n=0 n! x n=0 n! =

∞ X BF n

n=1

Next note that

n

(w n − 1)

xn−1 (n − 1)!

exp (x) − w1 expF (wx) 1 w − = 1 F expF (wx) expF (x) expF (x) expF (wx) w

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and that the numerator and denominator of this quotient can both be expressed as power series in expF (x) with coefficients in AF [w −1 ], both of which have leading term x. We thus have ∞ X w 1 − = γi (w)(expF (x))i expF (wx) expF (x) i=0

with each γi (w) stably F-numerical. Finally note that for any i if (expF (x))i is expanded in powers of x the result is a series of the form ∞ X xj rj j! j=1 F

with rj ∈ R. Combining these we see that Bnn (w n − 1) can be expressed as an R-linear combination of the γi ’s and so is in AF [w −1 ] The generalization of Von Staudt’s theorem to the BnF ’s has been established by Clarke ([10]), and gives an expression for the denominator of BnF /n in terms of the coefficients of logF (x). If AF is an algebra for which it is possible to determine directly the largest possible denominator, m(n), for which (w n − 1)/m(n) is stably numerical then this proposition can be used to establish divisibility results about the log coefficients. This is the case, for example, with the Euler formal group laws considered in §2. For algebras of numerical polynomials for rings of integers in quadratic number fields the function m(n) can be determined one prime factor at a time as in the rational case. Thus, for values of k, l, for which the the associated curve has complex multiplication, divisibility results about the associated log coefficients, which are special values of the Legendre polynomials, can be deduced. One consequence of this is a proof that such curves have supersingular reduction at rational primes which ramify or are inert in the number field and ordinary reduction at primes which split. References [1] J. F. Adams, A. S. Harris and R. M. Switzer, Hopf algebras of cooperations for real and complex K-theory, Proc. London Math. Soc. (3) 23, (1971), 385–408. [2] A. Baker, Combinatorial and Arithmetic Identities Based on Formal Group Laws, Algebraic Topology, Barcelona, 1986 (J.Aguade and R. Kane, eds.), Lecture Notes in Math. v. 1298, Springer Verlag, New York, 1987, pp. 17–34. [3] A. Baker, Operations and Cooperations in Elliptic Cohomology, Part 1 : Generalized Modular Forms and the Cooperation Algebra, New York Jour. Math. 1 (1995), 39–74. [4] A. Baker, F. W. Clarke, N.Ray, and L. Schwartz, On the Kummer Congruences and the Stable Homotopy of BU , Trans. Am. Math. Soc. 315 (1989), 591–603. [5] P. J. Cahen, Polynomes a Valeurs Entieres, Canad. J. Math. 24 (1972), 747–754. [6] L. Carlitz, The Coefficients of the Reciprocal of a Series, Duke Math. J. 8 (1941), 689–700. [7] L. Carlitz, Congruences Connected With The Power Series Expansions of the Jacobi Elliptic Functions, Duke Math. J. 20 (1953), 1–12.

Numerical Polynomials and Endomorphisms of Formal Group Laws 213 [8] L. Carlitz, The Coefficients of Singular Elliptic Functions, Math. Annalen 127 (1954), 162–169. [9] F. W. Clarke, On the determination of K∗ (K) using the Conner-Floyd isomorphism, preprint, Swansea, 1972. [10] F. W. Clarke, The Universal Von Staudt Theorems, Trans. Am. Math. Soc. 315 (1989), 591–603. [11] F. W. Clarke, J. Hunton and N. Ray, Extensions of Umbral Calculus II: Double Delta Operators, Leibniz Extensions and Hattori-Stong Theorems, Ann. Inst. Fourier 51 (2001), 297–336. [12] F. W. Clarke and K. Johnson, Cooperations in Elliptic Homology, LMS Lecture Notes 176 (1992), 131–143. [13] H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993. [14] M. Deuring, Die Typen der Multiplikatorenringe ellipisher Functionen korper, Abh. Math. Sem. Hamburg 14 (1941), 197–272. [15] L. Euler, De integratione aequationis differentialis √mdx = √ndy 4 , Novi Comm. 1−x4 1−y

[16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

Acad. Sci. Petropolitanae 6 (1756/7), 1761, 37–57; Collected Works, first series, Vol. 20, B. G. Teuber, 1912. M. Hazewinkel, Formal Groups and Applications, Academic Press, New York, (1973). D. Hensley, Polynomials which take Gaussian Integer Values at Gaussian Integers, J. Number Theory 9 (1977), 510–524. P. S. Landweber, Elliptic cohomology and modular forms, Elliptic curves and modular forms in topology, Princeton 1986, Lecture Notes in Math., vol. 1326, 1988, pp. 55–68. G. Laures, The Topological q-expansion Principle, Topology 38 (1999), 387–425. H. Miller, Universal Bernoulli Numbers and the S 1 Transfer, Current Trends in Algebraic Topology, CMS-AMS, Providence, R.I., 1982, pp. 437–449. A. Ostrowski, Uber Ganzwertige Polynome in Algebraischen Zahlkorper, J. Reine Angew. Math. (Crelle) 149 (1919), 117–124. G. Poyla, Uber Ganzwertige Polynome in Algebraischen Zahlkorper, J. Reine Angew. Math. (Crelle) 149 (1919), 97–116. D. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, New York, 1986. N. Ray, Extensions of Umbral Calculus: Penumbral Coalgebras and Generalized Bernoulli Numbers, Adv. in Math. 61 (1986), 49–100. N. Ray, Loops on the 3-sphere and Umbral Calculus, Contemp. Math. 96 (1989), 297–302.

Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 E-mail address: [email protected]

THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS NITU KITCHLOO AND JACK MORAVA Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new localization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.

Introduction If P → M is a principal bundle with structure group G then LP → LM is a principal bundle with structure group LG = Maps(S 1 , G), We will assume in the rest of the paper that the frame bundle of M has been refined to have structure group G, and that the tangent bundle of M is thus defined via some representation V of G. It follows that the tangent bundle of LM is defined by the representation LV of LG. The circle group T acts on all these spaces by rotation of loops. This is a report on the beginnings of a theory of differential topology for such objects. Note that if we want the structure group LG to be connected, we need G to be 1-connected; thus SU (n) is preferable to U (n). This helps explain why Calabi-Yau manifolds are so central in string theory, and this note is written assuming this simplifying hypothesis. Alternately, we could work over the universal cover of LM; then π2 (M ) would act on everything by decktranslations, and our topological invariants become modules over the Novikov ring Z[H2 (M )]. ¿From the point of view we’re developing, these translations may be relevant to modularity, but this issue, like several others, will be backgrounded here. Date: 9 March 2005. 1991 Mathematics Subject Classification. 22E6, 55N34, 55P35. NK is partially supported by NSF grant DMS 0436600, JM by DMS 0406461. 214

Thom prospectra for loopgroup representations

215

The circle action on the free loopspace defines a structure much closer to classical differential geometry than one finds on more general (eg) Hilbert manifolds; this action defines something like a Fourier filtration on the tangent space of this infinite-dimensional manifold, which is in some sense locally finite. This leads to a host of new kinds of geometric invariants, such as the Witten genus; but this filtration is unfamiliar, and has been difficult to work with [17]. The main conceptual result of this note [which was motivated by ideas of Cohen, Godin, and Segal] is the definition of a canonical ‘thickening’ L† M of a free loopspace, with the same equivariant homotopy type, but better differential-geometric and analytic properties: the usual tangent bundle to the free loopspace, pulled back over this ‘dressed’ model, admits a canonical filtration by finite-dimensional equivariant bundles. This thickening involves a contractible LG-space called the affine Tits building A(LG), which occurs under various guises in nature: it is a homotopy colimit of homogeneous spaces with respect to a finite collection of compact Lie subgroups of LG, and it is also the affine space of principal G-connections on the trivial bundle over S 1 ; we explore its structure in more detail in the appendix. The thickening L† M seems very natural, from a physical point of view: its elements are not the raw geometric loops of pure mathematics, but are rather loops endowed with a choice of connection (depending on the structure group G of interest). This is a conceptual distinction which means little in pure mathematics, but perhaps a lot in physics. In the first section below, we recall why the Spanier-Whitehead dual of a finite CW-space is a ring-spectrum, and sketch the construction (due to Milnor and Spanier, and Atiyah) of a model for that dual, when the space is a smooth compact manifold. Our goal is to produce an analog of this construction for a free loopspace, which captures as much as possible of its string-topological algebraic structure. In the second section, we introduce the technology used in our construction: pro-spectra associated to filtered infinite-dimensional vector bundles, and the topological Tits building which leads to the construction of such a filtration for the tangent bundle. In §3 we observe that recent work of Freed, Hopkins, and Teleman on the Verlinde algebra can be reformulated as a conjectural duality between LG-equivariant K-theory of a certain dualizing spectrum for LG constructed from its Tits building, and positive-energy representations of LG. In §4 we use a new strong localization theorem to study the equivariant K-theory of our construction, and we show how this recovers the Witten genus from a string-topological point of view. We plan to discuss actions of various string-topological operads [15] on our construction in a later paper; that work is in progress.

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Nitu Kitchloo and Jack Morava

We would like to thank R. Cohen, V. Godin, S. Stolz, and A. Stacey for many helpful conversations, and we would also like to acknowledge work of G. Segal and S. Mitchell as motivation for many of the ideas in this paper. 1. The Atiyah dual of a manifold If X is a finite complex, then the function spectrum F (X, S 0 ) is a ringspectrum (because S 0 is). If X is a manifold M, Spanier-Whitehead duality says that F (M+ , S 0 ) ∼ M −T M . If E → X is a vector bundle over a compact space, we can define its Thom space to be the one-point compactification X E := E+ . There is always a vector bundle E⊥ over X such that E ⊕ E⊥ ∼ = 1N is trivial, and following Atiyah, we write X −E := S −N X E⊥ . With this notation, the Thom collapse map for an embedding M ⊂ RN is a map ν N S N = RN M −T M , + → M = S Moreover, the Thom collapse for the diagonal embedding of M into the zero section of M+ ∧ M ν gives us N N M+ ∧ M ν → M ν⊕T M = M+ ∧ RN + → R+ = S

defining the equivalence with the functional dual. More generally, a smooth map f : M → N of compact closed orientable manifolds has a PontrjaginThom dual map fP T : N −T N → M −T M of spectra; in particular, the map S 0 → M −T M dual to the projection to a point defines a kind of fundamental class, and the dual to the diagonal of M makes M −T M into a ring-spectrum. The ring structure of M −T M has been studied by various authors (see for example [13], [25]). Prospectus: Chas and Sullivan [11] have constructed a very interesting product on the homology of a free loopspace, suitably desuspended, motivated by string theory. Cohen and Jones [15] saw that this product comes from a ring-spectrum structure on LM −T M := LM −e where e : LM → M

∗T M

Thom prospectra for loopgroup representations

217

is the evaluation map at 1 ∈ S 1 . Unfortunately this evaluation map is not T-equivariant, so the Chas-Sullivan Cohen-Jones spectrum is not in general a T-spectrum. The full Atiyah dual constructed below promises to capture some of this equivariant structure. The Chas-Sullivan Cohen-Johes spectrum and the full Atiyah dual live in rather different worlds: our prospectrum is an equivariant object, whose multiplicative properties are not year clear, while the CSCJ spectrum has good multiplicative properties, but it is not a Tspectrum. In some vague sense our object resembles a kind of center for the Chas-Sullivan-Cohen-Jones spectrum, and we hope that a better understanding of the relation between open and closed strings will make it possible to say something more explict about this.

2. Problems & Solutions For our constructions, we need two pieces of technology: Cohen, Jones, and Segal [16](appendix) associate to a filtration E : · · · ⊂ Ei ⊂ Ei+1 ⊂ . . . of an infinite-dimensional vector bundle over X, a pro-object X −E : · · · → X −Ei+1 → X −Ei → . . . in the category of spectra. [A rigid model for such an object can be constructed by taking E to be a bundle of Hilbert spaces, which are trivializable by Kuiper’s theorem. Choose a trivialization E ∼ = H × X and an exhaustive filtration {Hk } of H by finite-dimensional vector spaces; then we can define ⊥

X −Ei = lim S −Hk X Hk ∩Ei , with Ei⊥ the orthogonal complement of Ei in the trivialized bundle E.] This pro-object will, in general, depend on the choice of filtration. We will be interested in the direct systems associated to such a pro-object by a cohomology theory; of course in general the colimit of this system can be very different from the cohomology of the limit of the system of pro-objects. Example 2.1. If X = CP∞ , η is the Hopf bundle, and E is ∞η : · · · ⊂ (k − 1)η ⊂ kη ⊂ (k + 1)η ⊂ . . . then the induced maps of cohomology groups are multiplication by the Euler classes of the bundles Ei+1 /Ei , so −∞η H ∗ (CP∞ , Z) := colim{Z[t], t − mult} = Z[t, t−1 ] .

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Nitu Kitchloo and Jack Morava

We would like to apply such a construction to the tangent bundle of a free loopspace. Unfortunately, these tangent bundles do not, in general, possess any such nice filtration by finite-dimensional (T-equivariant) subbundles [17]! However, such a splitting does exist in a neighborhood of the constant loops: M = LM T ⊂ LM has normal bundle ν(M ⊂ LM) = T M ⊗C (C[q, q −1 ]/C) (at least, up to completions; and assuming things complex for convenience). Here small perturbations of a constant loop are identified with their Fourier expansions X an q n , n∈Z

with q = eiθ . The related fact, that T LM is defined by the representation LV of LG looped up from the finite-dimensional representation V of G, will be important below: for LV is not a positive-energy representation of LG. The main step toward our resolution of this problem depends on the following result, proved in §7 below. Such constructions were first studied by Quillen, and were explored further by S. Mitchell [26]. The first author has studied these buildings for a general Kac-Moody group [23]; most of the properties of the affine building used below hold for this larger class.

Theorem 2.2. There exist a certain finite set of compact ‘parabolic’ subgroups HI of LG (see 7.2), such that the topological affine Tits building A(LG) := hocolimI LG/HI ˜ of LG is T×LG-equivariantly contractible. In other words, given any com˜ pact subgroup K ⊂ T×LG, the fixed point space A(LG)K is contractible. Remark 2.3. The group LG admits a universal central extension LG by a circle group C. The natural action of the rotation group T on LG lifts to LG, and the T-action preserves the subgroups HI . Hence A(LG) admits an ˜ action of T×LG, with the center acting trivially. We can therefore express A(LG) as A(LG) = hocolimI LG/HI where HI is the induced central extension of HI .

Thom prospectra for loopgroup representations

219

Other descriptions of A(LG) This Tits building has other descriptions as well. For example: 1. A(LG) can be seen as the classifying space for proper actions with respect ˜ to the class of compact Lie subgroups of T×LG. That means that the space ˜ A(LG) is a T×LG-CW complex with the isotropy groups of the action being ˜ compact Lie groups. Moreover, given any compact Lie group K ⊂ T×LG, the K fixed point space A(LG) is contractible. It is not hard to show that these ˜ properties uniquely determine A(LG) upto T×LG-homotopy equivalence. 2. It also admits a more differential-geometric description as the smooth infinite dimensional manifold of holonomies on S 1 × G (see Appendix): Let S denote the subset of the space of smooth maps from R to G given by S = {g : R → G, g(0) = 1, g(t + 1) = g(t) · g(1)} ; then S is homeomorphic to A(LG). The action of h(t) ∈ LG on g(t) is given by hg(t) = h(t) · g(t) · h(0)−1 , where we identify the circle with R/Z. The action of x ∈ R/Z = T is given by xg(t) = g(t + x) · g(x)−1 . 3. The description given above shows that A(LG) is equivalent to the affine space A(S 1 × G) of connections on the trivial G-bundle S 1 × G. This identification associates to the function f (t) ∈ S, the connection f 0 (t)f (t)−1 . Conversely, the connection ∇t on S 1 × G defines the function f (t) given by transporting the element (0, 1) ∈ R × G to the point (t, f (t)) ∈ R × G using the connection ∇t pulled back to the trivial bundle R × G. Remark 2.4. These equivalent descriptions have various useful consequences. For example, the model given by the space S of holonomies says that given a finite cyclic group H ⊂ T, the fixed point space S H is homeomorphic to S. Moreover, this is a homeomorphism of LG-spaces, where we consider S H as an LG-space and identify LG with LGH in the obvious way. Notice also that S T is G-homeomorphic to the model of the adjoint representation of G defined by Hom(R, G). Similarly, the map S → G given by evaluation at t = 1 is a principal ΩG bundle, and the action of G = LG/ΩG on the base G is given by conjugation. This allows us to relate our work to that of Freed, Hopkins and Teleman in the following section. Finally, the description of A(LG) as the affine space A(S 1 × G) implies that the fixed point space A(LG)K is contractible for any compact subgroup ˜ K ⊆ T×LG. If E → B is a principal bundle with structure group LG, then (motivated by ideas of [14]) we construct a ‘thickening’ of B:

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Definition 2.5. The thickening of B associated to the bundle E is the space B † (E) = E ×LG A(LG) = hocolimI E/HI . We will omit E from the notation, when the defining bundle is clear from context. Remark 2.6. If P → M is a principal G bundle, then LP → LM is a principal LG bundle. In this case, the description above gives L† M := LM † (LP ) a smooth structure: L† M = {(γ, ω) | γ ∈ LM,

ω ∈ A(γ ∗ (P ))}

where A(γ ∗ (P )) is the space of connections on the pullback bundle γ ∗ (P ). ˜ Let T×LG be the extension of the central extension of LG by T, acting as rotations, and let U be a unitary representation of this group, of finite type. ˜ We propose to construct a Thom T×LG-prospectrum A(LG)−U , as a substitute for the nonexistent equivariant prospectrum defined by −U , regarded as a vector bundle over a point. The central extension of LG splits when restricted to the constant loops, ˜ so we have a torus T := T × C × T ⊂ T×LG, where T is a maximal torus of G, and C is the circle of the central extension; T is in fact a common ˇ be the character group maximal torus for the family HI of parabolics. Let T ˇ of this torus, and let Z[[T]] be its completed group ring. On restriction to ˜ I, the subgroup T×H ∼ U |T×H ˜ I = ⊕ UI (α) decomposes into a sum of finite dimensional representations; because U is of finite type, the isotypical summands appear only finitely often. Let ˇ char U |T×H ∈ Z[[T]] ˜ I

be its character. ˇ of characters, let For any finite subset R ⊂ T UI (R) = ⊕ {UI (α) | char UI (α) ⊂ ZhRi} ⊂ UI be the subresentation of UI with ‘support in R’, and let SI−U to be the Thom ˜ I -prospectrum associated to the filtered (equivariant) vector bundle R 7→ T×H UI (R) over a point. If I ⊂ J then HI maps naturally to HJ , and there is a corresponding morphism UJ → UI of filtered vector bundles, given by inclusions UJ (R) → UI (R). This defines a filtered system of vector bundles over the small category or diagram defined by inclusions of parabolics.

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˜ Example 2.7. Consider the standard representation of T×LSU (2) on LC2 . Let L1 and L2 denote the canonical coordinate summands of C2 , and let q denote the character of the rotation group T. The parabolic subgroups HI ⊂ LSU (2) in this case are: H1 = SU (2) ⊂ LSU (2) given by constant loops, and a bz −1 a b H0 = { } where ∈ SU (2) ; cz d c d

T = H∅ = H0 ∩H1 is the maximal torus of SU (2). Notice that the representation LC2 of the loop group is certainly not of positive energy. However, when ˜ I , it decomposes as a sum of representations: restricted to the subgroups T×H M LC2 = (L1 ⊕ L2 )q k k∈Z

˜ 1 , T×H ˜ ∅ and as for T×H

M

(L1 ⊕ qL2 )q k

k∈Z

˜ 0 . The resulting filtered vector bundle is in this case just a pushout. for T×H ˜ Definition 2.8. We define A(LG)−U to be the T×LG-prospectrum A(LG)−U = hocolimI LG+ ∧HI SI−U , where LG+ denotes LG, with a disjoint basepoint. Homotopy colimits in the category of prospectra can be defined in general, using the model category structure of [12]. Remark 2.9. Given any principal LG-bundle E → B, and a representation U of LG, we define the Thom prospectrum of the virtual bundle associated to the representation −U to be B!−U = E+ ∧LG A(LG)−U = hocolimI E+ ∧HI SI−U . In particular, if P is the refinement of the frame bundle of M via a representation V of G, then the tangent bundle of LM is defined by the representation LV of LG. Definition 2.10. The Atiyah dual LM −TLM of LM is the T-prospectrum L† M −LV . We will explore this object further in §6. Note that its underlying nonequivariant object maps naturally to (a thickening of) the Chas-Sullivan spectrum.

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The dualizing spectrum of a topological group K is defined [24] as the K-homotopy fixed point spectrum: DK = K+hK = F (EK+ , K+ )K where K+ is the suspension spectrum of the space K+ , endowed with a right K-action. The dualizing spectrum DK admits a K-action given by the residual left K-action on K+ . If K is a compact Lie group, then it is known [24] that DK is the one point compactification of the adjoint representation Ad(K)+ . It is also known that there is a K × K-equivariant homotopy equivalence K+ ∼ = F (K+ , DK ) . It follows from the compactness of K+ that for any free K+ -spectrum E, we have the K-equivariant homotopy equivalence E∼ = F (K+ , E ∧K DK ) . +

It is our plan to understand the dualizing spectrum for the (central extension of the) loop group. Theorem 3.1. There is an equivalence DLG ∼ = holimI LG+ ∧HI Ad(HI )+ of left LG-spectra. Proof. We have the sequence of equivalences: DLG = F (ELG+ , LG+ )LG ∼ = F (ELG+ ∧ A(LG)+ , LG+ )LG . The final space may be written as holimI F (ELG+ ∧HI LG+ , LG+ )LG = holimI F (ELG+ , LG+ )HI . Now recall the equivalence of HI × HI -spectra: (1) LG+ ∼ = F (HI + , LG+ ∧HI DHI ) . Taking HI -homotopy fixed points implies a left HI -equivalence F (ELG+ , LG+ )HI = (LG+ )hHI ∼ = LG+ ∧HI Ad(HI )+ ; where we have used equation (1) at the end. Replacing this term into the homotopy limit completes the proof. Similarly, we have: Theorem 3.2. There is an equivalence DLG ∼ = holimI LG+ ∧HI Ad(HI )+ of left LG-spectra.

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Remark 3.3. The diagram underlying DLG or DLG can be constructed in the category of spaces. Given an inclusion I ⊆ J, the orbit of a suitable element in Ad(HJ ) gives an embedding HJ /HI ⊂ Ad(HJ ), and the Pontrjagin-Thom construction for this embedding defines an HJ -equivariant map Ad(HJ )+ −→ HJ + ∧HI Ad(HI )+ which extends to the map LG+ ∧HJ Ad(HJ )+ −→ LG+ ∧HI Ad(HI )+ required for the diagram. Moreover, composites of these maps can be made compatible up to homotopy. A conjectural relationship with the work of Freed, Hopkins and Teleman The discussion below assumes the existence of LG-equivariant K-theory, as defined in [19] (see Appendix). Given a space X with a proper LGaction, Freed-Hopkins-Teleman define an LG-equivariant spectrum over X. The equivariant K-theory groups are defined as the homotopy groups of the space of sections of this spectrum. In this section, we will assume that these K-theory groups can be defined for spectra X with proper LG-action. We will primarily be interested with LG-CW spectra with proper isotropy. This hypothesis provides us with a convenient language. We expect to return to the underlying technical issues in a later paper. The center of LG acts trivially on DLG , defining a second grading on ∗ KLG (DLG ); we will use a formal variable z to keep track of the grading, so M ∗,n ∗ KLG (DLG ) = KLG (DLG )z n . n

The spectral sequence for the cohomology of a cosimplicial spectrum, in the ∗ case of KLG (DLG ), has E2i,j = colimiI KHj I (Ad(HI )+ ) .

This spectral sequence respects the second grading given by powers of z. As before, we may decompose KH∗ I (Ad(HI )+ ) under the action of the center of LG (contained in HI ) M ∗,n KH∗ I (Ad(HI )+ ) = KHI (Ad(HI )+ )z n n

∗,n We therefore have a spectral sequence converging to KLG (DLG ), with

E2i,j,n = colimiI KHj,nI (Ad(HI )+ ) . For n > 0, we will show presently that this spectral sequence collapses to give ∗,n KLG (DLG ) = colimI KH∗,n (Ad(HI )+ ) I

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Therefore, this group admits a natural Thom class given by the system {Ad(HI )+ }, of spinor bundles for the adjoint representations of the parabolics HI . We therefore have a (global) Thom isomorphism: K ∗+r+1,n (DLG ) ∼ = colimI Rep∗,n (HI ) . LG

where Rep (HI ) is the subgroup of Rep∗ (HI ) corresponding to KH∗,n I (Ad(HI )+ ) under the (local) Thom isomorphism. We will also show that the above colimit may naturally be identified with the free abelian group generated by the regular, dominant characters of level n. This free abelian group can further be identified with the Grothendieck group of level n−h positive energy representations of LG, where h denotes the dual Coxeter number. This Grothendieck group is known as the Verlinde algebra (of level n − h). ∗+r+1,n Therefore, we get an (abstract) isomorphism between KLG (DLG ) and the Verlinde algebra of level n − h. In [19], Freed-Hopkins-Teleman give a geometric meaning to the Thom isomorphism above, which explains the shift in level by showing that the Thom class has internal level h. ∗,n

Under the assumptions on KLG given above, we get: ∗,n Theorem 3.4. For n > 0, the groups KLG (DLG ) are two-periodic. We have a (Thom) isomorphism of groups: r+1,n (DLG ) Vn−h ∼ =K LG

where Vk is the Verlinde algebra of level k, h is the dual Coxeter number of r,n G, and r is its rank. Moreover, KLG (DLG ) = 0. Before we prove the collapse of the spectral sequence, let us consider an example: Example 3.5. To illustrate this in an example, recall the case of G = SU (2). In this case r = 1, h(G) = 2. Here the groups HI are given by H0 = SU (2) × S 1 ,

H1 = S 1 × SU (2),

H 0 ∩ H1 = T = S 1 × S 1 .

The respective representation rings may be identified by restriction with subalgebras of KT (pt) = Z[u±1 , z ±1 ] : KH0 (pt) = Z[u + u−1 , (z/u)±1 ],

KH1 (pt) = Z[z ±1 , u + u−1 ] .

Now consider the two pushforward maps involved in the colimit: ϕ0 : KT (pt) → KH0 (pt),

ϕ1 : KT (pt) → KH1 (pt)

A quick calculation shows that for k > 0, we have ( (z/u)k Symk (u + u−1 ), j = 0 ϕj (z k ) = zk , j=1 ( (z/u)k Symk−1 (u + u−1 ), j = 0 k −1 ϕj (z u ) = 0, j=1,

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where Symk (V ) denotes the k-th symmetric power of the representation V , e.g. Symk (u + u−1 ) = uk + · · · + u−k . The colimit is the cokernel of ϕ1 ⊕ ϕ0 : KT (pt) −→ KH1 (pt) ⊕ KH0 (pt) restricted to positive powers of z. Now consider the decomposition Z[u±1 , z] = Z[u + u−1 , z] ⊕ u−1 Z[u + u−1 , z] . It is easy to check from this that the cokernel for nontrivial powers of z is isomorphic to the cokernel of ϕ0 restricted to u−1 Z[u + u−1 , z] and hence is M Z[u + u−1 ] (z/u)k+2 k+1 (u + u−1 )i hSym k≥0 which agrees with the classical result [18].

We now get to the collapse of the spectral sequence. It is sufficient to establish: Proposition 3.6. Assume n > 0, then colimiI KHr+1,n (Ad(HI )+ ) is trivial if I i > 0. For i = 0, this group is isomorphic to the free abelian group generated by regular, dominant, level n characters of LG. Proof. To simplify the notation, we will abbreviate KHr+1,n (Ad(HI )+ ) by K(I), I hence the letter K denotes the functor I 7→ K(I). The strategy in proving the above proposition L is to show that the functor K decomposes into a sum of functors K = KJ , where all the functors KJ have trivial higher derived colimits. We then show that colimKJ is also trivial for all but one of the functors, and we identify its colimit as the free abelian group generated by the dominant regular characters of level n. Let R(T )(n) denote the free abelian group generated by characters of LG of level n. Recall that the fundamental domain for the action of the affine ˜ on the set of characters of level n, is the set of characters Weyl group W in the affine alcove ∆ (at height n). The affine alcove is an affine r-simplex in the (dual) Lie algebra of the maximal torus of LG. Hence we may index the walls of ∆ by the category C of proper subsets of the r + 1-element set {0, . . . , r}, where ∆I ⊆ ∆J if J ⊂ I. We get a corresponding decomposition: M R(T )(n) = RJ (T )(n) J∈C

where RJ (T )(n) is the free abelian group generated by the characters that ˜ -translates of characters in the interior of the face ∆I . Let RJ (T )(n) are W denote the free abelian group generated by the characters in the interior of the face ∆J , so we have an isomorphism: −1 ˜ /WJ ] ∼ RJ (T )(n) ⊗ Z[W = RJ (T )(n), eλ ⊗ w 7→ ew λ

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where WJ is the isotropy of the wall ∆J (and is also the Weyl group of HJ ). Consider the push forward map πI : R(T )(n) → K(I). By pushforward, we ˆ mean with respect to the A-orientation. Therefore, the character of πI (eλ ) is given by: P sgn(w) w(λ) Y e w∈WI (−1) , A(I) = (eα/2 − e−α/2 ), A(I) the product in the Weyl denominator A(I) being taken over the positive roots of HI . The map πI is surjective, and one may define KJ to be the functor I 7→ πI (RJ (T )(n)). It is not hard to see that direct sum decomposition of R(T )(n) remains direct when we push forward to K(I), hence we get a direct sum decomposition of functors: M K= KJ J∈C

We now proceed to calculate colim KJ by splitting it out of another functor LJ . We define the functor I 7→ LJ (I) by ˜ /WI ] LJ (I) = RJ (T )(n) ⊗ Z(sgn) ⊗Z[W ] Z[W i

J

where Z(sgn) denotes the sign representation of Z[WJ ]. One may check that the following map is a retraction of functors: LJ (I) → KJ (I),

eλ ⊗ 1 ⊗ w 7→ (−1)w πI (ew

−1 λ

)

where (−1)w denotes the sign of the element w. Hence, the groups colimi KJ are retracts of the homology of the complex RJ (T )(n) ⊗ Z(sgn) ⊗Z[WJ ] C∗ , where C∗ is the simplicial chain complex of the affine hyperplane given by ˜ orbit of ∆. Since WJ is a finite subgroup, this simplicial complex is the W WJ -equivariantly contractible. Thus C∗ is WJ -equivariantly equivalent to the constant complex Z in dimension zero. It follows that colimi KJ = 0 if i > 0, and colim KJ is a retract of RJ (T )(n) ⊗ Z(sgn) ⊗Z[WJ ] Z. Furthermore, if J is nonempty, then the groups RJ (T )(n)⊗Z(sgn)⊗Z[WJ ] Z are two torsion, hence the map to KJ is trivial. This shows that colim KJ = 0 if J is not the empty set. Finally, to complete the proof, one simply observes that for the empty set, the above retraction is an isomorphism, and hence colim Kφ = Rφ (T )(n), which is the free abelian group generated by the level n dominant regular weights of LG. Remark 3.7. We can calculate the equivariant K-homology KLG∗ (A(LG)) using the same spectral sequence. This establishes an isomorphism ∗ between KLG (DLG ) and KLG ∗ (A(LG)). Results above imply that the latter group calculates the Verlinde algebra. Infact, the results of [19] may be stated in terms of K-homology, and our spectral sequence argument may be seen as an alternate proof of their results. Recall also that A(LG) is the classifying space for proper actions (i.e. with compact isotropy) so these results also bear an interesting relationship to the Baum-Connes conjecture [7]

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Question. For the manifold LM, with frame bundle LP , we can construct a spectrum DLM := holimI LP+ ∧HI Ad(HI )+ It would be very interesting to understand something about KT (DLM ). 4. Localization Theorems If E is a T-equivariant complex-oriented multiplicative cohomology theory, and X is a T-space, we have contravariant (j ∗ ) and covariant (j ! ) homomorphisms associated to the fixedpoint inclusion j : XT ⊂ X , satisfying j ∗ j ! (x) = x · eT (ν) ; if the Euler class of the normal bundle ν is invertible, this leads to a close relation between the cohomology of X and X T . More generally, if f : M → N is an equivariant map, then its PontrjaginThom transfer is related to the analogous transfer defined by its restriction fT : MT → NT to the fixedpoint spaces, by a ‘clean intersection’ formula: ∗ ∗ jN ◦ f ! (−) = f T! (jM (−) · eT (ν(f )|M T )) .

Definition 4.1. The fixed-point orientation defined by the Thom class Th† (ν(f T )) = Th(ν(f T )) · eT (ν(f )|M T for the normal bundle of the inclusion of fixed-point spaces is the product of the usual Thom class with the equivariant Euler class of the full normal bundle restricted to the fixed-point space. Recall that we have the formula f T! (−) = fPT∗T (− · Th(ν(f T ))), where fPTT is the Thom collapse map: fPTT : Rk+ ∧ N+ → (M T )ν(f

T)

corresponding to f T (having first replaced f T by an embedding of M T into N T × Rk for large k). In our new notation the clean intersection formula becomes ∗ ∗ jN ◦ f ! = f T† ◦ jM with a new Pontrjagin-Thom transfer f T† (−) = fPT∗T (− · Th† (ν(f T ))) . In the case of most interest to us (free loopspaces), we identified the normal bundle above, in §2; using that description, we have Y eT (ν(M ⊂ LM)) = (e(Li ) +E [k](q)), 06=k∈Z;i

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where the Li are the line bundles in a formal decomposition of T M, q is the Euler class of the standard one-dimensional complex representation of T, and +E is the sum with respect to the formal group law defined by the orientation of E. It may not be immediately obvious, but it turns out that such a formula implies that the fixed-point orientation defined above will have good multiplicative properties. Such Weierstrass products sometimes behave better when ‘renormalized’, by dividing by their values on constant bundles [2]. If E is KT with the usual complex (Todd) orientation, we have e(L) +K [k](q) = 1 − q k L ; but for our purposes things turn out better with the Atiyah-Bott-Shapiro spin orientation; in that case the corresponding Euler class is (q k L)1/2 − (q k L)−1/2 . The square roots make sense under the simple-connectivity assumptions on G mentioned in the introduction: To be precise, let V be a representation of LG with an intertwining action of T. We restrict ourselves to finite type representations V which (for lack of a better name) we call symmetric, i.e. such that V is equivalent to the representation of LG obtained by composing V with the involution of LG which reverses the orientation of the loops. The restriction of the representation V ˜ to the constant loops T × G ⊂ T×LG has a decomposition X V = VT⊕ Vk q k k6=0

where Vk are representations of G, and q denotes the fundamental representation of T. Let V (m) be the finite dimensional subrepresentation X V (m) = V T ⊕ Vk q k ; 0 0, α0 (h) ≤ 1} . We may identify this space with ∆ using the roots αi . So, for example, the codimension 1 face ∆i for 1 ≤ i ≤ n is identified with the subset of the alcove {(1, h) | αi(h) = 0}, for i 6= 0, and ∆0 is identified with the subspace {(1, h) | α0(h) = 1}. General facts about Loop groups [22, 26] show that the surjective map Lalg G × ∆ −→ A,

(f (z), y) 7→ Adf (z) (y)

˜ alg Ghas isotropy HI on the subspace ∆I . Hence it factors through a T×L equivariant homeomorphism between A(Lalg G) and the affine space A. No˜ alg G admits a fixed point on tice that any compact subgroup K ⊂ T×L K A(Lalg G). Hence, the space A(Lalg G) is also affine. This completes the proof. Remark 7.6. The affine space A above should be thought of as the space of (algebraic) connections on the trivial G-bundle on S 1 . The action of the group Lalg G then corresponds to the action of the (algebraic) gauge group. This analogy can be taken one step further to define the space of (algebraic) holonomies: Salg = {g : R → G | g(t) = f (e2πit ) · exp(tX); f (z) ∈ Ωalg G, X ∈ Lie(G)} ˜ alg G given topologized as a quotient of Ωalg G × Lie(G), with the action of T×L by left multiplication (see discussion before 2.4). In fact, in [26] Mitchell shows that A(Lalg G) is Lalg G equivariantly homeomorphic to the space Salg (attributing the result to Quillen). We now define the smooth Tits building Definition 7.7. Let A(LG) be the homotopy colimit: A(LG) = hocolimI∈C LG/HI = LG ×Lalg G A(Lalg G) . ˜ Theorem 7.8. The smooth Tits building A(LG) is T×LG-equivariantly contractible.

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Proof. By the above remark, the space A(LG) is LG-equivariant homeomorphic to the space LG ×Lalg G Salg . This is easily seen to be LG-equivariantly homeomorphic to the space of smooth maps: S = {g : R → G | g(t) = f (e2πit ) · exp(tX); f (z) ∈ ΩG, X ∈ Lie(G)} Equivalently, we may describe S as S = {g : R → G, g(0) = 1, g(t + 1) = g(t) · g(1)} ; from which it follows (see 2.4) that S is LG-equivariantly homeomorphic to the space of connections on the trivial G-bundle on S 1 , which we denote by A(S 1 × G). This sequence of LG-equivariant homeomorphisms from A(LG) to A(S 1 × G) is compatible with the action of T. The proof is now complete, ˜ since the space of connections A(S 1 × G) is clearly T×LG-equivariantly contractible. References 1. F. Adams, A variant of E.H. Brown’s representability theorem, Topology, Vol. 10 (1971) 185-198. 2. M. Ando, J. Morava, A Renormalized Riemann-Roch formula and the Thom isomorphism for the free loopspace, in the Milgram Festschrift, Contemp. Math. 279 (2001) 3. ——, ——–, H. Sadofsky, Completions of Z/(p)-Tate cohomology of periodic spectra, Geometry & Topology 2 (1998) 145 - 174 4. —–, M. Hopkins, N. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Inv. Math. 146 (2001) 595 - 687 5. M. Artin, B. Mazur Etale homotopy, Springer LNM 100 (1969) 6. M. Atiyah, I. MacDonald, Commutative Algebra 7. P. Baum, A. Connes, and N. Higson. Classifying space for proper actions and K-theory of group C ∗ -algebras, p. 240 - 291 in C ∗ − algebras: 1943-1993, Contemporary Math 167, AMS (1994) 8. N. Bourbaki, Algebre Commutatif 9. T. Brocker, T. tom Dieck, Representations of Compact Lie groups, Springer GTM 98. 10. JL Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology 29 (1990) 461 - 480 11. M. Chas, D. Sullivan, String topology, available at math.AT/9911159 12. J.D. Christensen, D.C. Isaksen, Duality and prospectra (in progress) 13. R. Cohen, Multiplicative properties of Atiyah duality, available at math.AT/0403486 14. ———, V. Godin, A polarized view of string topology, available at math.AT/0303003 15. ——–, J.D.S Jones, A homotopy-theoretic realization of string topology, available at math.GT/0107187 16. ——–, ——–, G. Segal, Floer’s infinite-dimensional Morse theory and homotopy theory, in The Floer memorial volume, Progr. Math 133, Birkh¨ auser (1995) 17. ——–, A. Stacey, Fourier decomposition of loop bundles, available at math.AT/0210351 18. D. Freed, The Verlinde Algebra is Twisted Equivariant K-Theory, available at math.RT/0101038 19. D. Freed, M. Hopkins, C. Teleman, Twisted K-theory and loop group representations I, available at math.AT/0312155 20. M. Hovey, N.P. Strickland, Morava K-theories and localization, Mem. Amer. Math. Soc. 139 (1999), No.666.

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21. J.D.S Jones, S.B. Petrack, The fixed point theorem in equivariant cohomology, Transactions of the Amer. Math. Soc., Vol. 322, No. 1 (1990) 35-49. 22. V.G. Kac, Infinite dimensional Lie algebras, Cambridge University Press, 1985 23. N. Kitchloo, Topology of Kac-Moody groups, Thesis, M.I.T., 1998. 24. J. Klein, The dualizing spectrum of a topological group, Math. Ann 319 (2001) 421 456 25. G. Lewis, J.P. May, M. Steinberger, Equivariant stable homotopy theory, Springer LNM 1213 (1986) 26. S.A. Mitchell, Quillen’s theorem on buildings and the loops on a symmetric space, Enseign. Math. 34 (1988) 123-166 27. J. Morava, Forms of K-theory, Math Zeits. 201 (1989) 28. I. Rosu, Equivariant K-theory and equivariant cohomology, Math. Zeits. 243 (2003) 423-448. ´ 29. G. Segal, Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) 129-151. 30. S. Stolz, P. Teichner, What is an elliptic object? available at math.ucsd.edu 31. T. Torii, On degeneration of one-dimensional formal group laws and stable homotopy theory, AJM 125 (2003) 1037-1077 32. D. Zagier, Note on the Landweber-Stong elliptic genus, in Elliptic curves and modular forms in algebraic topology, ed. P. Landweber, Springer 1326 (1988) Department of Mathematics, UCSD, LaJolla, CA 92093. Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218. E-mail address: [email protected], [email protected]

RATIONAL VERTEX OPERATOR ALGEBRAS GEOFFREY MASON UNIVERSITY OF CALIFORNIA, SANTA CRUZ Abstract. We discuss the foundations of vertex operator algebras and their representations, concentrating on rational vertex operator algebras. We use the Witt-Grothendieck group of conformal field theories as a vehicle to describe results, in particular the connections with modular forms.

§1. Introduction Vertex operator algebras suggest themselves as objects that could play a rˆole in a geometric description of elliptic cohomology and related topics. As linear spaces with lots of symmetry they can participate in K-theoretic type constructions, and many (but not all) vertex operator algebras are endowed with a natural modular form as part of their structure. The incorporation of vertex operator algebras into topology is well under way (e.g. [Bo], [MS], [MSV], [T]), but it seems true to say that the underlying algebraic theory is not well understood by many potential users of the subject. What follows is an attempt to convey some of the basic ideas about vertex operator algebras, in particular the rapidly advancing theory of rational vertex operator algebras. These are the algebras most naturally associated to modular forms, and in the guise of RCFT (rational conformal field theory) they constitute an active area of research in physics ([FMS]) as well as mathematics. For more information and background the reader may refer to the following: [DM4], [FLM], [G], [K1], [KR], [QFS]. Additional references will be mentioned below. In my talk at the Newton Institute I emphasized questions about group actions (orbifold theory) and in particular how one can get information about finite group cohomology (e.g., for the Monster group) by looking at maps of equivariant Witt-Grothendieck groups of vertex operator algebras into group cohomology. In order to limit the length of the present exposition, however, we make no mention of groups or orbifolds here. §2. Vertex Operator Algebras and Locality Fix a complex linear space V , the Fock space. Elements of V are called states. A (quantum) field on V is an element X a(n)z −n−1 ∈ End(V)[[z, z −1 ]] a(z) = n∈Z

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with the property that v ∈ V ⇒ a(n)v = 0 for all large enough integers n. P (z is a formal variable.) One writes a(z)v = a(n)vz −n−1 , where only finitely many negative powers of z occur for a field a(z). The space of all fields on V is denoted by F (V ). The endomorphisms a(n) are the (Fourier) modes of the field a(z). A pair of fields a(z), b(z) ∈ F (V ) are mutually local in case the following holds: there is k ≥ 0 such that (y − z)k [a(y), b(z)] = 0 where y, z are independent variables. We write a(z) ∼ b(z) if a(z) and b(z) are mutually local fields and a(z) ∼k b(z) if one wishes to stress the degree of locality k, which depends on a(z) and b(z). Examples of Quantum Fields a. (Derivative of a field): If a(z) ∈ F (V ) then X ∂a(z) = a0 (z) = (−n − 1)a(n)z −n−2 ∈ F (V ), moreover b(z) ∈ F (V ) and a(z) ∼ b(z) ⇒ a0 (z) ∼ b(z).

b. (Tensor Products): Given Fock spaces V, W , there is a natural injection F (V ) ⊗ F (W ) −→ F (V ⊗ W ). P P (The map arises via a(z) ⊗ b(z) 7→ c(n)z −n−1 where c(n) = i∈Z a(i) ⊗ b(n − i − 1). Although this is an infinite sum of operators on V ⊗ W , it is well-defined because a(z), b(z) are fields: a state in V ⊗ W is annihilated by all but a finite number of the summands.) c. (Virasoro Algebra): The Virasoro algebra is the Lie algebra M V ir = CL(n) ⊕ Ck n∈Z

with relations [V ir, k] = 0 and

m3 − m δm+n,0 k. 12 Denote by V ir + the Lie subalgebra spanned by k together with the L(n), n ≥ 0. Let C0 be the 1-dimensional V ir + -module annihilated by all L(n), n > 0, and on which k, L(0) operate as prescribed scalars c, h respectively. Let U (c, h) be the Verma module IndVV ir+ (C0 ), and set X ω(z) = L(n)z −n−2 . [L(m), L(n)] = (m − n)L(m + n) +

n∈Z

(Note the change in grading convention in this case, which amounts to L(n) = ω(n − 1).) Then ω(z) ∈ F (U ), moreover ω(z) ∼4 ω(z).

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We now define a vertex operator algebra. It is a Z-graded Fock space M V = Vn n∈Z

with dimVn < ∞ and Vn = 0 for n δm,−n c. ˆ ∗ ˆ≥ We have the Verma module M(1, λ) = IndH = ˆ ≥ C0 . Here, λ ∈ H , H H H ⊗ C[t] ⊕ Ck and C0 is the 1-dimensional module in which H ⊗ tC[t] acts as 0, k acts as 1, and H = H ⊗ t0 acts through λ. Then M = M(1, 0) has the structure of a vertex operator algebra. We have c = dimH and Y 1 (5) ZM (q) = q −c/24 (1 − q n )−c = η(q)c n≥1

where η(q) is the Dedekind eta function. There C0 = P are identifications −n−1 M0 , H = M1 , and if h ∈ H then Y (h, z) = where h(n) n∈Z h(n)z is the natural action of h ⊗ tn on M. Physically, this theory models c free (noninteracting) bosons. d. Lattice Theories. Here we are given an even lattice L, i.e. a free abelian group L equipped with a positive-definite symmetric bilinear form : L ⊗ L → Z such that < x, x >∈ 2Z. Extend to H = C ⊗Z L and let M be the associated free boson theory. Then there is a vertex operator algebra with Fock space (6)

VL = M ⊗ C[L] = ⊕α∈L C ⊗ eα

where C[L] = ⊕α Ceα is the group algebra of L and eα eβ = eα+β . M = M ⊗e0 is a conformal subalgebra of VL . The partition function is θL (q) (7) ZVL (q) = η(q)rkL P /2 where θL (q) = is the theta-function of L. Physically, this α∈L q theory corresponds to rkL bosons with momenta restricted to lie on L. e. Tensor products. Given vertex operator algebras (V i , Yi ) of central charge ci , i = 1, 2, their tensor product is a vertex operator algebra (V 1 ⊗ V 2 , Y1 ⊗ Y2 ) with vacuum 11 ⊗ 12 , conformal vector ω1 ⊗ 12 + 11 ⊗ ω2 , central charge c1 + c2 and partition function ZV 1 ⊗V 2 (q) ∼ = ZV 1 (q)ZV 2 (q). For example, if M1 and M2 are free bosonic theories of central charge c1 , c2 respectively, then M1 ⊗ M2 is the free bosonic theory of central charge c1 + c2 . Similarly, given a pair of even lattices L1 , L2 , we have (8) V L ⊗ VL ∼ = VL ⊥L 1

2

where L1 ⊥ L2 is orthogonal direct sum.

1

2

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f. Moonshine Module. The most famous vertex operator algebra, it is tradition to follow Frenkel-Lepowsky-Meurman and denote it by V \ . We have c = 24 and ZV \ (q) = J(q) = q −1 + 196884q + 21493760q 2 + ... We give some different types of examples concerning important structural features of a vertex operator algebra. g. Connection with commutative algebras. Suppose that V has no negative weight spaces: Vn = 0 for n < 0 (a ubiquitous condition which does not, however, always hold). Then by skew-symmetry and associativity it follows that V0 admits the structure of a commutative, associative algebra with product u.v = u(−1)v, u, v ∈ V0 and identity 1. There are interesting examples in which V0 is the cohomology ring of a manifold (cf. Gorbounov’s lecture at this conference). h. Connection with affine Lie algebras. Suppose that V has a non-degenerate vacuum, that is V0 = C1. Then V has no negative weight spaces, and V1 carries the structure of Lie algebra via [u, v] = u(0)v, u, v ∈ V1 . Moreover V1 comes equipped with an invariant bilinear form : V1 ⊗ V1 → C where u(1)v =< u, v > 1. Fourier modes of the fields Y (u, z) for u ∈ V1 close on the corresponding affine Lie algebra (use the commutator formula). i. Connection with non-associative algebras. Suppose that V has a nondegenerate vacuum and that V1 = 0. Then V2 carries the structure of commutative, non-associative algebra via u.v = u(1)v, u, v ∈ V2 with identity ω/2. There is also an invariant form as in Example h. In the case of V \ the corresponding algebra is the famous Griess algebra with automorphism group the Monster [Gr] . We have turned history around here: it was Griess’s algebra and the numerical observations of Conway-Norton [CN] that led to the discovery of vertex operator algebras and construction of the Moonshine module at the hands of Borcherds [B] and Frenkel-Lepowsky-Meurman [FLM]. j. Connection with Poisson-Lie algebras. Define C2 (V ) = {u(−2)v|u, v ∈ V }. The quotient space P (V ) = V /C2 (V ) carries a natural Poisson-Lie algebra structure with multiplication and bracket induced by the same operations as Examples g and h, which are thus both incorporated into the larger structure. The reader will readily identify P (M), where M is a Heisenberg vertex operator algebra, with a well-known Poisson-Lie algebra. For the Moonshine module, P (V \ ) is a finite-dimensional Poisson-Lie algebra which admits the Monster simple group as automorphisms. This and similar examples seem not to have been closely studied thus far. (For example, I do not know the dimension of P (V \ )). Zhu’s influential paper [Z] was the first to make serious use of the finite-dimensionality of P (V ) (the so-called C2 -cofiniteness condition).

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See [DLM3] for more on the Poisson-Lie algebra-vertex operator algebra connection, and [ABD], [Bu], [GN] for results related to C2 -cofiniteness. §3 Representations of Vertex Operator Algebras There is a natural notion of module over a vertex operator algebra V , and the family of V -modules forms an abelian category V − mod. Representation theory is a fundamental part of vertex operator algebra theory, and an active subject of research at this time. Describing V −mod for a given V is generally a difficult task. A V -module is a pair (M, YM ) where M is a complex linear space M (9) M= Mn n∈C

graded into finite-dimensional pieces Mn indexed by a subset of C, together with a linear map YM : V → F (M), v 7→ YM (v, z).

There is a finiteness condition on the grading: if t ∈ C then Mt+n = 0 for all integers n k. 0, if n > k. Triviality (Corollary of Theorem 4.1) (Ravenel [Rav84]) • X: p-completed • X: p-completed harmonic: • #{n ∈ N | πn X 6= 0} < +∞, bounded below harmonic: • #{n ∈ N | K(n)∗ X 6= 0} < =⇒ X = ∗. +∞, =⇒ X = ∗. Vanishing (Corollary of Theorem 4.1) (Ravenel [Rav84]) • X: p-completed • X: p-completed harmonic: bounded below harmonic: • #{n ∈ N | πn Y 6= 0} < +∞, • #{n ∈ N | K(n)∗ Y 6= 0} < =⇒ [Y, X] = 0. +∞, =⇒ [Y, X] = 0. Recovery (Corollary of Theorem 4.1) (Ravenel [Rav84]) X: harmonic, X: bounded below harmonic, =⇒ The p-completion Xp may be re=⇒ The p-completion Xp may be covered from X(k,+∞) for any k = 0: recovered from Ck X for any k = 0: Xp = holimn L∞ X(k,+∞) ∧ M(pn ) Xp = L∞\{0,1,··· ,k} Ck X. where M(pn ) is the mod-pn Moore spectrum. However, this analogy is somewhat unusual because of the inclusion: {bounded below harmonic spectra} j {harmonic spectra} Thus, we are naturally led to: Question 4.3. Is there any relation between the chromatic tower and the Postnikov tower for bounded below harmonic spectra? Warning 4.4. The suspension behaves differently. More precisely, whereas the suspension does not change the chromatic filtration, the suspension shifts

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the Postnikov filtration: Lk (ΣX) = Σ(Lk X),

(ΣX)(−∞,k] = Σ X(−∞,k−1] .

This suggests we must fix some unstable condition to obtain such a relation. For this purpose, we restrict our attention to the Kriz’s case [Kri94]: Let X = T (f ∗ γ), the Thom spectrum of a virtual bundle f ∗ γ over a space S with classifying map f f ∗ γ −−−→ γ y y f

S −−−→ BU, where γ is the universal bundle over BU with dim γ = 0 so that T (γ) = MU f

(if p > 2, consider also S → BSO). Then we offer our speculation: Question . For any X = T (f ∗ γ), as above, is there an inclusion Ker(π∗ X → π∗ Lk X) j Ker(π∗ X → π∗ X(−∞,k] ) for any k ∈ Z≥0 ? In other words, although there is no commutative diagram

Lk X

X ? ?

6∃

?? ?? ?? ?? / X(−∞,k] ,

we speculate the injectivity of πk X → πk Lk X

k ∈ Z≥0 .

The Question is true for the simplest case X = S 0 = T (t), where t is the trivial map t : {a point} → BU. Proposition 4.5. πk S 0 → πk Lk S 0 is injective. Actually, this follows easily from [MRW77] and [Rav84] (see also [HS99] and [Min03] for related techniques to slightly extend Proposition 4.5). However, a very interesting special case is given by ∗ X = T (πBU γ) = MU ∧ S+ ,

where S is a space and πBU is the projection map πBU : BU × S → BU. In fact, for this case, the question (conjecture) essentially claims the injectivity of BPk S+ → (Lk BP )k S+ , and we easily arrive at the following:

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Proposition 4.6. Suppose the Johnson question is true for S, i.e. suppose the canonical map BPk S+ → vk−1 BPk S+ is injective. Then Question is true for X = MU ∧ S+ . Of course, this immediately follows from the commutative diagram: BPk S+ oo ooo o o w o

(Lk BP )k S+

OOO OOO O' / v −1 BP k

k S+

Although we can not prove the converse, it appears that the Question for the special case X = MU ∧ S+ asks almost as much as the Johnson question for the case S. In this way, the Question might be regarded as a conceptual explanation (and even an approach) for the Johnson question. For special cases and related matters concerning the Johnson question, consult e.g. [RW80, JW73, JWY94, HRW98]. References [Bou79a] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281. [Bou99] A. K. Bousfield, On K(n)-equivalence of spaces, Contemp. Math. 239 (1999), 85–89. [Dev98] E. S. Devinatz, A counterexample to a BP -analogue of the chromatic splitting conjecture, Proc. Amer. Math. Soc., 126 (1998) 907–911. [DHS88] E. Devinatz, M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory I, Annals of Math. 128 (1988) 207–241. [HRW98] M. J. Hopkins, D. C. Ravenel and W. S. Wilson, Morava Hopf algebras and spaces K(n)-equivalent to finite Postnikov systems, W. G. Dwyer, et. al., editors, Stable and Unstable Homotopy, Fields Institute Communications, 19 (1998) 137–163. [HS98] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II, Annals of Math. 148 (1998) 1–49. [Hov95] M. Hovey, Bousfield localization functors and Hopkins’ chromatic splitting conjecture, Contemp. Math. 181 (1995) 225–250. [HS99] M. Hovey and H. Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, J. London Math. Soc. 60 (1999), 284–302. [JW73] D. C. Johnson and W. S. Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353. [JWY94] D. C. Johnson, W. S. Wilson, and D. Y. Yan, Brown-Peterson homology of elementary p-groups, II, Topology and its applications, 59 (1994) 117–136. [JY80] D. C. Johnson and Z. Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980) 117–136. [Kri94] I. Kriz, All complex Thom spectra are harmonic, Contemp. Math. 158 (1994) 127– 134. [Kri97] I. Kriz, Morava K-theory of classifying spaces: some calculations, Topology 36 (1997) 1247–1273. [KL00] I. Kriz and K. P. Lee, Odd-degree elements in the Morava K(n)-cohomology of finite groups, Topology Appl. 103 (2000) 229–241.

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[MRW77] H. R. Miller, D. C. Ravenel and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Annals of Math. 106 (1977) 469–516. [Min02] N. Minami, From K(n+1)∗ (X) to K(n)∗ (X), Proc. Amer. Math. Soc. 130 (2002) 1557–1562. [Min03] N. Minami, On the chromatic tower, Amer. J. Math. 125 (2003), 449–473. [Min-a] N. Minami, On the horizontal vanishing line of the E(n)-based modified AdamsNovikov spectral sequence via injective resolutions, preprint. [Min-b] N. Minami, Some chromatic phenomena of bounded below harmonic spectra, preprint. [Mit90] S. A. Mitchell, The Morava K-Theory of Algebraic K-Theory Spectra, K-Theory 3 (1990) 607–626. [Mit92] S. A. Mitchell, Harmonic Localization of Algebraic K-Theory Spectra, Trans. Amer. Math. Soc., 332 (1992) 823–837. [Mor89] J. Morava, Forms of K-theory, Math. Z. 201 (1989) 401–428. [Rav84] D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math., 106 (1984), 351–414. [Rav87] D. C. Ravenel, The geometric realization of the chromatic resolution, Ann. of Math. Stud. 128, Princeton University Press, Princeton, (1987), 168–179. [Rav92] D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Stud. 128, Princeton University Press, Princeton, 1992. [RW80] D. C. Ravenel and W. S. Wilson, The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980) 691–748. [RWY98] D. C. Ravenel, W. S. Wilson and N. Yagita, Brown-Peterson Cohomology from Morava K-Theory, K-Theory 15 (1998), 147–199. [SY95] K. Shimomura and A. Yabe, The homotopy groups of π∗ (L2 S 0 ), Topology, 34 (1995) 261–289. [SW] K. Shimomura and X. Wang, The homotopy groups π∗ (L2 S 0 ) at the prime 3, preprint. [Wil99] W. S. Wilson, K(n + 1) equivalence implies K(n) equivalence, Contemp. Math. 239 (1999), 375–376. Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya 466-8555, JAPAN E-mail address: [email protected]

THE MOTIVIC THOM ISOMORPHISM JACK MORAVA Abstract. The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Deligne’s ideas on motivic Galois groups.

1. Introduction This talk is in part a review of some recent developments in Galois theory, and in part conjectural; the latter component attempts to fit some ideas of Kontsevich about deformation quantization and motives into the framework of algebraic topology. I will argue the plausibility of the existence of liftings of (the spectra representing) classical complex cobordism and K-theory to objects in some derived category of mixed motives over Q. In itself this is probably a relatively minor technical question, but it seems to be remarkably consistent with the program of Connes, Kreimer, and others suggesting the existence of a Galois theory of renormalizations. 1.1. One place to start is the genus of complex-oriented manifolds associated to the Hirzebruch power series z = zΓ(z) = Γ(1 + z) exp∞ (z) [25 §4.6]. Its corresponding one-dimensional formal group law is defined over the real numbers, with the entire function exp∞ (z) = Γ(z)−1 : 0 7→ 0 as its exponential. I propose to take seriously the related idea that the Gamma function Γ(z) ≡ z −1 mod R[[z]] defines some kind of universal asymptotic uniformizing parameter, or coordinate, at ∞ on the projective line, analogous to the role played by the exponential at the unit for the multiplicative group, or the identity function at the unit for the additive group. Date: 15 November 2003. 1991 Mathematics Subject Classification. 11G, 19F, 57R, 81T. The author was supported in part by the NSF. 265

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1.2. The second point of reference is a classical conjecture of Galois theory. The cyclotomic closure Qcyc of the rationals, defined by adjoining all roots of unity to Q, is the maximal extension of Q with commutative Galois group; ˆ × of profinite integers, that group, isomorphic to the multiplicative group Z plays an important role in work of Quillen and Sullivan on the Adams conjecture and in differential topology. Shafarevich (cf. [27]) conjectures that the Galois group Gal(Q/Qcyc ) is a free profinite group; in other words, the full Galois group of Q over Q fits in an exact sequence ˆ× → 1 . d → Gal(Q/Q) → Gal(Qcyc /Q) ∼ 1 → Gal(Q/Qcyc ) ∼ = Free =Z

What will be more relevant here is a related conjecture of Deligne [13 §8.9.5], concerning a certain motivic analog of the Galois group which I will denote Galmot (Q/Q), which is not a profinite but rather a proalgebraic groupscheme over Q; it is in some sense a best approximation to the classical Galois group in this category, which should contain the original group as a Zariski-dense subobject. [I should say that calling this object a Galois group is an abuse of terminology; it is more properly described (cf. §4.4) as the motivic Tate Galois group of Spec Z (without reference to Q).] In any case, this motivic group fits in a similar extension 1 → Fodd → Galmot (Q/Q) → Gm → 1 of groupschemes over Q, where Gm is the multiplicative groupscheme, and Fodd is the prounipotent groupscheme defined by a free graded Lie algebra fodd with one generator of each odd degree greater than one; the grading is specified by the action of the multiplicative group on the Lie algebra. The generators of this Lie algebra are thought to correspond with the odd zeta-values (via Hodge realization, [14 §2.14]) which are expected to be transcendental numbers (and thus outside the sphere of influence of Galois groups of the classical kind). Kontsevich introduced his Gamma-genus in the context of the Duflo - Kirillov theorem in representation theory. He argued that it lies in the same orbit, under an action of the motivic Galois group, as the analog of the clasˆ sical A-genus [3 §8.5]. How this group fits in the topological context is less familiar, and to a certain extent this paper is nothing but an attempt to find a place for that group in algebraic topology. The history of this question is intimately connected with Grothendieck’s theory of anabelian geometry, and it enters Kontsevich’s work through a conjectured Galois action on some form of the little disks operad. My impression these ideas are not yet very familiar to topologists, so I have included a very brief account of some of their history, with a few references, as an appendix below. I should acknowledge here that Libgober and Hoffman [25,40] have studied a genus with related, but not identical, properties, and that an attempt to

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understand their work was instrumental in crystallizing the ideas behind this paper. I owe many mathematicians - including G. Carlsson, D. Christensen, F. Cohen, P. Deligne, A. Goncharov R. Jardine, T. Kohno, and T. Wenger thanks for conversations about the material in this paper, and I am particularly indebted to a very knowledgeable and patient referee. In many cases they have saved me from mistakes and overstatements; but other such errors may remain, and those are the solely my responsibility. I also wish to thank the Newton Institute, and the Fields Institute program at Western Ontario, for support during the preparation of this paper. 2. The Gamma-genus 2.1. The Gamma-function is meromorphic, with simple poles at z = 0, −1, −2, . . . ; we might therefore hope for a Weierstrass product of the form Y z Γ(1 + z)−1 ∼ (1 + ) , n n≥1 from which we might hope to derive a power series expansion X X z z (− )k log Γ(1 + z) ∼ − log(1 + ) ∼ n n n≥1 n,k≥1 for its logarithm. Rearranging this carelessly leads to X ζ(k) X (−z)k 1 ∼ (−z)k , k k n k k≥1 n,k≥1

which is unfortunately implausible since ζ(1) diverges. In view of elementary renormalization theory, however, we should not be daunted: we can add ‘counter-terms’ to conclude that Y X ζ(k) z log (1 + ) e−z/n ∼ − (−z)k , n k n≥1 k≥2 and with a little more care we deduce the correct formula X ζ(k) (−z)k ) , Γ(1 + z) = exp(−γz + k k≥2

where γ is Euler’s constant. Reservations about the logic of this argument may perhaps be dispelled by observing that X ζ(2k) Γ(1 + z) Γ(1 − z) = exp( z 2k ) ; k k≥1 Euler’s duplication formula implies that the left-hand side equals πz , zΓ(z) Γ(1 − z) = sin πz

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consistent with the familiar evaluation of ζ at positive even integers in terms of Bernoulli numbers. 2.2. From this perspective, the Hirzebruch series X ζ(odd) πz 1 Γ(1 + z) = ( ) 2 exp(−γz + z odd ) ; sin πz odd for Kontsevich’s genus does in fact look like some kind of deformation of the ˆ A-genus; its values on a complex-oriented manifold will be polynomials in odd zeta-values, with rational coefficients. Similarly, the Witten genus Y x/2 φW (x) = [(1 − q n u)(1 − q n u−1 )]−1 sinh x/2 n≥1 ([57], with u = ex ) can be written in the form X xk exp(−2 gk ) , k! k≥1

where the coefficients gk are modular forms, with godd = 0: it is also a ˆ in another direction. deformation of A, [Behind the apparent discrepancies in these formulae is the issue of complex versus oriented cobordism: there are several possible conventions relating Chern and Pontrjagin classes. Hirzebruch expresses the latter as symmetric functions of indeterminates x2i , and writes the genus associated to the formal Q ˆ series Q(z) as Q(x2i ); thus for the A-genus, 1√ z Q(z) = 2 1 √ . sinh 2 z An alternate convention, used here, writes this symmetric function in the form Y 1 1 −xi /2 xi /2 )2 · ( )2 ) . (( sinh xi /2 sinh(−xi /2) The relation between the indeterminates x and z is a separate issue; I take z to be 2πix.] Kontsevich suggests that the values of the zeta function at odd positive integers (expected to be transcendental) are subject to an action of the moˆ tivic group Galmot (Q/Q), and that the A-genus and his Γ genus lie in the same orbit of this action. One natural way to understand this is to seek an action of that group on genera, and thus on the complex cobordism ring; or, perhaps more naturally, on some form of its representing spectrum. Before confronting this question, it may be useful to present a little more background on these zeta-values.

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3. Symmetric and quasisymmetric functions 3.1. The formula Y

(1 + xk z) =

k≥1

X

ek z k = exp(−

k≥0

X pk k

(−z)k ) ,

k≥1

where ek is the kth elementary symmetric function, and pk is the kth power sum, can be derived by formal manipulations very Q much like those in the preceding section, by expanding the logarithm of (1+ xn z); such arguments go back to Newton. The specialization xk 7→ k −2

(cf. the second edition of MacDonald’s book [43 Ch I §2 ex 21]) leads to Bernoulli numbers, but the map xk 7→ k −1 is trickier, because of convergence problems like those mentioned above; it defines a homomorphism from the ring of symmetric functions to the reals, sending pk to ζ(k) when k > 1, while p1 7→ γ [24]. Under this homomorphism the even power sums p2k take values in the field Q(π). 3.2. The Gamma-genus is thus a specialization of the formal group law with exponential Y X z =z (1 + xk z)−1 = (−1)k hk z k+1 Exp∞ (z) = e(z) k≥1 k≥0 having the complete symmetric functions (up to signs) as its coefficients. This is a group law of additive type: its exponential, and hence its logarithm, are both defined over the ring of polynomials generated by the elements hk . This group law is classical: it is defined by the Boardman-Hurewicz-Quillen complete Chern number homomorphism MU ∗ (X) → H ∗ (X, Z[h∗ ]) defined on coefficients by the homomorphism Lazard → Symm from Lazard’s ring which classifies the universal group law of additive type. The Landweber-Novikov Hopf algebra S∗ = Z[t∗ ] represents the prounipotent groupscheme D0 of formal diffeomorphisms X z 7→ t(z) = z + tk z k+1 k≥1

of the line, with coproduct

∆(t(z)) = (t ⊗ 1)((1 ⊗ t)(z)) ∈ (S∗ ⊗ S∗ )[[z]] .

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The universal group law t−1 (t(X) + t(Y )) of additive type is thus classified by the homomorphism Lazard → Lazard ⊗ S∗ → Z ⊗ S∗ → Symm representing the orbit of the Thom map Lazard → Z (which classifies the additive group law) under the action of D0 . This identifies the algebra S∗ with the ring of symmetric functions by tk 7→ (−1)k hk . 3.3. The symmetric functions are a subring Symm → QSymm of the larger ring of quasisymmetric functions, which is Hopf dual to the universal enveloping algebra ZhZ1 , . . . i of the free graded Lie algebra f∗ with one generator in each positive degree [23], given the cocommutative coproduct X ∆Zi = Zj ⊗ Z k . i=j+k

Standard monomial basis elements for this dual Hopf algebra, under specializations like those discussed above [6 §2.4, 23], map to polyzeta values X 1 ζ(i1 , . . . , ik ) = ik ∈ R ; i1 n · · · n 1 k n1 >···>nk ≥1 note that there are convergence difficulties unless i1 > 1. If we think of the Gamma-genus as taking values in the field Q(ζ) ⊂ R generated by such polyzeta values, then it is the specialization of a homomorphism Lazard → Symm → QSymm representing a morphism from the prounipotent groupscheme F with Lie algebra f∗ to the moduli space of one-dimensional formal group laws. Because we are dealing with group laws of additive type, there seems to be little loss in working systematically over a field of characteristic zero, where Lie-theoretic methods are available. Over such a field any formal group is of additive type: the localization of the map from the Lazard ring to the symmetric functions is an isomorphism. Similarly, over the rationals the Landweber - Novikov algebra is dual to the enveloping algebra of the Lie algebra of vector fields zk = z k+1 ∂/∂z , k ≥ 1 on the line, and the embedding of the symmetric in the quasisymmetric functions sends the free generators Zk to the Virasoro generators zk , corresponding to a group homomorphism F → D0 .

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3.4. It will be useful to summarize a few facts about Malcev completions and pro-unipotent groups [13 §9]. The rational group ring Q[G] of a discrete [ with group G has a natural lower central series filtration; its completion Q[G] respect to that filtration is a topological Hopf algebra, whose continuous dual represents a pro-unipotent groupscheme over Q. Applied to a finitelygenerated free group, for example, this construction yields a Magnus algebra of noncommutative formal power series. There are many variations on this theme: in particular, an action of the multiplicative group Gm defines a grading on a Lie algebra. The action tk 7→ uk tk , (u a unit) on the group of formal diffeomorphisms defines an extension of its Lie algebra by a new Virasoro generator v0 , corresponding to an extension S∗ [t± 0 ] of the Landweber-Novikov algebra. The group of formal diffeomorphisms is pro-unipotent, and this enlarged object is most naturally interpreted as a semidirect product D0 o Gm . Grading the free Lie algebra f similarly extends the homomorphism above to F o Gm → D o G m . 4. Motivic versions of classical K-theory and cobordism 4.1. There are now several (eg [39, 55]) good and probably equivalent constructions of a triangulated category DM(k) of motives over a field k of characteristic zero. The subject is deep and fascinating, and I know at best some of its vague outlines. Since this paper is mostly inspirational, I will not try to provide an account of that category; but as it is after all modelled on spectra, it is perhaps not too much of a reach to think that some of its aspects will look familiar to topologists. One approach to defining a motivic category DM(k) starts from a category whose morphisms are elements of a group of algebraic correspondences. At some later point it becomes useful to tensor these groups with Q, resulting in a category DMQ (k) whose Hom-objects are rational vector spaces. The underlying concern of this paper is the relation of such motivic categories to classical topology; but stable homotopy theory over the rationals is equivalent to the theory of graded vector spaces. This has the advantage of rendering some of the conjectures below almost trivially true – and the disadvantage of making them essentially contentless. Behind these conjectures, however, lies the hope that they might say something before rationalization, and for that reason I have outlined here a rough theory of integral geometric realizations of motives:

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4.2. The category DM(k) contains certain canonical Tate objects Z(n), defined [55 §2.1, but see also 13 §2] as tensor powers of a reduced version Z(1) of the projective line. Grothendieck’s original category of ‘pure’ motives, constructed from smooth projective varieties, is (in some generality [28]) semisimple, but categories of motives built from more general (non-closed) varieties admit nontrivial extensions. The (derived) category DMTQ (k) of mixed Tate motives can be defined as the smallest tensor triangulated subcategory of DMQ (k) containing the Tate objects. In this rationalized category, it is natural to denote the (images of) the generating objects by Q(n); however, I will be most interested here in the case k = Q and in a certain more subtle construction of a (rationalized, though I will now drop the subscript) subcategory DMT (Z) of DMTQ (Q) [14 §1.6], closely related to the motives over Spec Z ‘with integral coefficients’ in the sense of [13 §1.23, 2.1]. This is still a Q-linear category, but its objects have stronger integrality properties than one might naively expect. In particular: one of the foundation-stones of the theory of mixed motives is an isomorphism Ext1 (Q(0), Q(n)) ∼ = K2n−1 (Z) ⊗ Q M T (Z)

(cf. [2, 13 §8.2]. As the referee points out, one has to be careful here; the corresponding description for the category MT (Q) involves the algebraic Ktheory of Q, which is much larger than that of Z). The groups on the right have rank one for odd n > 1, and vanish otherwise, by work of Borel; the theory of regulators says that to some extent the zeta-values (n − 1)! ζ(n) (2πi)n (cf. [13 §3.7]) can be interpreted as natural generators for these groups. This is strikingly reminiscent to a homotopy-theorist of the identification (for n even) of the group Ext1 Adams (K(S 0 ), K(S 2n )) of extensions of modules over the Adams operations, with the cyclic subgroup of Q/Z generated by this zeta-number. These connections between the image of the J-homomorphism and the groups K4k−1 (Z), go back to the earliest days of algebraic K-theory [16, 50]. 4.3. This suggests that there might be some use for a notion of geometric or homotopy-theoretic realization for motives, which manages to retain some integral information. Aside from tradition (algebraic geometers usually work with cycles over Q, and topologists have been neglecting correspondences since Lefschetz), there seems to be no obstacle to the development of such a theory. Indeed, let E be a multiplicative (co)homology functor (ie a ringspectrum), supplied with a natural class of E-orientable manifolds: if E were stable homotopy, for example, we could use stably parallelizeable manifolds.

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In the case of interest below, however, E will be complex K-theory, and the manifolds will be smooth (proper) algebraic varieties over C. Such manifolds (X, Y, . . . ) define an additive category CorrE with E∗ (X × Y ) (suitably graded) as Hom-objects; composition of such morphisms can be defined using Pontrjagin-Thom transfers [45]. It is straightforward to check that X 7→ X+ ∧ E : CorrE → (Spectra) is a functor: the necessary homomorphism E∗ (X × Y ) = [S ∗ , X+ ∧ Y+ ∧ E] → [X+ ∧ E, Y+ ∧ E]∗ of Hom-objects is defined by the adjoint composition X+ ∧ Y+ ∧ E ∧ X+ ∧ E ∼ = X+ ∧ X+ ∧ Y+ ∧ E ∧ E → Y+ ∧ E built from the multiplication map of E and the composition X+ ∧ X+ ∧ E → X+ ∧ E → E of the transfer ∆! associated to the diagonal map, with the projection of X to a point. Following the pattern laid out by Voevodsky, we can now define a category of (topological) ‘E-motives’, and when E = K it is a classical fact [1] that an algebraic cycle defines a nice K-theory class. This allows us to associate to an embedding of k in C, a triangulated ‘realization’ functor Kmot : X 7→ X(C)+ ∧ K : DM(k) → (Spectra) . [Since algebraic cycles are triangulable (by Lojasiewicz), a theorem of Sullivan allows us to define these cycles in connective complex K-theory.] 4.4. When k is a number field, DMTQ (k) possesses a theory of truncations, or t-structures [14, 37], analogous to the Postnikov systems of homotopy theory; the heart of this structure is an abelian tensor category MTQ (k) of mixed Tate motives. Its existence permits us to think of DMTQ (k) as the derived category of MTQ (k); in particular, the (co)homology of an object of the larger category becomes in a natural way [31 §2.4] an object of MTQ (k). Similar considerations hold for the more rigid category DMT (Z), and since we are working in a rational, stable context, I will write π∗ for the homology groups of an object in this category, given this enriched structure. [For the purposes of this presentation I’ve reversed the logical order of construction: in fact in [14] the category MT (Z) is constructed first.] Now under very general conditions (involving a suitably rigid duality), an abelian tensor category with rational Hom-objects can be identified with a category of representations of a certain groupscheme of automorphisms of a suitable forgetful functor on the category; the resulting groupscheme is called a motivic (Galois) group. This theory applies to MT (Z), and as was noted in the introduction, Galmot (Q/Q) is the corresponding groupscheme [13

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§8.9.5, 14 §2; cf. also 2 §5.10]. In the preceding paragraph we constructed a homological functor Kmot := π∗ Kmot from DMQ to Q-vector spaces, together with a preferred lift of the functor to the category of spectra. It is easy to see that πodd Kmot = 0 , while π2n Kmot = Q(n) , as representations of Galmot (Q/Q), and thus that Kmot is represented by the mixed (Bott!)-Tate object ⊕n∈Z Q(n)[2n] = Q[b± ] ; in other words, we have constructed a lifting of the rationalized classical K-theory functor to the category of mixed Tate motives, with an action of Galmot (Q/Q) which factors through the multiplicative quotient. The possible existence of a descent spectral sequence for the automorphisms of the K-theoretic realization functor of §4.3 seems to be an interesting question, especially when restricted to some category of mixed Tate motives. 4.5. The main conjecture of this paper is that, similarly, a rational version of complex cobordism lifts to an object MUmot ∈ DMT (Z), with an action of Galmot (Q/Q) on π∗ MUmot defined by the obvious embedding Fodd o Gm → F o Gm followed by a homomorphism from the latter group to the diffeomorphisms of the formal line, cf. §2.4 above. This is a conjecture about an object characterized by its universal properties, so it can be reformulated in terms of the structures thus classified. The theory of Chern classes is founded on Grothendieck’s calculation of the cohomology of the projectification P (V ) of a vector bundle V over a scheme X. It follows immediately from his result that the cohomology of Atiyah’s model X V := P (V ⊕ 1)/P (V ) for a Thom space as a relative motive is free on one generator over that of X. Such a generator is a Thom class for V , but in the motivic context there seems to be no natural way to construct such a thing; this is related to the inconvenient nonexistence of abundantly many sections of vector bundles in the algebraic category. For a systematic theory of Thom classes it is enough, according to the splitting principle, to work with line bundles L, and in this context it is relevant that the Thom complexes −1 X L = P (L ⊕ 1)/P (L) ∼ = P (1 ⊕ L−1 )/P (L−1 ) = X L of a line bundle and its reciprocal are isomorphic objects. The conjecture about MUmot can be thus reformulated in terms of a theory of motivic

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Thom and Euler classes Umot (L), e(L) for line bundles L, satisfying a motivic Thom axiom Umot (L−1 ) = −Umot (L) , e(L−1 ) = −e(L) ; the conjecture is then the assertion that an element σ ∈ Galmot (Q/Q) sends Umot (L) to another Thom class X σ(Umot (L)) = [1 + σk e(L)k ] · Umot (L) k>0

for L, with coefficients σk depending only on σ. Since σ(Umot (L−1 )) = −σ(Umot (L)) , it follows that X X [1 + σk (−e(L))k ] · (−Umot (L)) = −[1 + σk e(L)k ] · Umot (L) , k>0

k>0

which entails that the classes σodd = 0, distinguishing the Hopf subalgebra Sev = Z[t2k | k > 0] which represents the group of odd diffeomorphisms of the formal line [46 §3.3]. Away from the prime two, classical complex cobordism is a kind of base extension MU[1/2] ∼ SO/SU ∧ MSO[1/2] of oriented cobordism, and I’m suggesting the existence of a similar splitting for the hypothetical motivic lift of complex cobordism. 4.6. After this paper had been submitted for publication, I became aware of the very elegant recent work of Levine and Morel [38], where an algebraic cobordism functor is characterized as a universal cohomology theory on the category of schemes, endowed with pullback and pushforward transformations satisfying certain natural axioms pf compatibility. [Voevodsky [56] has also considered a motivic version of the cobordism spectrum; its relation with their work is discussed briefly in the introduction to their paper.] I believe their work is fundamentally compatible with the conjectures made here, given a slight difference in framework and emphasis: they suppose a Thom isomorphism (or, equivalently, a system of covariant transfers) is to be given as part of the structure of a cohomology theory on schemes, while the spectrum hypothesized here is merely a ringspectrum, with no preferred choice of orientation.

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4.7. I should note that the trivial action of Fodd on π∗ (Kmot ), together with the usual action of Gm defined by the grading, is consistent with the existence of a spectral sequence ∗ ) =⇒ K ∗ (Z) ⊗ Q E2∗,∗ = Hc∗ (Galmot (Q/Q), Kmot

of descent type, as suggested above: using the Hochschild-Serre spectral sequence for the semidirect product decomposition of the Galois group (and confusing continuous cochain with Lie algebra cohomology) we start with ∗ Hc∗ (Fodd , Kmot )∼ = Qhe2k+1 | k ≥ 1i[b] ,

where angled brackets denote a vector space spanned by the indicated elements (with e2r+1 in degree (1, 0), and the Bott (-Tate?!) element b in degree (0, −2)). The Gm -action sends b to ub, where u is a unit in whatever ring we’re over; similarly, e2k+1 7→ u−2k−1 e2k+1 . Thus e2k+1 b2k+1 ∈ E21,−4k−2 is Gm -invariant, yielding a candidate for the standard generator in K4k+1 (Z) ⊗ Q. 5. Quantization and asymptotic expansions 5.1. The motivic Galois group appears in Kontsevich’s work through a conjectured action on deformation quantizations of Poisson structures [cf. [53]]. The framework of this paper suggests a plausibly related action on an algebra of asymptotic expansions for geometrically defined functionals on manifolds, interpreted in terms of the cobordism ring of symplectic manifolds [18, 19]. This is isomorphic to the complex cobordism (abelian Hopf) algebra MU∗ (BGm (C)) of circle bundles, and the dual (rationalized) Hopf algebra MUQ∗ (BGm (C)) ∼ = MUQ∗ [[}]] , P where } = k≥1 CPk−1 ek /k, can be interpreted [46] as an algebra generated by the coefficients of a kind of universal asymptotic expansion for geometrically defined heat kernels (or Feynman measures, via the Feynman-Kac formula [20 §3.2]), as the Chern class e of the circle bundle approaches infinity. A Poisson structure on an even-dimensional manifold V is a bivector field (a section of the bundle Λ2 TV ) satisfying a Jacobi identity modelled on that satisfied by the inverse of a symplectic structure. A symplectic manifold is thus Poisson, and although I am aware of no useful notion of Poisson cobordism (but cf. [7]) one expects a natural restriction map from asymptotic invariants Poisson manifolds to the corresponding ring for symplectic manifolds. If the conjectures above are correct, then one might further hope that (some motivic version of) such a restriction map would be equivariant, with respect to some motivic Galois action.

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5.2. Working in the opposite (local to global) direction, Connes and Kreimer have recently developed a systematic program for understanding classical quantum field theoretic renormalization in terms of its symmetries. The standard methods (eg dimensional regularization) for dealing with the singular integrals which appear in classical perturbation theory replaces them with certain meromorphic functions, and through work of Broadhurst, Kreimer, and others it has become more and more clear that the polar coefficients of these meromorphic functions are frequently elements of the polyzeta algebra. [Kontsevich has suggested that this is always so, but his program of proof fails, by the arguments of [4], and at present the question seems to be open.] Connes and Kreimer[12, 21, 26, 35 §12] have developed a systematic approach to the theory of Feynman integrals through certain Hopf algebras related to automorphism groups [42] of operads defined by graphs of various sorts. There are deep connections between the Grothendieck - Teichm¨ uller group and the Lie algebras of these automorphism groups [29, 54], and it seems likely that they (and the theory of quasisymmetric functions, via free Lie algebras) will eventually be understood to be intimately related; the appearance of polyzeta values in the theory of quantum knot invariants (cf. eg [36]) is another source of recent interest in this subject. Perhaps the deepest (and most precise) approach to the relations between these topics may be the work of Goncharov, who associates to a field F a certain Hopf algebra T• (F ) of F -decorated planar trivalent trees and a closely related Hopf algebra I• (F ) of motivic iterated integrals. According to the correspondence principle of [21 §7], the renormalization Hopf algebra corresponding to certain types of Feynman integrals should be closely related to a precisely defined subgroup of the motivic Tate Galois group of F . A slight strengthening of this correspondence principle would settle the question of the role of polyzeta values in perturbative expansions of Feynman integrals. 5.3. In exemplary cases Connes and Kreimer construct a very interesting representation of the prounipotent groupscheme underlying their renormalization algebra in the group of odd formal diffeomorphisms of the line [10 §1 eq. 20, §4 eq. 2]. It also seems quite possible that the action of their groupscheme on asymptotic expansions defined by Feynman measures associated to suitable Lagrangians [30] factor through an action of the motivic Galois group on cobordism, along the lines suggested in §4.5 above. appendix: motivic models for the little disk operad 1. In 1984 Grothendieck suggested the study of the action of the Galois group Gal(Q/Q) as automorphisms of the moduli of algebraic curves, understood as a collection of stacks linked by morphisms representing various geometrically natural fusion operations. There are remarkable analogies between his ideas and contemporary work in physics on conformal field theories, and in 1990 Drinfel’d [15] unified at least some of these lines of thought by constructing a

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pronilpotent group GT of automorphisms of certain braided tensor categories, together with a faithful representation of the absolute Galois group in that group. This program has been enormously productive; the LMS notes [42, 51] are one possible introduction to this area of research, but evolution has been extremely rapid. In the late 90’s Kontsevich [33, 34] recognized connections between these ideas, Deligne’s question on Hochschild homology, deformation quantization, and other topics, while physicists [9-12] interested in the algebra of renormalization were developing sophisticated Hopf-algebraic techniques, which are now believed [5 §8,9] to be closely related to the Hopf-algebraic constructions of Drinfel’d. The point of this appendix is to draw attention to a central conjecture in this circle of ideas: that the Lie algebra of GT, which acts as automorphisms of the system of Malcev completions of the braid groups, is a free graded Lie algebra, with one generator in each odd degree greater than one. The braid groups in fact form an operad, and I want to propose the related problem of identifying the automorphisms of the operad of Lie algebras defined by the braid groups (cf. [8]), in hope that this will shed some light on this question, and the closely related conjecture that Deligne’s motivic group acts faithfully on the unipotent motivic fundamental group [13] of P)1 −{0, 1, ∞} (with nice tangential base point). 2. For the record, an operad (in some reasonable category) is a collection of objects {On , n ≥ 2} together with composition morphisms Y cI : Or(I) × Oi → O|I| i∈I

P where I = i1 , . . . , ir is an ordered partition of |I| = ik with r(I) parts. These compositions are subject to a generalized associativity axiom, which I won’t try to write out here; moreover, the operads in this note will be permutative, which entails the existence of an action of the symmetric group Σn on On , also subject to unspecified axioms. Not all of the operads below will be unital, so I haven’t assumed the existence of an object O1 ; but in that case, and under some mild assumptions, an operad can be described as a monoid in a category of objects with symmetric group action, with respect to a somewhat unintuitive product [cf. eg. [17]]. The moduli {M0,n+1 } of stable genus zero algebraic curves marked with n + 1 ordered smooth points form such an operad: if the final marked point is placed at infinity, then composition morphisms are defined by gluing the points at infinity of a set of r marked curves to the marked points away from infinity on some curve marked with r + 1 points. This is an operad in the category of algebraic stacks defined over Z, so the Galois group Gal(Q/Q) acts on many of its topological and cohomological invariants.

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By definition, a stable algebraic curve possesses at worst ordinary double point singularities. The moduli of smooth genus zero curves thus define a system M0,∗ ⊂ M0,∗ of subvarieties, but not a suboperad: the composition maps glue curves together at smooth points, creating new curves with nodes from curves without them. However, the spaces M0,n+1 (C) of complex points of these moduli objects have the homotopy types of the spaces of configurations of n distinct points on the complex line, which are homotopy-equivalent to the spaces of the little disks operad C2 . Behind the conjectures of Deligne, Drinfel’d, and Kontsevich lies an apparently unarticulated question: is the little disks operad defined over Q?; or, more precisely: is there a version of the little disks operad in which the morphisms, as well as the objects, lie in the category of algebraic varieties defined over the rationals? Thus in [33] (end of §4.4) we have The group GT maps to the group of automorphisms in the homotopy sense of the operad Chains(C2 ). Moreover, it seems to coincide with Aut(Chains(C2 )) when this operad is considered as an operad not of complexes but of differential graded cocommutative coassociative coalgebras . . . and in [34] (end of §3) There is a natural action of the Grothendieck-Teichm¨ uller groups on the rational homotopy type of the [Fulton-MacPherson version of the] little disks operad . . . although the construction in [34 §7] is not (apparently) defined by algebraic varieties. It may be that the question above is naive [cf. [53]], but a positive answer would imply the existence of a system of homotopy types with action of the Galois groups, whose algebras of chains would have the properties claimed above; moreover, the system of fundamental groups of these homotopy types would yield an action of the Galois group on the system of braid groups, suitably completed. 3. This is probably a good place to note that Gal(Q/Q) acts by automorphisms of the etal´e homotopy type of a variety defined over Q, which is not at all the same as a continuous action on the space of complex points of the variety. In fact one expects to recover classical invariants of a variety (the cohomology, or fundamental group, of its complex points, for example) only up to some kind of completion. Various kinds of invariants [etal´e, motivic, Hodge-deRham, . . . ] each have their own associated completions, some of which are still quite mysterious; the fundamental group, in particular, comes in profinite [49] and prounipotent versions. Drinfel’d works with the latter, which corresponds to the Malcev completion used in rational homotopy theory. A free group on n generators corresponds to the graded Lie algebra defined by n noncommuting polynomial generators, and the Lie algebra pn defined by the pure braid group Pn on n

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strands [the fundamental group of the space of ordered configurations of n points in the plane] is generated by elements xik , 1 ≤ i < k ≤ n subject to the relations [xik , xst ] = 0 if i, k, s, t are all distinct, and [xik , xis ] = −[xik , xks ] if i, k, s are all distinct [32]. The fundamental groups of the little disks operad define a (unital) operad {Pn } (the symmetric group action requires some care [47 §3]); in particular, cabling Y cI : Pr(I) × Pi → P|I| i∈I

of pure braids defines composition operations which extend to homomorphisms Y cI : pr(I) × pi → p|I| i∈I

of Lie algebras, defining an operad {pn } in that category as well. The natural product of Lie algebras is the direct sum of underlying vector spaces, so cI is a sum of two terms, the second defined by the juxtaposition operation pi1 ⊕ · · · ⊕ pir → pi1 +···+ir . The remaining information is contained in a less familiar homomorphism c0I : pr(I) → p|I| defined on the first component by any partition I of |I| in r parts. It is not hard to see that X c0I (xst ) ≡ xpq + . . . ; p∈is ,q∈it

it would be very useful to know more about this expansion . . .

4. Groups act on themselves by conjugation, and thus in general to have lots of (inner) automorphisms; Lie algebras act similarly on themselves, by their adjoint representations. It would also be useful to understand something about the relations between the automorphisms of an operad in groups or Lie algebras (as monoids in a category of objects {O∗ } with {Σ∗ }-action, as in §2 above), and systems of inner automorphisms of the objects On : thus the adjoint action of a system of elements φn ∈ pn defines an operad endomorphism if c0I (φ∗ ) = φ|I| for all partitions I. From some perspective, the classification of such endomorphisms is really part of the theory of symmetric functions. This may be relevant, because the action GT on the completed braid groups is relatively close to inner. Drinfel’d [15 §4] describes elements of GT as pairs (λ, f ), where λ is a scalar (ie, an element of the field of definition for the kind of Lie algebras we’re working with: in our case, Q, for simplicity), and f

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lies in the commutator subgroup of the Malcev completion of the free group on two elements. The pairs (λ, f ) are subject to certain restrictions which I’ll omit here, on the grounds that the corresponding conditions on the Lie algebra are spelled out below. It is useful to regard elements f = f (a, b) of the free group as noncommutative functions of two parameters a, b; if σi is a standard generator of the braid group, and yi = σi−1 · · · σ2 σ12 σ2 . . . σi−1 ∈ Pn for 2 ≤ i ≤ n (with y1 = 1), then the action of GT on the braid group is defined by (λ, f )(σi ) = f (σi2 , yi )σiλ f (σi2 , yi )−1 ; the omitted conditions imply that when i = 1, this reduces to an exponential automorphism σ1 7→ σ1λ defined in the Malcev completion. 5. Here are some technical details about the Lie algebra of GT, reproduced from [15 §5]. Drinfel’d observes [remark before Prop. 5.5] that the scalar term λ can be used to define a filtration on this Lie algebra, and he describes the associated graded object grt. The following formalism is useful: fr is the free formal Q-Lie algebra defined by power series in two noncommuting generators A, B: it is naturally filtered by total polynomial degree, with the free graded Lie algebra on two generators as associated graded object. The algebra grt consists of series ψ = ψ(A, B) ∈ fr which are antisymmetric [ψ(A, B) = −ψ(B, A)] and in addition satisfy the relations ψ(C, A) + ψ(B, C) + ψ(A, B) = 0 and [B, ψ(A, B)] + [C, ψ(A, C)] = 0 when A + B + C = 0, as well as a third relation asserting that ψ(x12 , x23 + x24 ) + ψ(x13 + x23 , x34 ) − ψ(x12 + x13 , x24 + x34 ) equals ψ(x23 , x34 ) + ψ(x12 , x23 ) , assuming that the xik satisfy the relations defining p. The bracket in grt is hψ1 , ψ2 i = [ψ1 , ψ2 ] + ∂ψ2 (ψ1 ) − ∂ψ1 (ψ2 ) , where ∂ψ is the derivation of fr given by ∂ψ (A) = [ψ, A], ∂ψ (B) = 0. Drinfel’d shows [15 §5.6] that this Lie algebra is in fact isomorphic to the Lie algebra of GT [omitting the subalgebra corresponding to the scalars, used in this description to define the grading], but the isomorphism is defined inductively, so describing its action on the braid groups is not immediate. Nevertheless, Ihara (cf. [15 §6.3]) has shown that for each odd n > 1 there are elements ψn ∈ grt such that X n ψn (A, B) ≡ (adA)m−1 (adB)n−m−1 [A, B] m 1≤m≤n−1

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modulo [fr0 , fr0 ] (where fr0 is the derived Lie algebra of fr]. It is conjectured that grt is free on these generators. Aside from [41], the cabling described in §3 doesn’t seem to have been considered very closely, in this context.

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28. U. Jannsen, Motives, numerical equivalence, and semisimplicity, Inv. Math. 107 (1992) 447 - 452 29. R. Kaufmann, On spineless cacti, Deligne’s conjecture, and Connes-Kreimer’s Hopf algebra, available at math.QA/0308005 30. D. Kazhdan, Introduction to QFT, in Quantum fields and strings 377 - 418, AMS (1999) 31. R. Kiehl, R. Weissauer, Weil conjectures, perverse sheaves, and l-adic Fourier transform, Springer Ergenbisse III no. 42 (2001) 32. T. Kohno, S´erie de Poincar´e-Koszul associ´e aux groupes de tresses pures, Invent. Math. 82 (1985) 33. M. Kontsevich, Operads and motives in deformation quantization, available at math.QA/990405 34. ——–, Y. Soibelman, Deformations of algebras over operads and Deligne’s conjecture, available at math.QA/000115 35. D. Kreimer, Knots and Feynman diagrams, Cambridge Lecture Notes in Physics 15, CUP (2000) 36. TTQ Le, J. Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions. Top. Appl. 62 (1995) 193 - 206 37. M. Levine, Tate motives and the vanishing conjectures for algebraic K-theory, in Algebraic K-theory and algebraic topology, Lake Louise 1991, Kluwer (1993) 38. ——–, F. Morel, Algebraic cobordism, available at www.math.uiuc.edu/K-theory 39. ——–, Mixed Motives, AMS Surveys 57 (1998) 40. A. Libgober, Chern classes and the periods of mirrors, Math. Res. Lett 6 (1999) 141 149 41. P. Lochak, L. Schneps, The Grothendieck-Teichm¨ uller group and automorphisms of braid groups, p. 323 - 358 in [42] below 42. ——–, ——–, Geometric Galois actions I, London Math Soc. notes 242, CUP (1997) 43. I. MacDonald, Symmetric functions and Hall algebras, 2nd ed, OUP 44. I. Moerdijk, On the Connes-Kreimer construction of Hopf algebras, available at math.phy/9907010 45. J. Morava, Smooth correspondences, in Prospects in topology (Browder Festschrift), Ann. of Math. Studies 138, Princeton (1995). 46. ———, Cobordism of symplectic manifolds and asymptotic expansions, in Proc. Steklov Inst. Math. 225 (1999) 261 - 268 47. ——–, Braids, trees, and operads, available at math.AT/0109086 48. ——–, Heisenberg groups in algebraic topology, to appear in the Segal Festschrift; available at math.AT/0305250 49. H. Nakamura, L. Schneps, On a subgroup of the Grothendieck-Teichm¨ uller group acting on the tower of profinite Teichm¨ uller modular groups, Invent. Math. 141 (2000) 503 560 50. D. Quillen, letter to Milnor, in Algebraic K-theory (Northwestern 1976), Lecture Notes in Math. 551 (Springer) 51. L. Schneps (ed.), The Grothendieck theory of dessins d’enfants, London Math Soc. notes 200, CUP (1994) 52. ——–, The Grothendieck-Teichm¨ uller group: a survey, p. 183 - 204 in [51] above 53. D. Tamarkin, Action of the Grothendieck-Teichm¨ uller group on the operad of Gerstenhaber algebras, available at math.QA/0202039 54. P. van der Laan, I. Moerdijk, The renormalization bialgebra and opeads, available at hep-th/0210226

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55. V. Voevodsky, Triangulated categories of motives over a field, in E. Friedlander, A. Suslin, V. Voevodsky, Cycles, transfers, and motivic homology theories, Annals of Maths. Studies 143, Princeton (2000) 56. ———, Open problems in the motivic stable homotopy theory I, available at www.math.uiuc.edu/K-theory 57. D. Zagier, A note on the Landweber-Stong elliptic genus, in Elliptic curves and modular forms in algebraic topology, ed. P. Landweber, Lecture Notes in Mathematics 1326, Springer (1988) Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: [email protected]

TOWARD HIGHER CHROMATIC ANALOGS OF ELLIPTIC COHOMOLOGY DOUGLAS C. RAVENEL Abstract. We show that the Jacobian of a certain Artin-Schreier curve over the field Fp has a a 1-dimensional formal summand of height (p − 1)f for any positive integer f . We give two proofs, the classical one which was known to Manin in 1963 and which requires knowledge of the zeta function of the curve, and a new simpler one using methods of Honda. This is the first s tep toward contructing the cohomology theories indicated in the title.

1. Introduction A starting point for elliptic cohomology is a homomorphism ϕ from a cobordism ring to some ring R, which is often called an R-valued genus. When the cobordism theory is MU∗ , we know by Quillen’s theorem [Qui69] that ϕ is equivalent to a 1-dimensional formal group law over R. It is also known that the functor X 7→ MU∗ (X) ⊗ϕ R

is a homology theory if ϕ satisfies certain conditions spelled out in Landweber’s Exact Functor Theorem [Lan76]. Now suppose E is an elliptic curve defined over R. It is a 1-dimensional algebraic group, and choosing a local paramater at the identity leads to a b the formal completion of E. Thus we can apply the formal group law E, machinery above and get an R-valued genus. For example, the Jacobi quartic, defined by the equation y 2 = 1 − 2δx2 + x4 , is an elliptic curve over the ring R = Z[1/2, δ, ]. The resulting formal group law is the power series expansion of p √ x 1 − 2δy 2 + y 4 + y 1 − 2δx2 + x4 ; F (x, y) = 1 − x2 y 2

Date: October 26, 2006. 2000 Mathematics Subject Classification. Primary: 55N34; Secondary: 14H40, 14H50, 14L05, 55N22.

286

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this calculation is originally due to Euler. The resulting genus is known to satisfy Landweber’s conditions [LRS95], and this leads to one definition of elliptic cohomology. The rich structure of elliptic curves leads to interesting calculations with the cohomology theory and to the theory of topological modular forms due to Hopkins et al, [HM] and [AHS01]. In [HM] they consider the elliptic curve defined by the Weierstrass equation y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 . Under the affine coordinate change x 7→ x + r

and

y 7→ y + sx + t

we get

(1.1)

a6 → 7 a6 + a4 r + a 3 t + a2 r 2 +a1 r t + t2 − r 3 7 a4 + a3 s + 2 a 2 r a4 → +a1 (r s + t) + 2 s t − 3 r 2 a3 7→ a3 + a1 r + 2 t 2 a2 7→ a2 + a1 s − 3 r + s a1 7→ a1 + 2 s.

This can be used to define an action of the affine group on the ring A = Z[a1 , a2 , a3 , a4 , a6 ]. Its cohomology is the E2 -term of a spectral sequence converging to the homotopy of tmf, the spectrum representing topological modular forms. However it is known that the formal group law associated with an elliptic curve over a finite field can have height at most 2; see Corollary 2.4 below. Hence elliptic cohomology cannot give us any information about vn -periodic phenomena for n > 2. The purpose of this paper is to suggest a way to construct similar cohomology theories that go deeper into the chromatic tower. Suppose we have an algebraic curve C of genus g. Then its Jacobian J(C) is an abelian variety b of dimension g. J(C) has a formal completion J(C) which is a g-dimensional b formal group. If J (C) has a 1-dimensional summand, then Quillen’s theorem gives us a genus associated with the curve C. A result in this direction is the following. Theorem 1.2. Let C(p, f ) be the Artin-Schreier curve over Fp defined by y e = xp − x

where e = pf − 1

for a positive integer f . Then the Jacobian J(p, f ) of this curve (possibly after extension of scalars) has a 1-dimensional formal summand of height (p − 1)f .

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This result was stated by Manin in [Man63]. The case f = 1 was treated by Gorbunov and Mahowald in [GM00]. Most of what is needed for the proof can be found in Katz’s 1979 Bombay Colloquium paper [Kat81]. Koblitz’ Hanoi notes [Kob80] covers much of the same material in a less formal way. In §2 we will sketch the original proof since some of the ideas behind it will be needed later. In §3 we will give a new proof using some methods developed by Honda. It has the advantage of being both simpler and more flexible than the classicial proof. In a future paper we will use it to explore deformations of the Artin-Schreier curve and changes of coordinates leading to formulae analogous to (1.1). It is a pleasure to thank the Isaac Newton Institute for their generous hospitality during the preparation of this paper, and to thank Neil Strickland, Spencer Bloch and Mike Hopkins for helpful conversations. 2. The classical proof of Theorem 1.2 An important tool in studying commutative formal groups in characteristic p is the theory of Dieudonn´e modules. Theorem 2.1. [Die55] The category of commutative formal groups over a finite field k is equivalent to the category of modules over the ring D(k) = W(k)hF, V i/(F V = V F = p)

where W(k) is the ring of Witt vectors over k, w 7→ w σ is its Frobenius automorphism, F w = w σ F and V w σ = wV for w ∈ W(k). F is the Frobenius or pth power map, and V is the Verschiebung, the dual of F . A W(k)-module equipped with an action of such an F is called an F crystal, and a similar module over Q ⊗ W(k) is called an F -isocrystal. D(k) is the endomorphism ring of a certain projective object P in the category of commutative formal groups, and the Dieudonn´e module D(G) of a formal group G is group of homomorphisms from P to G. There is also a contravariant Dieudonn´e module D ∗ (G) defined as the set of morphisms from G to a certain injective object I also having D(k) as its endomorphism ring. See Hazewinkel [Haz78] for more information. Here are some examples. • The Dieudonn´e module for the formal group associated with the nth Morava K-theory is D(Fp )/(V − F n−1 ),

so in it we have F n = p. • More generally, for m and n relatively prime, let Gm,n = D(k)/(V m − F n ).

It corresponds to an m-dimensional formal group of height m + n. Theorem 2.2. [Dieudonn´e [Die57]]

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(i) Structure theorem. Any simple Dieudonn´e module M is isogenous over W(Fp ) to some Gm,n . (This means there is a map M → Gm,n with finite kernel and cokernel.) (ii) Let the characteristic polynomial for F in M be X Q(T ) = T m + ci T m−i i>0

for ci ∈ W(k). If its Newton polygon has a line segment of horizontal length n and slope j/n, then up to isogeny over W(k), M has a summand of the form Gj,n−j .

The Newton polygon is the lower convex hull of the set of points {(i, ordp (ci )) : 0 ≤ i ≤ m} , where c0 = 1. The condition on Q(T ) above is equivalent to the existence of n roots having p-adic valuation j/n. For more information, see [Kob80, pp. 19–23]. There are severe restrictions on the formal group attached to an abelian variety, as the following result indicates. Theorem 2.3. (i) [Manin [Man63]] Riemann symmetry condition. If A is an abelian b and its Dieudonn´e module D(A) b variety with formal completion A, has a summand Gm,n up to isogeny over W(Fp ), then it also has a summand Gn,m . (ii) [Tate [Tat66]] More precisely, if A has dimension g and is defined over Fq with q = pa , then the characteristic polynomial for F a has the form X Qa (T ) = T 2g + ci T 2g−i + q g 0 2, then the dimension of A is at least n. The key to finding the Dieudonn´e module of the formal completion of the Jacobian of a curve C is the following fortunate isomorphism. Theorem 2.5. [Grothendieck, Berthelot] Let C be a smooth curve of genus g over Fq , where q = pa . Then its crystalline (or de Rham) H 1 is a free W(Fq )-module of rank 2g isomorphic to the contravariant Dieudonn´e module b of its Jacobian D ∗ (J(C)), with the induced action of the Frobenius F˜ relative to Fq coinciding with the action of F a .

This result is originally due to Grothendieck in 1966. A proof can be found in [MM74, Appendix 2] and in [Ill79]. The crystalline cohomology of C is the same as the de Rham cohomology of a lifting of C to characteristic 0, i.e., from Fq to W(Fq ). For the curve C(p, f ), this H 1 is the free Zp -module with basis xi y j dx ωi,j = e−1 : 0 ≤ i ≤ p − 2, 0 ≤ j ≤ e − 2 , y the genus of the curve being g = (p−1)(e−1)/2. This can be derived from the fact that the curve is a d-fold branched cover of the projective line with p + 1 branch points. We will have occasion to extend scalars in the ground field, in which case H 1 gets tensored with the appropriate ring of Witt vectors. The space of holomorphic 1-forms on C(p, f ) is spanned by (2.6)

{ωi,j : ei + pj < (e − 1)(p − 1) − 1} .

This space is isomorphic to the cotangent space of the Jacobian. We will need it later in an alternate proof of Theorem 1.2. At this point is useful to state the Weil conjectures, although we will only need them in the case of algebraic curves. Given a smooth d-dimensional variety X over Fq , its zeta function is defined by ! X Tn Z(X, T ) = exp |X(Fqn )| . n n>0

where |X(Fqn )| denotes the number of points of X defined over Fqn . The following statements were conjectured by Weil in 1949 [Wei49] and proved by the indicated authors. Expository accounts have been given by Katz [Kat76] and Mazur [Maz75]. Theorem 2.7. (i) (Dwork [Dwo60]) Z(X, T ) is a rational function of T .

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(ii) Z(X, T ) satisfies a functional equation 1 Z X, d = ±q dχ/2 T χ Z(X, T ), q T where χ denotes the Euler-Poincar´e characteristic of X. (iii) (Artin, Grothendieck et al, [SGA73], [Gro95], [Gro77] and Lubkin [Lub68]) More precisely, Z(X, T ) =

P1 (T )P3 (T ) · · · P2d−1 (T ) P0 (T )P2 (T ) · · · P2d (T )

where P0 (T ) = 1 − T , P2d (T ) = 1 − q d T , and for 0 < i < 2d, Pi (T ) is a polynomial whose degree is the rank of H i (X) suitably defined. (iiv) Riemann hypothesis in characteristic p. (Deligne [Del74] and [Del80]) Each reciprocal root of Pi (T ) has absolute value q i/2 . (v) If Xis the reduction of a variety X defined over a number field K, then Pi (T ) = det(1 − T F˜ |H i (X(C))) where F˜ is the Frobenius relative to Fq . In particular the degree of Pi (T ) is the rank of H i (X(C)), the ith Betti number of X. Hence (ii) follows from an analog of the Lefschetz fixed point formula. The formula (iii) follows from an analog of the Lefschtez Fixed Point Theorem once one has defined a cohomology theory for varieties in charatcteristic p with suitable properties. These statements were proved for curves by Weil in [Wei48]. If X is a smooth curve of genus g, then Z(X, T ) =

P1 (T ) , (1 − T )(1 − qT )

where the factors (1 − T )−1 and (1 − qT )−1 correspond to H 0 and H 2 . P1 (T ), which corresponds to H 1 , has degree 2g with X P1 (T ) = 1 + ci T i + q g T 2g , 00 where Cnγ is the number of points in x in X(Fp ) satisfying γ(x) = F˜ n (x).

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Observe that if the action of G on X is trivial, and ρ is irreducible, then ! X 1 X Tn L(X, ρ, T ) = exp Tr(ρ(γ)) |X(Fqn )| |G| γ∈G n n>0 ! X T n X Tr(ρ(γ)) n = exp |X(Fq )| n γ∈G |G| n>0 1 if ρ is nontrivial = Z(X, T ) if ρ is trivial.

If ρ is the regular representation, then |G| if γ = e Tr(ρ(γ)) = 0 otherwise,

so L(X, ρ, T ) is just the zeta function. We also have

L(X, ρ1 ⊕ ρ2 , T ) = L(X, ρ1 , T )L(X, ρ2 , T ).

Recall that the regular representation decomposes as a sum of irreducible representations X degree(ρ)ρ, ρ irreducible

so

Z(X, T ) =

Y

L(X, ρ, T )degree(ρ) .

ρ irreducible

Deligne proved an alternating product formula for L(X, ρ, T ) similar to Weil’s for Z(X, T ), in which Piρ (T ) is the characteristic polynomial of F˜ restricted to HomG (ρ, H i (X) ⊗W(Fq ) K). Now suppose X is a curve of genus g and A is a finite abelian group with action defined over Fq such that in H 1 (X) ⊗W(Fq ) K, each character of A occurs with multiplicity at most 1. A always acts trivially on H 0 and H 2 , so we have P1 (T ) Z(X, T ) = (1 − T )(1 − qT ) Y ρ 1 = P (T ) (1 − T )(1 − qT ) ρ 1 Y 1 L(X, ρ, T ) = (1 − T )(1 − qT ) ρ ! Y 1 1 X = 1+ Tr(ρ(a))C1a T , (1 − T )(1 − qT ) ρ |A| a∈A

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where the product is over the 2g 1-dimensional representations ρ of A that occur as summands of H 1 (X) ⊗W(Fq ) K. Since P1 (T ) has degree 2g, each of its factors P1ρ (T ) must be linear. Our curve C(p, f ) has an action of a certain finite group G on defined over the field Fpn , where n = (p − 1)f . The group is the semidirect product G = Fp o µ(p−1)e .

It fixes the point at infinity and acts on the rest of the curve via (x, y) 7→ (ζ e x + a, ζ p y),

where ζ and a are generators of µ(p−1)e (the group of (p − 1)eth roots of unity) and Fp (regarded as an additive group) respectively. (G is isomorphic to a maximal finite subgroup of the extended Morava stabilizer group Gn . It will follow from the isomorphism above that G acts on the 1-dimensional formal summand as expected.) The smallest field containing these roots is Fpn , and its Frobenius F n respects the eigenspace decomposition associated with the action of µ(p−1)e . Under this action we have X i ωi,j 7→ ai−k ζ 1+ek+pj ωk,j . k 0≤k≤i

It is easily seen to have following properties. (a) In the restriction to the subgroup µ(p−1)e , each character which is nontrivial on µe occurs with multiplicity 1. Hence the subspace spanned by each ωi,j is also an eigenspace for F n . (b) The subspace spanned by the ωi,j for a fixed j is an eigenspace (with nontrivial eigenvalue) for the subgroup µe . We will denote it by H 1,χ where χ is the corresponding nontrivial character of µe . The action of µ(p−1)e is the induction (from µe to µ(p−1)e ) of χ. (c) In the restriction to the subgroup Z/(p), each subspace H 1,χ is an irreducible representation of degree p − 1 isomorphic to the augmentation ideal in the group ring of Z/(p), i.e., to the sum of the p − 1 nontrivial characters of Z/(p). (d) In the restriction to the abelian subgroup A = Z/(p) × µe , each character which is nontrivial on both factors occurs with multiplicity 1. We will denote by H 1,ψ,χ ⊂ H 1,χ the subspace corresponding to the nontrivial characters ψ and χ on Z/(p) and µe . Note that these 1dimensional eigenspaces are not the same as those for the subgroup µ(p−1)e . (e) Each H 1,χ is an irreducible representation of the full group G.

Next we need to define some Gauss sums associated with the characters ψ and χ. Let q = pf = e + 1. Let L be a number field containing the epth roots of unity. We extend the characters ψ and χ to F× q m in the following way. We

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compose the additive character ψ on Fp with the trace map Fqm → Fp , and × we compose the multiplicative χ with with the map F× q m → Fq sending x to m x(q −1)/(q−1) . We denote these composite characters by ψqm and χqm . Then our Gauss sum is X gqm (ψ, χ) = − ψqm (x)χqm (x). x∈F× qm

The action of F f commutes with the action of the subgroup Z/(p) × µe , so it respects the corresponding eigenspace decomposition of H 1 . This means that the action of F f n on H 1,ψ,χ is multiplication by a scalar, and that scalar is known to be gqm (ψ, χ). This is originally due to Hasse-Davenport [HD34] and is explained by Katz in [Kat81, Lemma 2.1]. The proof involves counting certain points in C(p, f ). It follows that the characteristic polynomial for the action of F f n on H 1 is Y (2.7) Pqm (T ) = (T − gqm (ψ, χ)), ψ,χ

where the product is over all nontrivial ψ and all nontrivial χ. Recall that in light of Grothendieck’s isomorphism, this is also the characteristic polynomial F f n acting on the Dieudonn´e module for the Jacobian of our curve. The numerator of the zeta function of C(p, f ), regarded as a curve over Fqm is Y T 2g Pqm (T −1 ) = (1 − gqm (ψ, χ)T ). ψ,χ

It turns out that for our purposes all we need to know about the gq (ψ, χ) is their p-adic valuations. These were originally determined by Stickelberger [Sti90], but we will derive them from the Gross-Koblitz formula. To state it we need to pick a p-adic place p in L. Its residue field is isomorphic to Fq . This choice allows us to identify (via reduction mod p) the target of χ with F× q . This means that χ can be defined as the ath power map for some integer a with 0 < a < e, so we will denote χ by χa . The choice of ψ amounts to choosing a primitive pth root of unity λ, and for each such λ there is a unique solution π to the equation π p−1 = −1 such that λ ≡ 1 + π mod

(π)2 ,

so we will write such a ψ as ψπ . Finally, let α(k) denote the sum of the digits in the p-dic expansion of k. Then the Gross-Koblitz formula says (2.8)

g (ψπ , χa ) = u(a) qm

m

qm π α((qm −1)a/e)

.

295

Higher Chromatic Analogs Here u(a) is the p-adic unit u(a) = (−1)

f

Y

0≤j 0. Suppose that K has an endomorphism σ such that there is a power q of p with aσ ≡ aq modulo m for any a ∈ A. Let Aσ hhT ii be the ring of noncommutative power series in T over A subject to the rule T a = aσ T . Let Mn (Am ) denote the ring of n × n-matrices over Am , and define the ring Mn (Am )σ hhT ii in a similar way. F is characterized by its logarithm f , which is a vector of n power series f1 , . . . , fn over K in n variables x1 , . . . , xn with fi ≡ xi modulo term of degree 2. (Honda calls f the transformer of F .) F is given by the formula F (x, y) = f −1 (f (x) + f (y)), where x and y are n-dimensional vectors.

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i

Let f σ be the power series from f by applying σ i to each coeffiP obtained i cient. Given a matrix H = i Ci T in Mn (Am )σ hhT ii, define X i i (H ∗ f )(x) = Ci f σ (xq ). i

If f is the logarithm for F , we say that H is a Honda matrix for F (or for the vector f ) and that F is of type H, if H ≡ πIn modulo T (where π is a uniformizing element for m and In is the n × n identity matrix) and (H ∗ f )(x) ≡ 0 modulo m. (Honda calls such a matrix special with respect to F .) We say that two matrices H1 , H2 ∈ Mn (Am )σ hhT ii are equivalent if H1 = U H2 for an invertible matrix U ∈ Mn (Am )σ hhT ii. Theorem 3.1. (i) [Hon70, Theorem 4] Suppose that the field K as above is unramified at p, i.e., that Am is the ring of Witt vectors W (k) and m = (p). Then the strict isomorphism classes of n-dimensional formal group laws over A correspond bijectively to the equivalence classes of matrices H ∈ Mn (Ap )σ hhT ii congruent to pIn modulo degree 1. H and f are related by the formula f (x) = (H −1 ∗ p)(x). (ii) [Hon70, Corollary to Theorem 4] Moreover, given a set B ⊂ A of representatives of the residue field A/(p), for each F there is a unique Honda matrix of the form X H = pIn + Ci T i i>0

such that each Ci has coefficients in B. (iii) [Hon70, 5.5] Suppose that the coefficients of Ci are all invariant under the endomorphism σ, let ξ denote the Frobenius endomorphism of the mod p reduction F of the formal group F , and let X det H = pn + ci T i . i>0

Then this is also the characteristic polynomial (or power series) of the Frobenius endomorphism of F . Here are some examples of Honda matrices. Example 3.2. For n = 1 and A = Z, let H be the 1 × 1 matrix with entry u = p − T h for a positive integer h. Then X −1 u−1 = p−1 1 − p−1 T h = p−1 p−i T hi i≥0

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so f (x) =

X xphi i≥0

pi

and F is the formal group law for the Morava K-theory K(h)∗ . More generally the mod p reduction of a 1-dimensional formal group law over Z, Z (p) or Zp has height h iff u is congruent to a unit multiple of T h modulo (p, T h+1 ). Example 3.3. Let A = Zp [[u1 , u2 , . . . uh−1 ]] for a positive integer h, and let uσi = upi . Let H be the 1 × 1 matrix with entry X u = p − Th − ui T i . 00

Then f (x) is the logarithm for the universal p-typical formal group law, i.e., the usual formal group law over BP∗ . Example 3.5. The Honda matrix for the Dieudonn´e formal group Gn,m is

where

H = pIn − C1 T − Cm+1 T m+1

C1 =

0 .. . 0 0 0

1 ··· .. . 0 ··· 0 ··· 0 ···

0 0 .. . 1 0 0 1 0 0

and

Cm+1

0 ... = 0

0 ··· 0 .. .. . . . 0 ··· 0

1 0 ··· 0

Details can be found in [Hon70, 5.2]. Next we recall some results from [Hon73] about the formal completion of the Jacobian J of an algebraic curve C of genus g over K. Let {ω1 , . . . ωg } be a basis of the holomorphic 1-forms on C. (For our curve C(p, f ), such a basis is given in (2.6).) Choose a local parameter z at some point P ∈ K(C) and denote by ωi (z) the expansion of ωi at P . There are power series ψi (z) over K with ψi (0) = 0 and dψi (z) = ωi (z). Let y = (y1 , . . . , yg ) be a system of local parameters at the origin of the Jacobian J(C), and let ν1 , . . . , νg be the invariant differentials of J such that νi ◦ Λ = ωi , where Λ : C → J is the canonical map corresponding to the point P . Let νi (y) denote the local expansion of νi at the origin. There are

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power series φi (y) over K with φi (0) = 0 and dφi (y) = νi (y). Then the vector b φ(y) = (φi (y)) is the logarithm for the formal completion J(C) of J(C) with respect to the local parameters y, whichl we denote by F (C).

Theorem 3.6 (Honda [Hon73]). With notation as above there is a finite set S of prime ideals in K such that if m ∈ / S and H ∈ Mg (Am )σ hhT ii is a Honda matrix for the vector ψ(z), then it is also a Honda matrix for the logarithm φ(y), so F (C) is of type H as a formal group law over Am . This means there is a close connection between power series expansions of holomorphic 1-forms on the curve C and the formal group law for its Jacobian J(C). For the curve C(p, f ) we will use y as a local parameter about the origin. A basis for the space of holomorphic 1-forms is given in (2.6). Since y e = xp − x, we can solve for x and get a power series expansion of the form x = y e g0 (y m )

where m = (p − 1)e,

for some power series g0 . It follows that dx = y e−1 g1 (y m )dy, xi y j dx ωi,j = y e−1 = y ei+j g˜i,j (y m )dy, and

ψi,j = y ei+j+1gi,j (y m )

for some power series g1 , g˜i,j ∈ Zp [[y m ]] and gi,j ∈ Qp [[y m ]]. These conditions on the ψi,j put some restrictions on the Honda matrix. Let (3.7)

E = {ei + j + 1 : i, j ≥ 0, ei + pj < (p − 1)(e − 1) − 1} ⊂ Z/(m).

Note that these conditions on i and j are the same as those in (2.6). In particular this set has g elements, where the genus g is (p − 1)(e − 1)/2. The following description of it is useful. Lemma 3.8. The set E of (3.7) is the image of p−1−i −1 ei + j + 1 : 0 ≤ i ≤ p − 2, 0 ≤ j < e p Proof. By definition E is the image of [ {ei + j + 1 : ei ≤ ei + pj < (p − 1)(e − 1) − 1} 0≤i≤p−2

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We have ei + pj < pe − p − e + 1 − 1 = m − p pj < (p − 1 − i)e − p p−1−i j < e − 1, p and the result follows.

Thus for k ∈ E, the values of i and j are uniquely determined. We will denote ψi,j and gi,j by ψei+j+1 and gei+j+1 respectively. Let hs,t ∈ ZhhT ii denote the coefficient in the tth column of the sth row of H(p, f ), where it is understood that some rows and columns are empty. Thus we have Lemma 3.9. With notation as above, hs,t (for appropriate s and t) is nonzero only if s ≡ pn t mod (m) for some integer n, and in that case X hs,t = hs,t,n T n , n≥0

where the sum is over all such nonnegative integers n. This implies that H(p, f ) has a block decomposition as follows. The pth power map acts on the group Z/(m) ∼ = µm of eigenvalues. The set E ⊂ µm is the union of its intersections with the orbits of the action on µm . The cardinality of each orbit is a divisor of (p − 1)f . Each nonempty intersection of cardinality g 0 corresponds to a g 0 × g 0 factor of H(p, f ), and hence to a g 0 -dimensional formal summand of the Jacobian J(p, f ). Example 3.10. Consider the case (p, f ) = (3, 2), for which m = 16. The values of ei + j + 1 for the integrals of the seven holomorphic 1-forms ω i,j listed in (2.6) are indicated in the following table. j 0 1 2 3 4 i=0 1 2 3 4 5 i = 1 9 10 Meanwhile, the orbits in Z/(m) under the multiplication by p include {1, 3, 9, 11} , {15, 13, 7, 5} , {2, 6} , {14, 10} , and {4, 12} . Each value listed in the table is in one of these. The first orbit contains three such elements, and the other orbits contain one each.

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It follows that the Honda matrix H (with empty rows and columns omitted) has the form b h1,1 (T 4 ) 0 T 3b h1,3 (T 4 ) 0 0 T 2b h1,9 (T 4 ) 0 2 b 0 h2,2 (T ) 0 0 0 0 0 Tb 4 4 3b 4 b 0 h3,3 (T ) 0 0 T h3,9 (T ) 0 h3,1 (T ) 2 b 0 0 0 h4,4 (T ) 0 0 0 4 b 0 0 0 0 h5,5 (T ) 0 0 4 4 4 b b T 2b h9,1 (T ) 0 T h9,3 (T ) 0 0 h9,9 (T ) 0 b 0 0 0 0 0 0 h10,10 (T 2 )

where each power series b hi,i has constant term p, so

det(H) = b h2,2 (T 2 )b h4,4 (T 2 )b h5,5 (T 4 )b h10,10 (T 2 )D, where D = b h1,1 (T 4 )b h3,3 (T 4 )b h9,9 (T 4 ) + T 8b h1,3 (T 4 )b h3,9 (T 4 )b h9,1 (T 4 )

+T 4b h1,9 (T 4 )b h3,1 (T 4 )b h9,3 (T 4 ) − T 4b h9,1 (T 4 )b h3,3 (T 4 )b h1,9 (T 4 ) −T 4b h9,3 (T 4 )b h3,9 (T 4 )b h1,1 (T 4 ) − T 4b h9,9 (T 4 )b h3,1 (T 4 )b h1,3 (T 4 ).

We will study this by computing the various factors modulo (3, T n ) for appropriate n. Recall that H is congruent to 3I7 modulo T , so b h2,2 (T 2 ) ≡ h2,2,2 T 2 b h4,4 (T 2 ) ≡ h4,4,2 T 2 b h5,5 (T 2 ) ≡ h5,5,4 T 4 b h10,10 (T 2 ) ≡ h10,10,2 T 2 and D ≡ h1,3,0 h3,9,0 h9,1,0 T 4 so det(H) ≡ h2,2,2 h4,4,2 h5,5,4 h10,10,2 h1,3,0 h3,9,0 h9,1,0 T 14

mod(3, mod(3, mod(3, mod(3, mod(3, mod(3,

T 4 ), T 4 ), T 8 ), T 4 ), T 8) T 16 ),

where the 3-adic integer hi,j,k is the coefficient of T k in the matrix entry hi,j . On the other hand by Theorem 3.1(iii), det(H) is the characteristic polynomial of the Frobenius, so by Theorem 2.3(ii) it must have the form T 14 + · · · + 37 . It follows that h5,5,4 is a unit, which gives us the desired 1-dimensional formal summand of height 4. In order to generalize the calculation above we need the following three easy lemmas. Lemma 3.11. Each orbit of Z/(m) other than {ei} for 0 ≤ i ≤ p − 2 has a nonempty intersection with E. If a is in such an orbit, then k ∈ E iff e−k ∈ / E. Lemma 3.11 is needed for the following.

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Lemma 3.12. The determinant of a block of the Honda matrix H(p, f ) associated with an orbit of cardinality n is congruent to an integer multiple of T n modulo (p, T n+1 ), and the sum of all such cardinalities is 2g. Lemma 3.13. The orbit of (m−1) has (p−1)f elements, but its intersection with E has only one element, namely (m − 1)/p. Lemma 3.12 impies that det(H(p, f )) is congruent to an integer multiple of T 2g modulo (p, T 2g+1 ). By Theorems 3.1(iii) and 2.3(ii), the coefficient of T 2g is 1, so the coefficients in 3.12 are all units. In particular the one for the orbit of (m − 1)/p is a unit, which gives us the desired 1-dimensional formal summand of height (p − 1)f . This completes our alternate proof of Theorem 1.2. Proof of Lemma 3.11. The only singleton orbits are the ones cited since pk ≡ x modulo (m) implies that k is a multiple of d. These orbits do not intersect E. Given k not divisible by d with 0 < k < e, we write k = ei + j + 1 and e − k = ei0 + j 0 + 1. Then i0 = p − 2 − i and j 0 = e − 1 − j. Now k ∈ E iff p−1−i 0≤ j <e p p−1−i p − 1 − i0 0 e−1≥ j >e −1=e −1 p p which holds iff k − e ∈ / E.

Proof of Lemma 3.12. Suppose e1 ∈ E is in such an orbit. Define elements ei inductively by letting ei+1 ∈ E be congruent modulo (m) to pk ei for the smallest possible positive value of k. Hence the intersection of the orbit with E has the form {e1 , e2 , . . . es } ki ks with P ei+1 ≡ p ei for 1 ≤ i < s, and e1 ≡ p es . The cardinality n of the orbit is ki . The entries in the corresponding block of H(p, f ) are ( T ki +···+kj−1 b hei ,ej (T n ) if i ≤ j hei ,ej = n−kj −···−ki−1 b n T hei ,ej (T ) if j < i Since hei ,ei ,0 = 0, it follows that the determinant of this block is congruent to he1 ,e2 ,k1 he2 ,e3 ,k2 · · · hes−1 ,es ,ks−1 hes ,e1 ,ks T n

modulo (p, T 2n ) as desired. To find the sum of the cardinalities, we need Lemma 3.11. If an orbit intersecting E is self-conjugate, that is it contains both a and −a, then 3.11 implies that exactly half the elements in the orbit are in E. Otherwise if s out of n elements of an orbit are in E, then E also contains n − s elements

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in the conjugate orbit. In either case half of the elements in the union of an intersecting orbit and its conjugate are in E. Since E has g elements, the sum of the cardinalities is 2g. Proof of Lemma 3.13. The orbit of 1 is k p : 0 ≤ k < (p − 1)f = pk + ei : 0 ≤ k < f, 0 ≤ i ≤ p − 2 since modulo (m)

pf +k (d + 1)pk ≡ e + pk . It follows that the orbit of m − 1 is e − pk − ei : 0 ≤ k < f, 0 ≤ i ≤ p − 2 = ei + e − pk : 0 ≤ k < f, 0 ≤ i ≤ p − 2 ,

which has (p − 1)f elements. (m − 1)/p is among them since m−1 = pf − pf −1 − 1 = e − pf −1 . p We need to show that no other elements of this orbit are in E. Using 3.8 we see that ei + e − pk is in E only if p−1−i k e−p < e p −1 − i k −p < e p i+1 k , p > e p which means i = 0 and k = f − 1.

References [AHS01] M. Ando, M. J. Hopkins, and N. P. Strickland. Elliptic spectra, the Witten genus and the theorem of the cube. Invent. Math., 146(3):595–687, 2001. ´ [Del74] Pierre Deligne. La conjecture de Weil. I. Inst. Hautes Etudes Sci. Publ. Math., (43):273–307, 1974. ´ [Del80] Pierre Deligne. La conjecture de Weil. II. Inst. Hautes Etudes Sci. Publ. Math., (52):137–252, 1980. [Die55] Jean Dieudonn´e. Lie groups and Lie hyperalgebras over a field of characteristic p > 0. IV. Amer. J. Math., 77:429–452, 1955. [Die57] Jean Dieudonn´e. Groupes de Lie et hyperalg`ebres de Lie sur un corps de caract´eristique p > 0. VII. Math. Ann., 134:114–133, 1957. [Dwo60] Bernard Dwork. On the rationality of the zeta function of an algebraic variety. Amer. J. Math., 82:631–648, 1960. [GM00] V. Gorbounov and M. Mahowald. Formal completion of the Jacobians of plane curves and higher real K-theories. J. Pure Appl. Algebra, 145(3):293–308, 2000. [Gro77] Cohomologie l-adique et fonctions L. Springer-Verlag, Berlin, 1977. S´eminaire de G´eometrie Alg´ebrique du Bois-Marie 1965–1966 (SGA 5), Edit´e par Luc Illusie, Lecture Notes in Mathematics, Vol. 589.

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[Gro95] Alexander Grothendieck. Formule de Lefschetz et rationalit´e des fonctions L. In S´eminaire Bourbaki, Vol. 9, pages Exp. No. 279, 41–55. Soc. Math. France, Paris, 1995. [Haz78] M. Hazewinkel. Formal Groups and Applications. Academic Press, New York, 1978. [HD34] H. Hasse and H. Davenport. Die Nullstellensatz der Kongruenz zeta-funktionen in gewissn zyklischen F¨ allen. J. Reine Angew. Math., 172:151–182, 1934. [HM] M. J. Hopkins and M. A. Mahowald. From elliptic curves to homotopy theory. Preprint in Hopf archive at http://hopf.math.purdue.edu/Hopkins-Mahowald/eo2homotopy. [Hon68] Taira Honda. Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan, 20:83–95, 1968. [Hon70] Taira Honda. On the theory of commutative formal groups. J. Math. Soc. Japan, 22:213–246, 1970. [Hon73] Taira Honda. On the formal structure of the Jacobian variety of the Fermat curve over a p-adic integer ring. In Symposia Mathematica, Vol. XI (Convegno di Geometria, INDAM, Rome, 1972), pages 271–284. Academic Press, London, 1973. ´ [Ill79] Luc Illusie. Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. (4), 12(4):501–661, 1979. [Kat76] Nicholas M. Katz. An overview of Deligne’s work on Hilbert’s twenty-first problem. In Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), pages 537–557. Amer. Math. Soc., Providence, R. I., 1976. [Kat81] Nicholas M. Katz. Crystalline cohomology, Dieudonn´e modules, and Jacobi sums. In Automorphic forms, representation theory and arithmetic (Bombay, 1979), volume 10 of Tata Inst. Fund. Res. Studies in Math., pages 165–246. Tata Inst. Fundamental Res., Bombay, 1981. [Kob80] Neal Koblitz. p-adic analysis: a short course on recent work, volume 46 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1980. [Lan76] P. S. Landweber. Homological properties of comodules over M U∗ (M U ) and BP∗ (BP ). American Journal of Mathematics, 98:591–610, 1976. [LRS95] P. S. Landweber, D. C. Ravenel, and R. E. Stong. Periodic cohomology theories ˇ defined by elliptic curves. In Mila Cenkl and Haynes Miller, editors, The Cech Centennial, volume 181 of Contemporary Mathematics, pages 317–338, Providence, Rhode Island, 1995. American Mathematical Society. [Lub68] Saul Lubkin. A p-adic proof of Weil’s conjectures. Ann. of Math. (2) 87 (1968), 105-194; ibid. (2), 87:195–255, 1968. [Man63] Ju. I. Manin. Theory of commutative formal groups over fields of finite characteristic. Uspehi Mat. Nauk, 18(6 (114)):3–90, 1963. [Maz75] B. Mazur. Eigenvalues of Frobenius acting on algebraic varieties over finite fields. In Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pages 231–261. Amer. Math. Soc., Providence, R.I., 1975. [MM74] B. Mazur and William Messing. Universal extensions and one dimensional crystalline cohomology. Springer-Verlag, Berlin, 1974. Lecture Notes in Mathematics, Vol. 370. [Qui69] D. G. Quillen. On the formal group laws of oriented and unoriented cobordism theory. Bulletin of the American Mathematical Society, 75:1293–1298, 1969. [SGA73] Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 3. Springer-Verlag, Berlin, 1973. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1963–1964 (SGA

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4), Dirig´e par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305. ¨ [Sti90] L. Stickelberger. Uber eine Verallgemeinerung der Kreistheilung. Mathematische Annalen, 37:321–367, 1890. [Tat66] John Tate. Endomorphisms of abelian varieties over finite fields. Invent. Math., 2:134–144, 1966. [Wei48] Andr´e Weil. Vari´et´es ab´eliennes et courbes alg´ebriques. Actualit´es Sci. Ind., no. 1064 = Publ. Inst. Math. Univ. Strasbourg 8 (1946). Hermann & Cie., Paris, 1948. [Wei49] Andr´e Weil. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., 55:497–508, 1949. University of Rochester, Rochester, NY 14627

WHAT IS AN ELLIPTIC OBJECT? GRAEME SEGAL ALL SOULS COLLEGE, OXFORD

1. Elliptic cohomology A generalized cohomology theory is a sequence of contravariant functors {hi }i∈Z from spaces to abelian groups which are linked together in a wellknown way. The theories that arise in nature are of two types: K-theories, and cobordism theories. (Classical cohomology can be approached in so many different ways that I shall leave it aside for the moment.) On a compact space X the isomorphism classes of complex vector bundles form an abelian semigroup Vect(X) under the operation of direct sum, and K 0 (X) is the abelian group got by formally adjoining inverses to the semigroup Vect(X). Then K 0 is a homotopy functor, and the functors K −i , for i > 0, defined — roughly — by composing K 0 with the i-fold suspension functor, have the properties of “half” a cohomology theory. That much is true for any representable homotopy functor, but the functors K i are special because of the Bott periodicity theorem, which gives a canonical equivalence between K i and K i−2 for i ≤ 0, and enables us to define K i for all i ∈ Z by periodicity. There is a completely different reason, however, unrelated to Bott periodicity, why the functor K 0 forms part of a cohomology theory, and it applies in a much more general context. For any (discrete) ring A we have a contravariant functor X 7→ ModA (X), where ModA (X) is the semigroup of isomorphism classes of bundles of finitely generated projective A-modules on X. It is a representable homotopy functor, though not a very interesting one, as it sees only the fundamental group of X. But if, instead of making the semigroup ModA (X) into a group separately for each space X, we perform the group-completion on the representing space, i.e. we look for the universal abelian-group-valued representable homotopy functor F with a transformation ModA → F , then we obtain a much more interesting functor KA0 . In fact KA0 (X) = [X; ΩB|PA |]. Here |PA |, the “space” of the category PA of finitely generated projective A-modules, which is the representing space for ModA (X), is a topological semigroup under ⊕, and ΩB|PA | is the loop-space of the classifying space of |PA |. The remarkable thing is that for any category C with a composition law 306

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which, like ⊕, is commutative up to coherent canonical isomorphisms we can iterate the classifying space functor |C| 7→ B|C|, and can define a cohomology theory by KCi (A) = [X; ΩB i+1 |C|]. When C = PA this is Quillen’s algebraic K-theory for the ring A, and when C is the category of finite sets under disjoint union then we get stable cohomotopy theory [S1]. The theories KC∗ are the first basic class of cohomology theories. Classical cohomology with coefficients in an abelian group A is the case coming from the category whose objects are the elements of A, and which has no morphisms except identity morphisms. The second basic class of cohomology theories are cobordism theories. They arise as homology rather than cohomology theories.1 The basic example is oriented bordism, where hn (X) is defined as the cobordism classes of maps φ : M → X, where M is a compact oriented smooth n-manifold. (A cobordism between (M0 , φ0 ) and (M1 , φ1 ) is a cobordism W between M0 and M1 with a map φ : W → X restricting to φ0 and φ1 at the ends.) By considering manifolds M with various kinds of additional structure, or allowing manifolds with singularities of various specified kinds, we get a variety of different homology theories. For example, framed manifolds lead to stable homotopy theory, while classical integral homology corresponds to allowing singular manifolds with arbitrary singularities of codimension two. The example we shall be interested in is complex cobordism MU∗ , corresponding to manifolds with weakly almost-complex structure. A noticeable difference between cobordism theories and K-theories is that, with the former, classes of any degree n are equally well represented geometrically, not just those of degree 0. For any cobordism theory h∗ there is an appropriate space Mn of n-manifolds such that h−n (X) = [X; Mn ], and Mn has an interpretation even for n < 0. As far as I know, the two classes of theories just mentioned exhaust the known “natural” cohomology theories. But there are other classes of theories which can be constructed from them algebraically. The most important of these are complex-orientable theories. Definition 1.1. A multiplicative2 theory h∗ is complex-oriented if there is given an element c1 ∈ h2 (P∞ C ) which restricts to the canonical generator of 2 2 ∼ 0 ˜ h (S ) = h (point) . (Here S 2 = P1C ⊂ P∞ C .) For such a theory we have (see [Ad]) 1We

can pass at will between homology and cohomology theories, but it is significant that it is the homology classes which have a straightforward geometric interpretation. 2A theory h∗ is multiplicative if h∗ (X) is an anticommutative graded ring.

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Proposition 1.2. ∼ h∗ (P∞ C ) = A[[c1 ]], where A = h∗ (point). Because P∞ C is an H-space (its composition-law representing the tensor product of line bundles) the ring A[[c1 ]] is a Hopf algebra, and the diagonal map c1 7→ m(c1 ⊗ 1, 1 ⊗ c1 ) is a formal group-law associated to h∗ . In a complex-oriented theory h∗ there is a canonical Thom class for any complex vector bundle, and, in particular, a sequence of elements in h2n (MUn ) corresponding to a transformation MU ∗ → h∗ . This means that complex cobordism is universal among complex-oriented theories. Quillen proved that its formal group law is also universal, in the sense that a law over any graded ring R comes from that of MU ∗ by a ring-homomorphism AM U = MU ∗ (point) → R. Alternatively expressed, a formal group-law over R is the same thing as a genus for weakly almost-complex manifolds, for a genus is exactly such a homomorphism. Elliptic cohomology was conceived because of the discovery of the elliptic genus — actually, of the remarkable rigidity properties (see [L],[S2]) of a particular family of genera Φτ : AM U → C parametrized by elliptic curves Στ = C/(Z + τ Z). The Φτ can be assembled into Φ : AM U → R, where R is a ring of modular forms. Landweber observed that if we take R = Z[ 21 , δ, ε, ∆−1 ], where ∆ = ε(δ 2 − ε)2 , and δ and ε are the functions of the curve Στ which arise when its equation is written in the form y 2 = 1 − 2δx2 + εx4 , then Ell∗ (X) = MU ∗ (X) ⊗AM U R satisfies conditions that he had previously found which ensure that the functor MU ∗ ( ) ⊗AM U R is a cohomology theory. This was the original definition of elliptic cohomology. Since its proposal a great deal of work — especially by Hopkins [H] and his collaborators — has been devoted to finding an improved version, which ought not to require inverting the prime 2. It is now believed that the “correct” theory, which Hopkins calls tmf∗ , is not, in fact, quite complex-orientable, but that a tmf∗ -orientation of a manifold M should be a string structure on M in the sense described below. The coefficient ring tmf∗ (point) maps to the ring MZ of integral modular forms (i.e. modular forms whose expansion in terms of q = e2πiτ lies in Z[[q]]). If we tensor with the rational numbers Q then tmf ∗ (point) → MZ becomes an isomorphism, but tmf∗ (point) is a much more subtle and complicated ring than MZ , with a great deal of torsion. An n-manifold with a string structure has a genus in tmf−n (point) — the image of the fundamental class [M] under the map

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induced by M → (point) — and the image of this genus in MZ is the Witten genus. 2. Elliptic objects It is a very natural question whether the elliptic cohomology classes of a space have geometric representatives in the way that K-theory classes are represented by vector bundles. The main evidence that this may be true is Witten’s heuristic argument for the rigidity of the elliptic genus. The essential idea is that the elliptic genus of a compact manifold M is the Hilbert series Σq k dim(Vk ) of a (virtual) graded vector space V = ⊕Vk which is the index ker(DLM ) − coker(DLM ) of a Dirac-like differential operator DLM defined not on M but on its smooth loop-space LM. The grading on the index comes from the action of the circle T on LM by rotation of loops. This loop space perspective, however, does not by itself shed light on the modularity which is the basic property of the elliptic genus. The place where one “naturally” encounters graded vector spaces whose Hilbert series are modular is two-dimensional conformal field theory. Definition 2.1. 3 A conformal field theory is a Hilbert space H together with a trace-class operator UΣ,ξ : H⊗p → H⊗q associated to each pair (Σ, ξ), where Σ is a Riemann surface which is a cobordism from an “incoming” manifold Sp consisting of p parametrized circles to a similar “outgoing” manifold Sq , and ξ is a point of the Quillen determinant ¯ line DetΣ of the ∂-operator of Σ. The essential properties the operators UΣ,ξ are required to satisfy are (i) UΣ,0 ,ξ 0 ◦ UΣ,ξ = UΣ0 ∪Σ,ξ 0 ⊗ξ when the cobordisms Σ and Σ0 are concatenated, and (ii) UΣ,ξ ⊗ UΣ0 ,ξ 0 = UΣtΣ0 ,ξ⊗ξ 0 when the cobordisms are simply put side-by-side. If UΣ,ξ depends holomorphically on (Σ, ξ) then the theory is called chiral. If, furthermore, we have UΣ,λξ = λm UΣ,ξ for λ ∈ C× then we say the theory is of level m. If, finally, we have UΣ,ξ = UΣ∗ ∗ ,ξ for all (Σ, ξ), where Σ∗ is Σ with the conjugate complex structure, then the theory is called unitary. If the surface Σ is closed then UΣ,ξ is simply a complex number. For a chiral theory of level m, the restriction of UΣ,ξ to closed surfaces of genus 1 is a modular form of level m. 3For

more details, see [S3].

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Among the cobordisms from S1 to S1 there is a sub-semigroup formed by the annuli Aq = {z ∈ C : |q| ≤ |z| ≤ 1} for 0 ≤ |q| ≤ 1, with the boundary circles parametrized by θ 7→ eiθ , θ 7→ qeiθ . Its action on the Hilbert space H of a chiral theory gives H a grading H = ⊕k≥0 Hk by finite dimensional subspaces. Proposition 2.2. If Σq is the torus C× /q Z , then UΣq ,ξq = Σq k dim(Hk ) for any chiral theory of level m, where ξq is the canonical element coming from the annulus Aq .

4

of DetΣq

This is easily proved by regarding the cobordism Σq as the composite of two annuli Aq1 , Aq2 with q1 q2 = q, but I shall not give the details here . As chiral conformal field theories give us modular forms so naturally, we might first guess that elliptic cohomology is a K-theory made from bundles of field theories. The crudest approximation to an elliptic class is simply a graded complex vector bundle, and there is indeed a forgetful transformation tmf ∗ (X) → K ∗ (X)[[q]] corresponding to the q-expansion of a modular form. Nevertheless, to get further we must remember that the elliptic genus is the index of an operator not on X but on LX. We need the notion of a conformal field theory over X. There is no loss in assuming that X is a smooth manifold. Definition 2.3. A conformal field theory over X is a rule which assigns a vector space Hγ to each smooth loop γ : S 1 → X, and an operator UΓ,ξ : Hγ1 ⊗ . . . Hγp → Hγp+1 ⊗ . . . Hγp+q to each Riemann surface Σ which is a cobordism from the “incoming” loops γ1 , . . . , γp to the “outgoing” loops γp+1 , . . . , γp+q , and is equipped with a map Γ : Σ → X. As before, ξ ∈ DetΣ , and UΓ,ξ must have the properties (i) and (ii) of Definition 2.1. The basic example which motivates Definition 2.2 is the bundle of spinors on the loop space of an oriented Riemannian n-manifold M. The tangent bundle of LM has structure group LSOn , and this group has a projective unitary representation (see [PS] Chap.12) which is naturally regarded as its 4See

[S3] §6. The element ξq differs from the “more canonical” element of DetΣq which is unique only up to a 12th root of unity by multiplication by the square of the Dedekind η-function.

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“spin” representation.5 The condition that we can make a Hilbert space bundle on LM associated to its tangent bundle by this projective representation is the vanishing of the characteristic classes w2 and 12 p1 of M. The choice of such a spinor bundle on LM is called a string structure on M. A string structure automatically extends — using the Riemannian metric of M — to a conformal field theory of level n over M. The propagation operators UΓ,ξ form a kind of connection in the spinor bundle, though it should be remembered that even when Σ is a cylinder, i.e. Γ : Σ → M is a path in LM, the operator UΓ,ξ is a contraction operator, not a unitary isomorphism. In my talk [S2] I speculated whether a conformal field theory over X of level m defines a class in Ell−m (X). As far as I know, the question is still open. If something of the kind really is true then it seems to me quite remarkable, for, apart from cobordism theories, the only situations I know where we have geometric representatives for cohomology classes of all dimensions are real and complex K-theory (in virtue of Bott periodicity), and classical de Rham theory for smooth manifolds. The main evidence in support of the idea is that a level m theory over a compact 2n-manifold X with a string structure can be “integrated” to give a virtual conformal field theory of level m+2n, and hence a modular form in Ell−m−2n (point): the integration process is tensoring the theory with the Dirac operator on LX and forming the index of the resulting coupled operator. In the language of quantum field theory the Dirac operator in the spinor bundle on LM is a supersymmetry operator. To explain what this means we must first recall two more aspects of the formalism of conformal field theory. First, the group Diff(S 1 ) of diffeomorphisms of S 1 acts on the Hilbert space H of a conformal theory, and the action of annuli extends the action of its Lie algebra Vect(S 1 ) to the complexification, giving us a map L : VectC (S 1 ) → End(H). In the case of a chiral theory the map L is complex-linear, but in general we write L = L+ + L− , where L+ is C-linear and L− -antilinear. The maps L+ and L− define commuting (projective) actions of VectC (S 1 ) on H. The second point is that VectC (S 1 ) is the even part of a Lie superalgebra 1 V(S 1 ) whose odd part is the space Ω− 2 (S 1 ) of (− 21 )-forms on S 1 (two of which can be multiplied pointwise to give a vector field). The class of conformal field theories which are “half-supersymmetric” in the sense that there is given a Cantilinear action on V(S 1 ) extending the L− -action of VectC (S 1 ) is important 1 for elliptic cohomology, for the action of the odd element (dθ)− 2 of V(S 1 ) on 5More

precisely, there is a positive energy and a negative energy spin representation, differing by changing the orientation of the circle. We actually want the negative energy choice, which makes the theory antichiral, in the sense that the operators depend antiholomorphically on the complex structure of the surface.

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H has an index which is a virtual chiral conformal field theory. Indeed the space of these half-supersymmetric theories seems to be the correct model of the space of virtual chiral conformal theories, just as the space of Fredholm operators is the best model of the space of virtual finite-dimensional vector spaces. When we have a string structure {Hγ } on a manifold M the Dirac operator acts — in principle — on the Hilbert space H of sections of the spinor bundle {Hγ } over LM which are square-summable for a measure on LM which forces them to be concentrated in an extremely small neighbourhood of the point loops. The group Diff(S 1 ) acts on LM, and the bundle {Hγ } is equivariant with respect to it, so we expect Diff(S 1 ) to act on H. One would like to say that this action is part of a conformal field theory structure on H which is halfsupersymmetric in the above sense: the Dirac operator should be the action 1 on H on the element (dθ)− 2 of the superalgebra. In fact that is too much to hope for; but as far as homotopy theory is concerned one can proceed much more formally, replacing the space of sections of the bundle {Hγ } on LM by the space H of jets of sections along the subspace M of point loops.6 On this Hilbert space H one can much more plausibly define the half-supersymmetric conformal field theory structure, as I have attempted to sketch in [S2]. We can do this even after tensoring the spinor bundle with an arbitrary chiral conformal field theory over M as defined in 2.3. This is the “integration” operation referred to above. The Dirac operator itself is mapped to the Witten genus, while the original elliptic genus is the image of the Dirac operator tensored with the chiral — rather than antichiral — spinor bundle. If we could do this for a family of manifolds M rather than just a single one then we should have related the space of level m + n conformal field theories to the n-fold loop space of the space of level m theories. Unfortunately, one could not expect to use conformal field theories over X by themselves to define Ell∗ (X). The essential reason is that, like the loop space LX, they are not defined locally on X, and so do not have the basic Mayer-Vietoris property of a cohomology theory. Another disconcerting fact is that a chiral conformal field theory is a rigid object which is not determined up to isomorphism by its modular form (e.g. an even unimodular lattice of rank k gives rise to a conformal field theory of level k, and non-isomorphic lattices with the same modular form give non-isomorphic theories). It is plausible, however, that chiral theories with the same modular form are connected in the space of half-supersymmetric theories. Before describing the recent progress in understanding elliptic objects it seems worth mentioning one other hint — first pointed out by Grojnowski 6I

should mention at this point a large body of work by Gorbounov, Malikov, Schechtman, and Vaintrob, e.g. [MSV],[GMS], who have developed a notion of ”chiral de Rham complex” defined on the formal neighbourhood of M in LM .

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[G] — that the field theory approach may be on the right track. This is the question of G-equivariant elliptic cohomology for a compact group G: we know that the definition of equivariant K-theory in terms of G-vector-bundles is one of the things that makes K-theory such a useful tool. There is a general notion of a G-equivariant quantum field theory — in physicists’ language, a “theory with gauged G-symmetry” — which in the present situation reduces to the following. Definition 2.4. A two-dimensional G-equivariant chiral conformal field theory assigns a Hilbert space HS,P to each oriented 1-manifold S equipped with a principal G-bundle P , and an operator UΣ,Q,ξ : HS0 ,P0 → HS1 ,P1 to each conformal cobordism Σ from S0 to S1 equipped with a holomorphic GC bundle reducing to P0 and P1 at the ends. The operator should have properties analogous to those of definition 2.1, and should depend holomorphically on the pair (Σ, Q). As usual, ξ ∈ DetΣ,Q . When G is a connected group this means that HS,P is a positive-energy projective representation of the loop group LG. The attractive idea that Ell∗G (point) should be some kind of representation ring of LG has been pursued further by Devoto [Dv1] and Ando [An], but for lack of a satisfactory equivariant version of Landweber’s theorem there is still no real candidate for the equivariant elliptic theory Ell∗G . For finite groups G the field theory point of view seems to fit with what is known ([HKR],[Dv2]) about Ell∗ (BG), but again I know no definite theorem. The work of Stolz and Teichner [ST] has moved the idea of an elliptic object forward in several important ways. One is their focus on the space of the halfsupersymmetric theories already mentioned. But their main contribution concerns the Mayer-Vietoris property. The problem with the definition 2.1 of a conformal field theory is that it does not incorporate any sense in which the Hilbert space H associated to the circle S 1 is local with respect to S 1 . If H could be reconstructed from objects HI associated to small subintervals I of S 1 then we might be able to think of a conformal field theory over X as a local object on X, and could hope to construct a cohomology theory. The simplest sense in which H could be local would be if one could associate a Hilbert space HI to each closed subinterval I of the circle so that H ∼ = ⊗HIi when the circle is the union of intervals Ii meeting only at their ends. Locality of this simple kind — which would hold, for example, if H were the symmetric or exterior algebra on L2 (S 1 ) — is easily seen to be impossible in conformal field theory. In the simplest conformal field theories, the space H is a “renormalized” symmetric or exterior algebra on a space of functions such as L2 (S 1 ), where the renormalization depends on a polarization L2 (S 1 ) = L2 (S 1 )+ ⊕ L2 (S 1 )−

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of L2 (S 1 ) into positive and negative frequency parts. The projection operators defining the polarization need to be given only up to Hilbert-Schmidt perturbations: they are singular integral operators on S 1 with kernels whose supports can be chosen in an arbitrarily small neighbourhood of the diagonal in S 1 × S 1 , but which cannot be supported exactly on the diagonal. The effect for the locality of H is that if S 1 = I ∪ J, where I and J are open intervals, then H can be reconstructed from Hilbert spaces HI , HJ and a von Neumann algebra AI∩J associated to I ∩ J which acts on both HI and HJ . The reconstruction is by means of Connes’s notion of the tensor product of bimodules over von Neumann algebras. If A, B, C are von Neumann algebras, M is an (A, B)-bimodule, and N is a (B, C)-bimodule, then Connes defines a (A, C)-bimodule M ∗B N . Two important features of his theory are the existence of a neutral B-bimodule B0 with the property that M ∗B B0 ∼ =M ∼ and B0 ∗B N = N , and the fact that a (B, B)-bimodule gives us a Hilbert space M∗B = M ∗(B op ⊗B) B0 .7 The relevance of the Connes tensor product to the locality of loop group representations, and hence to two-dimensional conformal field theory, was first realized by Wassermann [W]. In the light of his work the following definition — essentially that of Stolz and Teichner — seems appropriate. Definition 2.5. A three-tier conformal field theory over X consists of the data of Definition 2.2 together with (i) a bundle of von Neumann algebras {Ax }x∈X on X, and (ii) an (Ax , Ay )-bimodule Hγ for each path γ from x to y. The properties the bimodules must have are that Hγ ∼ = Hγ ∗A Hγ 2

z

1

if the path γ from x to y is the concatenation of γ1 from x to z and γ2 from z to y, and that Hγ0 ∼ = Hγ ∗Ax if the path γ from x to x is regarded as a closed path γ0 . One can presumably construct a cohomology theory based on 3-tier conformal field theories of any chosen level, but, as far as I know, little has yet been proved, especially about why the theories at different levels should be related by suspension. Apart from the Stolz-Teichner programme there is another quantum field theory approach to elliptic cohomology which has been proposed by Baas, Dundas, and Rognes [BDR]. In one important sense it is much less ambitious: it aims only to construct elliptic objects of degree zero, relying on the 7I

have taken the notation M ∗B from Quillen, who uses it in an algebraic setting for the quotient of M which equalizes the left and right B-actions.

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machinery of algebraic K-theory to produce the theory in other dimensions. I shall try to say briefly how it fits in to the general philosophy of this talk. We can give a definition of a general d-dimensional quantum field theory along the lines of 2.1, but using manifolds equipped with a Riemannian rather than just a conformal structure. (Of course we shall have a Hilbert space HS assigned to each compact oriented Riemannian (d − 1)-manifold S, subject to HS1 ⊗ HS2 ∼ = HS1 tS2 .) Among these theories are the conformal ones, and — much more specially still — the so-called topological field theories, for which the vector spaces and the operators depend only on the smooth structure of the manifolds, without any metric at all. On the space of all quantum field theories we have the renormalization group flow: a theory is a functor from a cobordism category to vector spaces, and for any t > 0 we can compose the theory with the functor from the cobordism category to itself which multiplies the metric of every manifold by t. When d = 1 a quantum field theory is precisely a semigroup of trace-class operators in a Hilbert space — self-adjoint operators if the theory is unitary. The renormalization group flow retracts the space of 1-dimensional theories (with its natural topology) to the subspace of topological theories, which is simply the space of finite-dimensional complex vector spaces. We can also define supersymmetric unitary 1-dimensional theories. Such a theory is a mod 2 graded Hilbert space with a trace-class semigroup whose generator is given as the square of a self-adjoint operator of degree 1. Up to homotopy, this is the space of Fredholm operators Z × BU , i.e. the representing space for K-theory. When d = 2 we can not assume that the space of quantum field theories is homotopy equivalent to the space of topological theories. It nevertheless seems interesting to consider the space of 2-dimensional topological theories, and better, in the light of the discussion above, the space of “3-tier” unitary topological theories. The general definition of a 3-tier d-dimensional quantum field theory — of which Definition 2.5 is a specialization — is as a structure that assigns (i) (ii)

a linear category CZ to each closed (d − 2)-manifold Z, a functor FY : CZ0 → CZ1 to each (d − 1)-dimensional cobordism from Z0 to Z1 , and (iii) a transformation of functors UX : FY0 → FY1 to each d-dimensional cobordism X between cobordisms Y0 and Y1 from Z0 to Z1 . These data must satisfy natural conditions which I shall not spell out. (There is a discussion of the 3-dimensional case in Lecture 3 of [S4].) In Definition 2.5 the category associated to a point x is the category of modules for the von Neumann algebra Ax , and the functors are defined in the usual way by bimodules. Now in the topological 2-dimensional case we expect the whole structure to be determined by the category assigned to a point, so that a theory reduces to a semisimple C-linear category, i.e. a “module” over the

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category of finite dimensional vector spaces, or, in the language of [BDR] a “two vector space”. These objects define module spectra over the complex K-theory spectrum, and the K-theory of that ring-spectrum is the elliptic theory proposed in [BDR].

References [Ad ] Adams, J.F., Stable homotopy and generalised cohomology. Univ. of Chicago Press, 1974. [An ] Ando, M., The sigma orientation for analytic circle-equivariant elliptic cohomology. Geom. Topology 7(2003), 91-153. [BDR ] Baas, N., B. Dundas, and J. Rognes, Two vector spaces and forms of elliptic cohomology. Topology, Geometry, and Quantum Field Theory. Proc. Oxford 2002. Oxford Univ. Press 2004. [Dv1 ] Devoto, J.A., An algebraic description of the elliptic cohomology of classifying spaces. J. Pure Appl. Alg. 130(1998), 237-264. [Dv2 ] Devoto, J.A., Equivariant elliptic cohomology and finite groups. Michigan Math. J. 43(1996), 3-32. [GMS ] Gorbounov, V., F. Malikov, and V. Schechtman, Gerbes of chiral differential operators. Math. Res. Letters 7(2000), 55-66. AG/9906117. [G ] Grojnowski, I., A delocalized form of equivariant elliptic cohomology. Preprint available fron the author’s home page at www.maths.cam.ac.uk. [H ] Hopkins, M.J., Topological modular forms, the Witten genus, and the theorem of the cube. Proc. Internat. Cong. Mathematicians, Zurich 1994, 554-565. Birkh¨auser 1995. [HKR ] Hopkins, M., N. Kuhn, and D. Ravenel, Group characters and complex oriented cohomology theories. J. Amer. Math. Soc. 13(2000), 553-594. [L ] Landweber, P.S. (Ed.), Elliptic curves and modular forms in algebraic topology. Springer Lecture Notes in Mathematics 1326, 1988. [MSV ] Malikov, F., V. Schechtman, and A. Vaintrob, Chiral de Rham complex. Comm. Math. Phys. 204(1999), 439-473. AG/9803041. [PS ] Pressley, A., and G. Segal, Loop groups. Oxford Univ. Press, 1986. [S1 ] Segal, G., Categories and cohomology theories. Topology 13 (1974), 293-312. [S2 ] Segal, G., Elliptic cohomology. S´eminaire Bourbaki no.695, 1988. Ast´erisque 161-162 (1989),187-201. [S3 ] Segal, G., The definition of conformal field theory. Topology, Geometry, and Quantum Field Theory. Proc. Oxford 2002. Oxford Univ. Press 2004. [S4 ] Segal, G., Notes of lectures at Stanford, available from www.cgtp.duke.edu/ITP99/segal

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[ST ] Stolz, S., and P. Teichner, What is an elliptic object? Topology, Geometry, and Quantum Field Theory. Proc. Oxford 2002. Oxford Univ. Press 2004. [W ] Wassermann, A., Operator algebras and conformal field theory. Proc. Internat. Cong. Mathematicians, Zurich 1994, 966-979. Birkh¨auser 1995.

SPIN BORDISM, CONTACT STRUCTURE AND THE COHOMOLOGY OF p-GROUPS C.B. THOMAS

Among the interesting mathematical developments of the last twenty years has been the use of the topological technique of surgery in differential geometry. This has led, for example, to the establishment of necessary and sufficient conditions for the existence of a metric of positive scalar curvature on a simply-connected manifold, and to the construction of contact forms on certain classes of manifold satisfying the necessary tangential condition. There are also a few scattered results for metrics of positive Ricci curvature, together with an outline argument (due to S. Stolz) that a necessary condition for the existence of such a metric on a simply-connected spin manifold is the vanishing of the Witten genus. In part Stolz’ argument consists in transferring a problem about Ricci curvature on M to one of scalar curvature on the loop space LM, whose ‘K-Theory’ is elliptic cohomology. For a survey of what is known, and additional references, see [6]. The aim of the present paper is to formulate a programme for extending these results for simply-connected manifolds to manifolds having finite fundamental group. Leaving aside a few special cases this has already achieved for groups with periodic cohomology (p-rank = 1), and there is evidence that similar results hold when the p-rank of G = π1 M equals 2. For such groups the even-dimensional cohomology subring H even (G, Z) is generated by ‘transfered Euler classes’, and many of them satisfy the stronger condition Ch(G) = H even (G, Z). Given the results and conjectures already mentioned for positive scalar and positive Ricci curvature (using the Aˆ and the Witten genus respectively) our use of group cohomology suggests that for nonsimply-connected manifolds the proper setting may involve a G-equivariant genus taking values in KO ∗ (BG) or Ell∗ (BG). Note that for groups of low p-rank the Morava building blocks K(n)∗ (BG) for several cohomology theories can be described in terms of Chern or transfered Euler classes of complex representations, see [10] and [8]. The links between contact geometry and ‘bundle-like’ cohomology theories are more tenuous. As J. Devoto explains elsewhere in these proceedings 3dimensional contact manifolds appear to play a role in the construction of

Date: 7 May 2003.

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K3-objects in the v4 -periodic theories generalising elliptic cohomology. Another clue may be provided by the relationship between contact automorphism groups of regular contact manifolds fibering over a K3-surface and the Mathieu simple group M23 . We explain a little of this below. The results of this paper are stated for 5-dimensional contact manifolds. There are easily formulated analogues for 5-manifolds with positive scalar curvature, and in contrast to the contact case there is no geometric obstruction to extending these results to higher dimensions. However here more delicate calculations in cohomology may be needed, especially if part of H even does not come from transfered Euler classes. This work is part of an on-going collaboration with Hansj¨org Geiges. 1. Basic Definitions As usual we write Ωspin n (X) for the bordism group of n-dimensional closed spin manifolds together with maps into the space X. For small values of n the coefficients are as follows: 1 2 Z/2 Z/2 S 1 (S 1 )2

3 0 —

4 Z K3

5,6,7 8 L 0 Z Z — HP2 Bott

More can be said: there is a natural transformation Ωspin n (X) → kon (X) which is a 2-local isomorphism in dimensions 6 7. We will be concerned with the case X = BG, the classifying space for the finite group G (usually of order pt ). We also recall that a contact form α on M 2n−1 is a 1-form such that α∧ (dx)n 6= 0. Such forms arise naturally in theoretical physics, being defined for example on (suitable) constant energy levels in Hamiltonian systems. The form α, or more generally the scalar multiple λα, is associated with a codimension one distribution in TM of symplectic type, called a contact structure. Contact structures are examples of surgery-compatible geometric structures, others being metrics of positive scalar curvature and perhaps positive Ricci curvature. We restrict attention to dimension 5 because: (a) Ωspin 5 (BG) is open to calculation, partly because of the relation with connective K-theory, and (b) we can sidestep framing problems in contact surgery. Technically this amounts to requiring the first Chern class c1 to evaluate as zero on 2-spheres. Contact surgery along any 1-sphere is then possible for any topological framing, and for 2-surgeries we have no choice since π2 (SO3 ) vanishes. There is some hope that the argument may extend to dimension 7, but in higher dimensions more restrictions may be needed.

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Note that the spin assumption (w2 = 0) implies that the tangential structural group of an oriented 5-manifold reduces from SO5 to U2 ⊕ 1 (almost contact condition). Given the possibility of doing contact surgery the problem of integrating some almost contact structure on M 5 reduces to finding a spin-bordant N 5 ∈ Cont5 (G) ⊆ Ωspin 5 (BG). Here Cont5 (G) = set of classes with representatives of the form (f : N → BG, σ), where N admits a contact structure ξ defining the orientation given by the spin structure σ and c1 (ξ) = 0. Definition . G is said to be a contact group if Cont 5 (G) = Ωspin 5 (BG). If M ∼ (N, α) with c1 (ξ) a torsion class, G is a weak contact group. One justification for restricting attention to the class of p-groups is that Cont5 (·) inherits what one can call homological Frobenius reciprocity from equivariant spin bordism. It is this reciprocity which reduces the determination of Cont5 (G) from the group itself to a representative family of Sylow subgroups. Since this use of subgroups to detect the existence of geometric structure also applies to certain classes of p-group it is worthwhile recalling the definitions needed here. Let h : G → G0 be a homomorphism. There is an induced homomorphism spin 0 (Bh)∗ : Ωspin 5 (BG) → Ω5 (BG ), (f : V → BG, σ) 7→ ((Bh)f : V → BG0 , σ),

and, if h is an inclusion, a transfer homomorphism in the reverse direction spin 0 (Bh)! : Ωspin 5 (BG → Ω5 (BG).

Here σ stands for spin structure. To define transfer note that, given (f 0 : 0 V → BG0 , σ 0 ) in Ωspin 5 (BG ), we have a pull-back diagram f

V −−−→ BG y yBh f0

V 0 −−−→ BG0 .

Lift the spin structure σ 0 on V 0 to σ on V , via the covering V → V 0 , and then define (Bh)! (f 0 : V 0 → BG0 , σ 0 ) = (f : V → BG, σ). It is easy to see that both Bh∗ and Bh! map Cont5 to Cont5 , although care must be taken with both their compositions. For G0 the map (Bh)∗ (Bh)! can be studied by means of its analogue in ordinary cohomology; for G the map (Bh)! (Bh)∗ is a little more transparent, at least when G is normal in G0 . Reciprocity properties of this kind are discussed at some length, but with reference to the calculation of bordism groups, rather than to contact or curvature questions by P. Conner and E. Floyd in [2].

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2. Known Results t Proposition 1. The spin bordism group Ωspin 5 (BZ/p )(p = odd) is gener5 t 5 t ated by L (p ; 2, 1, 1) and L (p ; 2, 2, 2) with their canonical spin and Z/pt structures. In particular (for p > 5) Ωspin (BZ/q) ∼ = Z/q ⊕ Z/q. 5

The prime p = 3 is anomalous in that Ω5 ∼ = Z/9. Hence Z/pt is certainly a weak contact group. Calculation with Pontrjagin numbers shows that Proposition 2. If p > 5 or G ∼ = Z/3 (i.e. t = 1), Z/pt is a contact group. As one might expect the prime 2 is more complicated/interesting. Proposition 3. Ωspin (Z/2) = 0. For k > 2, Ωspin (Z/2k ) ∼ = Z/2k ⊕ Z/2k−2 , 5

5

generated by fibre spaces of the form L3 −→ N 5 −→ S 2 . π

Corollary . Z/4 is a weak contact group. (Ω5 is generated by L3 (4, 1, 1) → N 5 → S 2 , which can be shown to be contact with c1 a torsion class.) More tantalising is the result for the quaternion group Q2k : ∼ Proposition 4. Ωspin 5 (BQ2k ) = Z/2 ⊕ Z/2 (k > 3). One of the Z/2-factors is generated by the map f : (S 3 /Q2k ) × T 2 → BQ2k , where on the first factor f classifies the standard linear action and on the second factor f is trivial. (The product admits a contact form, by a branched cover construction.) For these results see [3]. What is suggestive about the proof of Proposition 4 (using the AtiyahHirzebrach spectral sequence for bordism) is that the factors correspond to 2 2 E4,1 and E3,2 . Looking at the low-dimensional spin bordism generators listed on page 3 this suggests that we consider a K3-surface with its natural symplectic form and the contact manifold V 5 obtained by the ‘regular contact construction’ over it. If X is a K3-surface then X admits a nowhere vanishing holomorphic 2form ω. The group G of automorphisms of X which preserve ω is the group of (complex) symplectic automorphisms of X. This group is finite and contained in the Mathieu group M23 , see [5, table on page 184]. Here are some explicit examples: • For the standard quartic x4 + y 4 + z 4 + t4 = 0 in CP 3 G is isomorphic to the extension (42 ) o S4 of order 384. • For the surface in CP 4 defined by the pair of equations x31 + · · · + x34 = 0 x1 x2 + x3 x4 + x25 = 0

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G is a semi-direct product of (32 ) and D8 . • There is a double cover of CP 2 with branch defined by x6 + y 6 + z 6 − 10(x3 y 3 + y 3 z 3 + z 3 x3 ) = 0 and with group isomorphic to the small solvable Mathieu group M9 ∼ = (32 ) · Q8 . The form ω can be written as the sum of the real symplectic forms ω1 +iω2 , and hence using the regular hypercontact/contact construction we obtain a hypercontact form on an S 3 -bundle over X 4 , and (in two ways) a contact form on the total space N 5 of an S 1 -fibration over X 4 . Furthermore, using a construction of A. Banyaga [1] symplectic automorphisms of X 4 lift to strict contact automorphisms of N 5 . The explicit examples given above strongly suggest that, at least for small values if k, a contact generator for the second 2 factor Z/2 ∼ in Ωspin = E4,1 5 (BQ2k ) can be constructed by this method. Full details of the groups and manifolds arising from Mukai’s examples will appear elsewhere [9]. 3. Cohomology of p-Groups, p > 5. At least in principle Ωspin 5 (BG) can be calculated from the terms of total degree 5 on the E 2 -page of the Atiyah-Hirzebruch spectral sequence Hr· (G, Ωspin s (pt)) ⇒ spin Ωr+s (BG). Since |G| is odd, universal cycles are carried by subquotients of H5 (G, Z) and H1 (G, Z) = Gab . [In the case of cyclic groups both these summands are cyclic of order pt , partly explaining Proposition 1.] For finite groups G it is easier to look at cohomology rather than homology, and we have H 6 (G, Z) ∼ = Hom(H6 , Z) ⊕ Ext(H5 , Z), with the first summand equal to zero. The subring H even (G, Z) is finitely generated as a module over the possibly smaller subring Ch(G) generated by Chern classes of irreducible complex representations. For the sake of clarity we will consider this ring rather than the larger and in general more useful one generated by transfered Euler classes of representations of proper subgroups. From [7] we recall the following: Let H(1) (G, Z) be the subring of H even (G, Z) generated by elements of the form {h∗ xj for x ∈ H 2 and some inclusion G1 → G}. Then Proposition 5. If G is a p-group and k < p − 1, H(1) (G, Z) ∩ H 2k (G, Z) = Chk (G). The proof uses Blichfeldt’s theorem on the representations of supersolvable groups together with the ‘Riemann-Roch’ formula for the Newton polynomials in the Chern classes of an induced representation. Dualising from cohomology to spin bordism this allows us to construct contact generators in Ωspin 5 (BG) spin by a transfer process applied to contact generators in Ω5 (BG1 ). The programme now is to construct contact generators for subquotients of the two

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2 2 homology groups H5 (G, Z) ∼ and H1 (G, Z) ∼ . For the first, at = E5,0 = E1,4 6 3 least when H (G, Z) = Ch (G), we use transfer from proper subgroups; for the second a variant of the construction of a regular contact manifold over a K3-surface. If G does not belong to the restricted class of subgroups of M23 considered there, we will need to look for a symplectic manifold bordant to K3 admitting G as a group of automorphisms. Even for groups such that Ch(G) is properly contained in H even , the first part of the argument may apply, since we are only looking at one low dimension. One hint that Chern classes may suffice is provided by the relation between spin bordism and connective K-theory quoted in section one.

An example will bring us down from speculation to rigorous calculation. 2

Using the ‘Atlas’ notation: p1+2 = {a, b : bp = ap = 1, aba−1 = b1+p }, i.e. − 1+2 p− is metacyclic with normal subgroup of index p generated by b. H ∗ (p1+2 − , Z) = Z[α, χ, ζ, β1 . . . βp−1 ], deg α = 2, deg βi = 2i, deg ζ = 2p, deg χ = 2p + 1 with relations p2 ζ = pα = pχ = pβi = 0, χ2 = 0, βi α = βi χ = βi βj = 0 (all i, j). If β ∈ H 2 (< b >, Z) generates the cohomology of the normal subgroup , < b >, then we may take βi = corestriction(β i ). For the details see [4]. 1+2 It follows from the last sentence that H even (p1+2 − , Z) = Ch(p− ).

Let us interpret these calculations for spin bordism, noting first that because of our assumption that p > 5, odd dimensional cohomology arises at 3 worst in dimension 11. H 2 ∼ = Z/αp ⊕ Z/βp 3 . Here = Z/αp ⊕ Z/βp 1 and H 6 ∼ β1 = cor(β), β3 = cor(β 3 ). The relations imply that the monomials α2 β1 , αβ12 , β13 , αβ2 and β1 β2 all vanish. Next recall that for the subgroups < a > and < b > we have already proved that Ωspin is the sum of two copies of a cyclic group. 5 For the subgroup < b > Ω5 ∼ = 2(Z/p2 ), both factors being generated by lens spaces having c1 (ξ) = 0. The p1+2 − action on the transfered 5-sphere can be described in a similar way to an induced representation module. The generator b acts on the disjoint union of p copies of S 5 by (some conjugate of) the original action, while the generator a permutes the factors. This non-connected manifold admits the standard contact form on each sphere. By naturality and another application of the Riemann-Roch formula, cor(c1 (ξ)) = c1 (i∗ ξ) = 0. For the non-normal subgroup generated by a two arguments are available. Transfer can again be used, although the non-normality of the subgroup makes it less transparent. On the other hand we can note that < a > is a retract of p1+2 − , so that the naturality of Cont 5 implies that Cont5 (B < a >),

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as with ordinary cohomology, splits off as a direct summand. Motivation for the transfer argument was provided to the author by [11]. Proposition 6. If p > 5 the metacyclic group p1+2 is a contact group. − A similar argument applies to other split metacyclic p-groups. More interesting is the group p1+2 of exponent p, the finite Heisenberg group. Its even + cohomology, again see [4], is generated by Chern classes. However there are • odd dimensional generators of degree 3 (in contrast to 2p + 1), and • elements which restrict to zero on every proper subgroup. Along with other minimal non-abelian groups the Heisenberg group provides the first test for the validity of the general argument sketched above. References [1] A. Banyaga, Sur le groupe de diff´eomorphismes qui pr´eservent une forme de contact r´eguli`ere, CR Acad. Sci. Pari.; 281 (1975) A707–A709. [2] P.E. Conner, E.E. Floyd, Differentiable Periodic Maps, Ergebruisse der Mathematik 33 (1964) (Springer, Heidelberg). [3] H. Geiges, C.B. Thomas, Contact structures, equivariant spin bordism, and periodic fundamental groups, Math. Annalen 320 (2001) 685–670. [4] G. Lewis, The integral cohomology rings of groups of order p 3 , Trans. Amer. Math. Soc. 132 (1968) 501–529. [5] S. Mukai, Finite groups of automorphisms of K3-surfaces and the Mathieu group, Invent. Math. 94 (1988) 183–221. [6] J. Rosenberg, Surgery Theory Today - what it is and where it’s going, Annals of Math. Studies 149 (Princeton University Press 2001) 3–47. [7] C.B. Thomas, An integral Riemann - Roch formula for flat line bundles, Proc. London Math. Soc. (3) (1977) 87–101. [8] C.B. Thomas, Morava K-theory for groups of small p-rank, (in preparation). [9] C.B. Thomas, Automorphisms of hypercontact manifolds fibering over K3-surfaces, (in preparation). [10] N. Yagita, Cohomology for Groups of rank G = 2 and Brown-Peterson Cohomology, J. Math. Soc. Japan 45 (1993) 627–644. [11] N. Yagita, Complex cobordism of the dihedral group, J. Math. Soc. Japan 48 (1996) 195–203.

University of Cambridge May 2003 [email protected]

BRAVE NEW ALGEBRAIC GEOMETRY AND GLOBAL DERIVED MODULI SPACES OF RING SPECTRA ¨ AND GABRIELLE VEZZOSI BERTRAND TOEN Abstract. We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, brave new algebraic geometry. We discuss various model topologies on the model category of commutative algebras in S, and their associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry). Two examples of geometric S-stacks are given: a global moduli space of associative ring spectrum structures, and the stack of elliptic curves endowed with the sheaf of topological modular forms.

Key words: Sheaves, stacks, ring spectra, elliptic cohomology. MSC-class: 55P43; 14A20; 18G55; 55U40; 18F10. Contents 1. Introduction 2. Brave new sites 2.1. The brave new Zariski topology 2.2. The brave new ´etale topology 2.3. Standard topologies 3. S-stacks and geometric S-stacks 3.1. Some descent theory 3.2. The S-stack of perfect modules 3.3. Geometric S-stacks 4. Some examples of geometric S-stacks 4.1. The brave new group scheme RAut(M) 4.2. Moduli of algebra structures 4.3. Topological modular forms and geometric S-stacks References

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Homotopical Algebraic Geometry is a kind of algebraic geometry where the affine objects are given by commutative ring-like objects in some homotopy theory (technically speaking, in a symmetric monoidal model category); these affine objects are then glued together according to an appropriate homotopical modification of a Grothendieck topology (a model topology, see [To-Ve 1, 4.3]). More generally, we allow ourselves to consider more flexible objects like stacks, in order to deal with appropriate moduli problems. This theory is developed in full generality in [To-Ve 1, To-Ve 2] (see also [To-Ve 3]). Our motivations for such a theory came from a variety of sources: first of all, on the algebro-geometric side, we wanted to produce a sufficiently functorial language in which the so called Derived Moduli Spaces foreseen by Deligne, Drinfel’d and Kontsevich could really be constructed; secondly, on the topological side, we thought that maybe the many recent results in Brave New Algebra, i.e. in (commutative) algebra over structured ring spectra (in any one of their brave new symmetric monoidal model categories, see e.g. [Ho-Sh-Sm, EKMM]), could be pushed to a kind of Brave New Algebraic Geometry in which one could take advantage of the possibility of gluing these brave new rings together into an actual geometric object, much in the same way as commutative algebra is helped (and generalized) by the existence of algebraic geometry. Thirdly, on the motivic side, following a suggestion of Y. Manin, we wished to have a sufficiently general theory in order to study algebraic geometry over the recent model categories of motives for smooth schemes over a field ([Hu, Ja, Sp]). The purpose of this paper is to present the first steps in the second type of applications mentioned above, i.e. a specialization of the general framework of homotopical algebraic geometry to the context of stable homotopy theory. Our category S − Aff of brave new affine objects will therefore be defined as the the opposite model category of the category of commutative rings in the category S of symmetric spectra ([Ho-Sh-Sm]). We first define and study various model topologies defined on S − Aff. They are all extensions, to different extents, of the usual Grothendieck topologies defined on the category of (affine) schemes, like the Zariski and ´etale ones. With any of these model topologies τ at our disposal, we define and give the basic properties of the corresponding model category of S-stacks, understood in the broadest sense as not necessarily truncated presheaves of simplicial sets on S − Aff satisfying a homotopical descent (i.e. sheaf-like) condition with respect to τ -(hyper)covers. A model topology on S − Aff is said to be subcanonical if the representable simplicial presheaves, i.e. those of the form Map(A, −), for some commutative ring A in S, Map being the mapping space in S − Aff op , are S-stacks.

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As in algebraic geometry one finds it useful to study those stacks defined by smooth groupoids (these are called Artin algebraic stacks), we also define a brave new analog of these and call them geometric S-stacks, to emphasize that such S-stacks host a rich geometry very close to the geometric intuition learned in algebraic geometry. In particular, given a geometric S-stack F , it makes sense to speak about quasi-coherent and perfect modules over F , about the K-theory of F , etc.; various properties of morphisms (e.g. smooth, ´etale, proper, etc.) between geometric S-stacks can likewise be defined. Stacks were introduced in algebraic geometry mainly to study moduli problems of various sorts; they provide actual geometric objects (rather than sets of isomorphisms classes or coarse moduli schemes) which store all the fine details of the classification problem and on which a geometry very similar to that of algebraic varieties or schemes can be developed, the two aspects having a fruitful interplay. In a similar vein, in our brave new context, we give one example of a moduli problem arising in algebraic topology (the classification of A∞ -ring spectrum structures on a given spectrum M) that can be studied geometrically through the geometric S-stack RAssM it represents. We wish to emphasize that instead of a discrete homotopy type (like the ones studied, for different moduli problems, in [Re, B-D-G, G-H]), we get a full geometric object on which a lot of interesting geometry can be performed. The geometricity of the S-stack RAssM , with respect to any fixed subcanonical model topology, is actually the main theorem of this paper (see Theorem 4.2.1). We also wish to remark that the approach presented in this paper can be extended to other, more interested and involved, moduli problems algebraic topologists are interested in, and perhaps this richer geometry could be of some help in answering, or at least in formulating in a clearer way, some of the deep questions raised by the recent progress in stable homotopy theory (see [G]). In this direction, we will explain in §4.3 how topological modular forms give rise to a natural geometric S-stack which is an extension in the brave new direction of the moduli stack of elliptic curves (see Theorem 4.3.1). This fact seems to us a very important remark (probably much more interesting than our Theorem 4.2.1), and we think it could be the starting point of a very interesting research program. We also present a brave new analog of the stack of vector bundles on a scheme, called the S-(pre)stack Perf of perfect modules (Section 3.2), and we expect it to be a key tool in brave new algebraic geometry. The prestack Perf is a stack if and only if the model topology we are working with is subcanonical (Thm. 3.2.1 whose proof is postponed to [To-Ve 2]). This is another instance of the relevance of the descent problem, i.e. the question whether a given model topology is subcanonical or not (see Section 3.1). Though we prove that some of the model topologies we introduce (namely the standard and the semi-standard ones, Section 2.3) are subcanonical, at present we are not

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able to settle (nor in the positive nor in the negative) the descent problem for the three most promising model topologies we define, namely the Zariski, ´etale and thh-´etale ones. Though this is at the moment quite unsatisfactory, we believe that the descent problem for these topologies is a very interesting question in itself even leaving outside its crucial role in brave new algebraic geometry. For the Zariski model topology, we have a partial positive result in this direction. By definition, for a model topology τ the property of being subcanonical depends on the notion of stacks we consider; if instead of defining a stack as a prestack (i.e. a simplicial presheaf) satisfying homotopical descent with respect to all homotopy τ -hypercovers, we simply require descent with ˇ respect to all Cech τ -hypercovers (i.e. those arising as nerves of τ -covers), ˇ we obtain a notion of Cech stacks, recently considered by J. Lurie ([Lu]) and Dugger-Hollander-Isaksen ([DHI]). We prove (Corollary 3.1.4) that the Zariski model topology is in fact subcanonical with respect to the notion of ˇ Cech stacks. Moreover, by replacing in Theorem 4.2.1 the word “stack” with ˇ the weaker “Cech stack”, the statement remains true for any model topology ˇ which is subcanonical with respect to the notion of Cech stacks. It is therefore natural to ask why we did not choose to formulate everyˇ thing only in terms of Cech stacks. We believe that at this early stage of development of homotopical algebraic geometry and, in particular, of brave new algebraic geometry, it is not advisable to make choices that could prevent some applications or obscure some of the properties of the objects involved, while it is more useful to keep in mind various options, some of which can be more useful in one context than in others. For example, it is clear that knowing that a given, geometrically meaningful, simplicial presheaf is a stack ˇ and not only a Cech stack adds a lot more informations, in fact exactly the descent property with respect to unbounded hypercovers ([DHI, Thm. A.6]). ˇ Moreover, Cech stacks fail in general to satisfy an analog of Whitehead theˇ orem: a pointed Cech stack may have vanishing πi sheaves for any i ≥ 0 without being necessarily contractible. This last fact is a very inconvenient ˇ property of Cech stacks, that makes Postnikov decompositions and spectral ˇ sequences arguments uncertain. On the other hand, the Cech descent condition is usually much easier to establish than the full descent condition, and as we have already remarked, some natural model topologies are easily seen ˇ to be subcanonical with respect to the notion of Cech stacks while it might be tricky to show that they are actually subcanonical. Finally, we would like to mention that in our experience we have never met serious troubles by using one or the other of the two notions, and in many interesting contexts it does not really matter which notion one uses, as the rather subtle differences actually tend not to appear in practice.

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Acknowledgments. We wish to thank the organizers of the INI Workshop on Elliptic Cohomology and Higher Chromatic Phenomena (Cambridge UK, December 2003) for the invitation to speak about our work and Bill Dwyer for his encouraging comments. We would also like to thank Michael Mandell, Peter May, Haynes Miller, John Rognes, Stefan Schwede and Neil Strickland for helpful discussions and suggestions. Notations. To fix ideas, we will work in the category S := Sp Σ of symmetric spectra (see [Ho-Sh-Sm]), but all the constructions of this paper will also work, possibly with minor variations (see [Sch]), for other equivalent theories (e.g for the category of S-modules of [EKMM]). We will consider S as a symmetric monoidal simplicial model category (for the smash product − ∧ −) with the Shipley-Smith positive S-model structure (see [Shi, Prop. 3.1]). We define S − Alg as the category of (associative and unital) commutative monoids objects in S, endowed with the S-model structure of [Shi, Thm. 3.2]; we will simply call them commutative S-algebras instead of the more correct but longer, commutative symmetric ring spectra. For any commutative Salgebra A, we will denote by A − Alg the under-category A/S − Alg, whose objects will be called commutative A-algebras. Finally, if A is a commutative S-algebra, A − Mod will be the category of A-modules with the A-model structure ([Shi, Prop. 3.1]). This model category is also a symmetric monoidal model category for the smash product − ∧A − over A. For a morphism of commutative S-algebras, f : A −→ B one has a Quillen adjunction f ∗ : A − Mod −→ B − Mod

A − Mod ←− B − Mod : f∗ ,

where f ∗ (−) := − ∧A B is the base change functor. We will denote by Lf ∗ : Ho(A−Mod) −→ Ho(B−Mod)

Ho(A−Mod) ←− Ho(B−Mod) : Rf∗

the induced derived adjunction on the homotopy categories. Our references for model category theory are [Hi, Ho]. For a model category M with equivalences W , the set of morphisms in the homotopy category Ho(M) := W −1 M will be denoted by [−, −]M , or simply by [−, −] if the context is clear. The (homotopy) mapping spaces in M will be denotedby MapM (−, −). When M is a simplicial model category, the simplicial Hom’s (resp. derived simplicial Hom’s) will be denoted by Hom M (resp. RHomM ), or simply by Hom (resp. RHom) if the context is clear. Recall that in this case one can compute MapM (−, −) as RHomM (−, −). Finally, for a model category M and an object x ∈ M we will often use the coma model categories x/M and M/x. When the model category M is not left proper (resp. is not right proper) we will always assume that x has been replaced by a cofibrant (resp. fibrant) model before considering x/M (resp. M/x). More generally, we will not always mention fibrant and cofibrant replacements and suppose implicitly that all our objects are fibrant

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and/or cofibrant when required. Since we wish to concentrate on applications to stable homotopy theory, some general constructions and details about homotopical algebraic geometry will be omitted by referring to [To-Ve 1]. For a few of the results presented we will only give here sketchy proofs; full proofs will appear in [To-Ve 2].

2. Brave new sites In this section we present two model topologies on the (opposite) category of commutative S-algebras. They are brave new analogs of the Zariski and ´etale topologies defined on the category of usual commutative rings and will allow us to define the brave new Zariski and ´etale sites. We denote by S − Aff the opposite model category of S − Alg. If M is a model category we say that an object x in M is finitely presented if, for any filtered direct system of objects {zi }i∈J in M, the natural map colimi MapM (x, zi ) −→ MapM (x, colimi zi ) is an isomorphism in the homotopy category of simplicial sets. Definition 2.0.1. A morphism A → B of commutative S-algebras is finitely presented if it is a finitely presented object in the model under-category A/(S − Alg) = A − Alg; in this case, we also say that B is a finitely presented commutative A-algebra. An A-module E is finitely presented or perfect if it is a finitely presented object in the model category A − Mod. Perfect A-modules can also be characterized as retracts of finite cell Amodules (see [EKMM, Thm. III-7.9]); in particular, there are plenty of them. If A is a commutative S-algebra, then the free commutative A-algebra on a finite number of generators (or, more generally, on any perfect A-module) is a finitely presented A-algebra. The reader will find other examples of finitely presented morphisms of commutative S-algebras in Lemma 2.1.6. 2.1. The brave new Zariski topology. Definition 2.1.1. • A morphism f : A −→ B in S − Alg is called a formal Zariski open immersion if the induced functor Rf∗ : Ho(B − Mod) −→ Ho(A − Mod) is fully faithful. • A morphism f : A −→ B is a Zariski open immersion if S − Alg is it is a formal Zariski open immersion and of finite presentation (as a morphism of commutative S-algebras).

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• A family {fi : A −→ Ai }i∈I of morphisms in S − Alg is called a (formal) Zariski open covering if it satisfies the following two conditions. – Each morphism A −→ Ai is a (formal) Zariski open immersion. – There exist a finite subset J ⊂ I such that the family of inverse image functors {Lfj∗ : Ho(A − Mod) −→ Ho(Aj − Mod)}j∈J is conservative (i.e. a morphism in Ho(A − Mod) is an isomorphism if and only if its images by all the Lfj∗ ’s are isomorphisms). Example 2.1.2. If A ∈ S − Alg and E is an A-module such that the associated Bousfield localization LE is smashing (i.e. the natural transformation LE (−) → LE A ∧LA (−) is an isomorphism), then A → LE A (which is a morphism of commutative S-algebras by e.g. [EKMM, §VIII.2]) is a formal Zariski open immersion. This follows immdiately from the fact that Ho(LE A − Mod) is equivalent to the subcategory of Ho(A − Mod) consisting of LE -local objects, by [Wo]. It is easy to check that (formal) Zariski open covering families define a model topology in the sense of [To-Ve 1, §4.3] on the model category S − Aff. For the reader’s convenience we recall what this means in the following lemma. Lemma 2.1.3. • If A −→ B is an equivalence of commutative S-algebras then the one element family {A −→ B} is a (formal) Zariski open covering. • Let {A −→ Ai }i∈I be a (formal) Zariski open covering of S-algebras and A −→ B a morphism. Then, the family of homotopy push-outs {B −→ B ∧LA Ai }i∈I is also a (formal) Zariski open covering. • Let {A −→ Ai }i∈I be a (formal) Zariski open covering of S-algebras, and for any i ∈ I let {Ai −→ Aij }j∈Ji be a (formal) Zariski open covering of S-algebras. Then, the total family {A −→ Aij }i∈I,j∈Ji is again a (formal) Zariski open covering. Proof: Left as an exercise.

2

By definition, Lemma 2.1.3 shows that (formal) Zariski open coverings define a model topology on the model category S − Aff and so, as proved in [To-Ve 1, Prop. 4.3.5], induce a Grothendieck topology on the homotopy category Ho(S − Alg). This model topology is called the brave new (formal) Zariski topology, and endows S − Aff with the structure of a model site in the sense of [To-Ve 1, §4]. This model site, denoted by (S − Aff, Zar) for the brave new Zariski topology, and (S − Aff, fZar) for the brave new formal Zariski topology. They will be called the brave new Zariski site and the brave new formal Zariski site.

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Let Alg be the category of (associative and unital) commutative rings. Let us recall the existence of the Eilenberg-Mac Lane functor H : Alg −→ S − Alg, sending a commutative ring R to the commutative S-algebra HR such that π0 (HR) = R and πi (HR) = 0 for any i 6= 0. This functor is homotopically fully faithful and the following lemma shows that our brave new Zariski topology does generalize the usual Zariski topology. Lemma 2.1.4. (1) Let R −→ R0 be a morphism of commutative rings. The induced morphism HR −→ HR0 is a Zariski open immersion of commutative S-algebras (in the sense of Definition 2.1.1) if and only if the morphism Spec R0 −→ Spec R is an open immersion of schemes. (2) A family of morphisms of commutative rings, {R −→ Ri0 }i∈I , induces a Zariski covering family of commutative S-algebras {HR −→ HRi0 }i∈I (in the sense of Definition 2.1.1) if and only if the family {Spec Ri −→ Spec R}i∈I is a Zariski open covering of schemes. Proof: Let us start with the general situation of a morphism f : A −→ B of commutative S-algebras such that the induced functor Rf∗ : Ho(B − Mod) −→ Ho(A − Mod) is fully faithful. Let L = Rf∗ ◦ Lf ∗ , which comes with a natural transformation Id −→ L. Then, the essential image of Rf∗ consist of objects M in Ho(A − Mod) such that the localization morphism M −→ LM is an isomorphism. The Quillen adjunction (f ∗ , f∗ ) extends to a Quillen adjunction on the category of commutative algebras f ∗ : A − Alg −→ B − Alg

A − Alg ←− B − Alg : f∗ ,

also with the property that Rf∗ : Ho(B − Alg) −→ Ho(A − Alg) is fully faithful. Furthermore, the essential image of this last functor consist of all objects C ∈ Ho(A − Alg) such that the underlying A-module of C satisfies C ' LC (i.e. the underlying A-module of C lives in the image of Ho(B − Mod)). ¿From these observations, we deduce that for any commutative A-algebra C, the mapping space RHomA−Alg (B, C) is either empty or contractible; it is non-empty if and only if the underlying A-module of C belongs to the essential image of Rf∗ . To prove (1), let us first suppose that f : Spec R0 −→ Spec R is an open immersion of schemes. The induced functor on the derived categories f∗ : D(R0 ) −→ D(R) is then fully faithful. As there are natural equivalences ([EKMM, IV Thm. 2.4]) Ho(HR − Mod) ' D(R)

Ho(HR0 − Mod) ' D(R0 )

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this implies that the functor Rf∗ : Ho(HR0 − Mod) −→ Ho(HR − Mod) is also fully faithful. It only remains to show that HR −→ HR0 is finitely presented in the sense of Definition 2.0.1. We will first assume that R0 = Rf for some element f ∈ R. The essential image of Rf∗ : Ho(HR0 − Mod) −→ Ho(HR − Mod) then consists of all objects E ∈ Ho(HR − Mod) ' D(R) such that f acts by isomorphisms on the cohomology R-module H ∗ (E). By what we have seen at the beginning of the proof, this implies that for any commutative HR-algebra C the mapping space RHomHR−Alg (HR0 , C) is contractible if f becomes invertible in π0 (C), and empty otherwise. ¿From this one easily deduces that RHomHR−Alg (HR0 , −) commutes with filtered colimits, or in other words that HR0 is a finitely presented HR-algebra in the sense of Definition 2.0.1. In the general case, one can write Spec R0 as a finite union of schemes of the form Spec Rf for some elements f ∈ R. A bit of descent theory (see §3.1) then allows us to reduce to the case where R0 = Rf and conclude. Let us now assume that HR −→ HR0 is a Zariski open immersion of commutative S-algebras. By adjunction (between H and π0 restricted on connective S-algebras) one sees easily that R −→ R 0 is a finitely presented morphism of commutative rings. The induced functor on (unbounded) derived categories f∗ : D(R0 ) ' Ho(HR0 − Mod) −→ D(R) ' Ho(HR − Mod) is fully faithful. Through the Dold-Kan correspondence, this implies that the Quillen adjunction on the model category of simplicial modules (see [G-J]) f ∗ : sR − Mod −→ sR0 − Mod

sR − Mod ←− sR0 − Mod : f∗

is such that Lf ∗ ◦ f∗ ' Id. Let sR − Alg and sR0 − Alg be the categories of simplicial commutative R-algebras and simplicial commutative R0 -algebras, endowed with their natural model structures (equivalences are and fibration are detected in the category of simplicial modules). Then, the Quillen adjunction f∗ : sR0 − Alg −→ sR − Alg

sR0 − Alg ←− sR − Alg : f ∗

also satisfies Lf ∗ ◦ f∗ ' Id, as this is true on the level on simplicial modules. In particular, for any simplicial R0 -module M, the space of derived derivations LDerR (R0 , M) := RHomsR−Alg/R0 (R0 , R0 ⊕M) ' RHomsR0 −Alg/R0 (R0 , R0 ⊕M) ' ∗ is acyclic (here R0 ⊕ M is the simplicial R0 -algebra which is the trivial extension of R0 by M). As a consequence one sees that Quillen’s cotangent complex LR0 /R is acyclic, which implies that the morphism R −→ R0 is an ´etale morphism of rings. Finally, using the fact that the functor on the category of modules R 0 − Mod −→ R − Mod is fully faithful, one sees that Spec R0 −→ Spec R is a monomorphism of schemes. Therefore, the morphism of schemes Spec R 0 −→

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Spec R is an ´etale monomorphism, and so is an open immersion by [EGA-IV, Thm. 17.9.1]. Finally, point (2) is clear if one knows (1) and that Ho(HR − Mod) ' D(R). 2 Remark 2.1.5. The argument at the beginning of the proof of Lemma 2.1.4 shows that if f : A → B is a Zariski open immersion, the functor L(f ) := Rf∗ Lf ∗ is a localization functor on the homotopy category of A-modules in the sense of [HPS, Def. 3.1.1]. And it is also clear by definition that L(f ) is also smashing ([HPS, Def. 3.3.2]). Let us call a localization functor L on Ho(A − Mod) a formal Zariski localization functor over A if L ' L(f ) for some formal Zariski open immersion f . Let us also say that a localization functor L on Ho(A−Mod) is a smashing algebra Bousfield localization over A if L ' LB for some A-algebra B such that LB is smashing (over A). Then it is easy to verify that in the set of equivalence classes of localization functors on Ho(A − Mod), the subset consisting of formal Zariski localization functors over A coincides with the subset consisting of smashing algebra Bousfield localizations over A. In fact, if f : A → B is a Zariski open immersion, LB denotes the Bousfield localization with respect to the A-module B, and `B/A : A → LB A the corresponding morphism of commutative A-algebras, we have L(f ) ' LB ' L(`B/A ) because all three localizations have the same category of acyclics. Conversely, if LC is a smashing algebra Bousfield localization over A, and `C/A : A → LC A is the corresponding morphism of commutative A-algebras, one has LB ' L(`C/A ). Let Aff be the opposite category of commutative rings, and (Aff, Zar) the big Zariski site. The site (Aff, Zar) can also be considered as a model site (for the trivial model structure on Aff ). Lemma 2.1.4 implies in particular that the functor H : Aff −→ S − Aff induces a continuous morphism of model sites ([To-Ve 1, Def. 4.8.4]). In this way, the site (Aff, Zar) becomes a sub-model site of (S − Aff, Zar). To finish with the Zariski topology we will now describe a general procedure in order to construct interesting open Zariski immersions of commutative S-algebras using the techniques of Bousfield localization for model categories. Let A be a commutative S-algebra and M be a A-module. We will assume that M is a perfect A-module (in the sense of Definition 2.0.1), or equivalently that it is a strongly dualizable object in the monoidal category Ho(A−Mod). As already noticed, perfect A-modules are exactly the retracts of finite cell A-modules, see [EKMM, Thm. III-7.9]). Let M[n] = S n ⊗L M be the n-th suspension A-module of M, for n ∈ Z.

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We denote by D(M[n]) the derived dual of M[n], defined as the derived internal Hom’s of A-modules D(M[n]) := RHOMA−Mod (M[n], A). Consider now the (derived) free commutative A-algebra over D(M[n]), LFA (D(M[n])), characterized by the usual adjunction [LFA (D(M[n])), −]A−Alg ' [D(M[n]), −]A−Mod . The model category A − Alg is a combinatorial and cellular model category, and therefore one can apply the localization techniques (see e.g. [Hi, Sm]) in order to invert the natural augmentations FA (D(M[n])) −→ A for all n ∈ Z. One checks easily that, since M is strongly dualizable, the local objects for this localization are the commutative A-algebras B such that M ∧LA B ' 0 in Ho(B − Mod). The local model of A for this localization will be denoted by AM . By definition, it is characterized by the following universal property: for any commutative A-algebra B, the mapping space RHomA−Alg (AM , B) is contractible if B ∧LA M ' 0 and empty otherwise. In other words, for any commutative S-algebra B the natural morphism RHomS−Alg (AM , B) −→ RHomS−Alg (A, B) is equivalent to an inclusion of connected components and its image consists of morphisms A −→ B in Ho(S − Alg) such that B ∧LA M ' 0. Lemma 2.1.6. With the above notations, the morphism A −→ AM is a Zariski open immersion. Proof: Let us start by showing that AM is a finitely presented commutative A-algebra. Let {Bi }i∈I be a filtered system of commutative A-algebras and B = colimi Bi . We assume that B ∧LA M ' 0, and we need to prove that there exists an i ∈ I such that Bi ∧LA M ' 0. By assumption, the two points Id and 0 are the same in π0 (REndB−Mod (M ∧LA B)). But, as M is a perfect A-module one has π0 (REndB−Mod (M ∧LA B)) ' colimi∈I π0 (REndBi −Mod (M ∧LA Bi )). This implies that there is some index i ∈ I such that Id and 0 are homotopic in REndBi −Mod (M ∧LA Bi ), and therefore that M ∧LA Bi is contractible. It remains to prove that the induced functor Ho(AM − Mod) −→ Ho(A − Mod) is fully faithful. To see this, one first notice that by definition the functor Ho(AM − Alg) −→ Ho(A − Alg) is fully faithful, and therefore so is Ho(AM − Alg/AM ) −→ Ho(A − Alg/AM ). For any two AM -modules N and P , one consider the trivial extensions AM ∨ N and AM ∨ P of AM by N and P (these are AM -augmented commutative AM -algebras). Then, one has

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a natural equivalence RHomAM −Alg/AM (AM ∨ N, AM ∨ P ) ' RHomAM −Mod (N, P ). Furthermore one has a natural fiber sequence RHomA−Mod (N, P ) → RHomA−Alg/AM (AM ∨ N, AM ∨ P ) → RHomA−Alg/AM (AM , AM ∨ P ). But, as RHomA−Alg/AM (AM , AM ∨ P ) ' RHomAM −Alg/AM (AM , AM ∨ P ) ' ∗ this shows that the natural morphism RHomAM −Mod (N, P ) → RHomA−Mod (N, P ), is an equivalence and therefore that Ho(AM − Alg) −→ Ho(A − Alg) is fully faithful. 2 An important property of the localization A −→ AM is the following fact. Lemma 2.1.7. Let A be a commutative S-algebra, and M be a perfect Amodule. Then the essential image of the fully faithful functor Ho(AM − Mod) −→ Ho(A − Mod) consists of all A-modules N such that M ∧LA N ' D(M) ∧LA N ' 0. Note that since M is perfect, then for any A-module N , M ∧LA N ' 0 iff D(M) ∧LA N ' 0, so the two conditions in the lemma are actually one. Moreover, AM ' AD(M ) in Ho(A − Alg). Proof: As every AM -module N can be constructed by homotopy colimits of free AM -modules and − ∧LA M commutes with homotopy colimits, it is clear that AM ∧LA M ' 0 implies N ∧LA M ' 0. Since AM ∧LA D(M) ' D(AM ∧LA M) ' 0 (here the second derived dual is in the category of AM -modules), the same argument shows that N ∧LA D(M) ' 0. Conversely, let N be an A-module such that N ∧LA M ' N ∧LA D(M) ' 0. By definition, the commutative A-algebra A −→ AM is obtained as a local model of A → A when one inverts the set of morphisms LFA (D(M[n])) −→ A, for any n ∈ Z. It is well known (see e.g. [Hi, §4]) that such a local model can be obtained by a transfinite composition of homotopy push-outs of the form AO α ∂∆p ⊗L LFA (D(M[n])) /

/

Aα+1 O

∆p ⊗L LFA (D(M[n]))

in the category of A-algebras. ¿From this description, and the fact that −∧LA M commutes with homotopy colimits, one sees that the adjunction morphism N −→ N ∧LA AM is an equivalence because by assumption on N , the natural morphism N ' N ∧LA A −→ N ∧LA LFA (D(M[n])) is an equivalence. 2 Lemma 2.1.7 allows us to interpret geometrically AM as the open complement of the support of the A-module M. Lemma 2.1.7 also has a converse whose proof is left as an exercise.

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Lemma 2.1.8. Let f : A −→ B be a morphism of commutative S-algebras and M be a perfect A-module. We suppose that the functor Rf∗ : Ho(B − Mod) −→ Ho(A−Mod) is fully faithful and that its essential image consists of all A-modules N such that N ∧LA M ' 0. Then, the two commutative Aalgebras B and AM are equivalent (i.e. isomorphic in Ho(A − Alg)). Remark 2.1.9. One should note carefully that even though if the EilenbergMac Lane functor H embeds (Aff, Zar) in (S − Aff, Zar) as model sites, there exist commutative rings R and Zariski open coverings HR −→ B in S − Aff such that B is not of the form HR0 for some commutative R-algebra R0 . One example is given by taking R to be C[X, Y ], and considering the localized commutative HR-algebra (HR)M (in the sense above), where M is the perfect R-module R/(X, Y ) ' C. If (HR)M were of the form HR0 for a Zariski open immersion Spec R0 −→ Spec R, then for any other commutative ring R00 , the set of scheme morphisms Hom(Spec R00 , Spec R0 ) would be the subset of Hom(Spec R00 , A2 ) consisting of morphisms factoring through A2 − {0}. This would mean that Spec R0 ' A2 − {0}, which is not possible as A2 − {0} is a not an affine scheme. This example is of course the same as the example given in [To, §2.2] of a 0-truncated affine stack which is not an affine scheme. These kind of example shows that there are many more affine objects in homotopical algebraic geometry than in usual algebraic geometry. Remark 2.1.10. (1) Note that Lemma 2.1.7 shows that the localization process (A, M) /o /o /o / AM is in some sense “orthogonal” to the usual Bousfield localization process (A, M) /o /o /o / LM A in that the local objects for the former are exactly the acyclic objects for the latter. To state everything in terms of Bousfield localizations, this says that LAM -local objects are exactly LM -acyclic objects (compare with Remark 2.1.5). Note that however, while the Bousfield localization is always defined for any A-module M, the commutative A-algebra AM probably does not exist unless M is perfect. (2) Let Sp be the p-local sphere. If f : Sp → B is any formal Zariski open immersion then L := Rf∗ Lf ∗ is clearly a smashing localization functor in the sense of [HPS, §3]. Its category C of perfect1 acyclics (i.e. perfect objects X in Ho(Sp − Mod) such that LX is null) is then a localizing thick subcategory of the homotopy category Ho(Sp −Modperf ) of the category of perfect Sp -modules, and therefore by [H-S] it is equivalent to the category Cn of perfect E(n)-acyclics, for some 0 ≤ n < ∞, where E(n) is the n-th Johnson-Wilson Sp -module (see e.g. [Rav]); in other words L and Ln := LE(n) are both smashing localization functors on Ho(Sp − Mod) having the same subcategory of finite acyclics. Therefore, if we assume (one of the form of) the 1The

word finite instead of perfect would be more customary in this setting.

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Bertrand To¨en and Gabrielle Vezzosi Telescope conjecture (see [Mil]), we get that Ln and L have equivalent categories of acyclics and so have equivalent categories of local objects. But the category of local objects for L is equivalent to the category Ho(B − Mod) (since Rf∗ is fully faithful by hypothesis) and the category of local objects for Ln is equivalent to the category Ho((Ln Sp ) − Mod), by [Wo] since Ln is smashing. This easily implies that the two commutative Sp -algebras B and Ln Sp are equivalent (i.e. isomorphic in Ho(Sp − Alg)). In conclusion, one sees that if the Telescope conjecture is true, then, up to equivalence of Sp -algebras, the only (non-trivial) formal Zariski open immersions for Sp are given by the family U := {Sp → Ln Sp }0≤n

290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 321 322 323 324 325 326 328 329 330 331 332 333 334 335 336 337 338 339 340 341

Quantum groups and Lie theory, A. PRESSLEY (ed) Tits buildings and the model theory of groups, K. TENT (ed) A quantum groups primer, S. MAJID Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK Introduction to the theory of operator spaces, G. PISIER Geometry and Integrability, L. MASON & YAVUZ NUTKU (eds) Lectures on invariant theory, I. DOLGACHEV The homotopy category of simply connected 4-manifolds, H.-J. BAUES Higher operads, higher categories, T. LEINSTER Kleinian Groups and Hyperbolic 3-Manifolds Y. KOMORI, V. MARKOVIC & C. SERIES (eds) Introduction to Möbius Differential Geometry, U. HERTRICH-JEROMIN Stable Modules and the D(2)-Problem, F.E.A. JOHNSON Discrete and Continuous Nonlinear Schrödinger Systems, M. J. ABLORWITZ, B. PRINARI & A. D. TRUBATCH Number Theory and Algebraic Geometry, M. REID & A. SKOROBOGATOV (eds) Groups St Andrews 2001 in Oxford Vol. 1, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2001 in Oxford Vol. 2, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Peyresq lectures on geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) Surveys in Combinatorics 2003, C. D. WENSLEY (ed) Topology, geometry and quantum field theory, U. L. TILLMANN (ed) Corings and Comodules, T. BRZEZINSKI & R. WISBAUER Topics in Dynamics and Ergodic Theory, S. BEZUGLYI & S. KOLYADA (eds) Groups: topological, combinatorial and arithmetic aspects, T. W. MÜLLER (ed) Foundations of Computational Mathematics, Minneapolis 2002, FELIPE CUCKER et al (eds) Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC Structured ring spectra, A. BAKER & B. RICHTER (eds) Linear Logic in Computer Science, T. EHRHARD et al (eds) Advances in elliptic curve cryptography, I. F. BLAKE, G. SEROUSSI & N. SMART Perturbation of the boundary in boundary-value problems of Partial Differential Equations, D. HENRY Double Affine Hecke Algebras, I. CHEREDNIK Surveys in Modern Mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N. C. SNAITH (eds) Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) Singularities and Computer Algebra, C. LOSSEN & G. PFISTER (eds) Lectures on the Ricci Flow, P. TOPPING Modular Representations of Finite Groups of Lie Type, J. E. HUMPHREYS Fundamentals of Hyperbolic Manifolds, R. D. CANARY, A. MARDEN & D. B. A. EPSTEIN (eds) Spaces of Kleinian Groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) Noncommutative Localization in Algebra and Topology, A. RANICKI (ed) Foundations of Computational Mathematics, Santander 2005, L. PARDO, A. PINKUS, E. SULI & M. TODD (eds) Handbook of Tilting Theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) Synthetic Differential Geometry 2ed, A. KOCK The Navier-Stokes Equations, P. G. DRAZIN & N. RILEY Lectures on the Combinatorics of Free Probability, A. NICA & R. SPEICHER Integral Closure of Ideals, Rings, and Modules, I. SWANSON & C. HUNEKE Methods in Banach Space Theory, J. M. F. CASTILLO & W. B. JOHNSON (eds) Surveys in Geometry and Number Theory, N. YOUNG (ed) Groups St Andrews 2005 Vol. 1, C.M. CAMPBELL, M. R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2005 Vol. 2, C.M. CAMPBELL, M. R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Ranks of Elliptic Curves and Random Matrix Theory, J. B. CONREY, D. W. FARMER, F. MEZZADRI & N. C. SNAITH (eds)

Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues HAYNES R. MILLER Massachusetts Institute of Technology DOUGLAS C. RAVENEL University of Rochester

CAMBRIDGE

UNIVERSITY

PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521700405 © Cambridge University Press 2007 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2007 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloguing in Publication data ISBN 978-0-521-70040-5 paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface Charles Thomas, 1938–2005

page vii xiii 1

1.

Discrete torsion for the supersingular orbifold sigma genus Matthew Ando and Christopher P. French

2.

Quaternionic elliptic objects and K3-cohomology Jorge A. Devoto

26

3.

The M-theory 3-form and E8 gauge theory Emanuel Diaconescu, Daniel S. Freed and Gregory Moore

44

4.

Algebraic groups and equivariant cohomology theories John P. C. Greenlees

89

5.

Delocalised equivariant elliptic cohomology (with an introduction by Matthew Ando and Haynes Miller) Ian Grojnowski

111

6.

On finite resolutions of K(n)-local spheres Hans-Werner Henn

122

7.

Chromatic phenomena in the algebra of BP∗ BP-comodules Mark Hovey

170

8.

Numerical polynomials and endomorphisms of formal group laws Keith Johnson v

204

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Thom prospectra for loopgroup representations Nitu Kitchloo and Jack Morava

214

10.

Rational vertex operator algebras Geoffrey Mason

239

11.

A possible hierarchy of Morava K-theories Norihiko Minami

255

12.

The motivic Thom isomorphism Jack Morava

265

13.

Toward higher chromatic analogs of elliptic cohomology Douglas C. Ravenel

286

14.

What is an elliptic object? Graeme Segal

306

15.

Spin cobordism, contact structure and the cohomology of p-groups C. B. Thomas

318

Brave New Algebraic Geometry and global derived moduli spaces of ring spectra Bertrand Toen and Gabriele Vezzosi

325

9.

16.

17.

The elliptic genus of a singular variety Burt Totaro

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A workshop entitled “Elliptic Cohomology and Chromatic Phenomena” was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, on 9–20 December, 2002. The workshop attracted over 75 participants from thirteen nations. The event, an EU Workshop, was the final one in INI’s program New Contexts for Stable Homotopy Theory held in the fall of that year. During the first week nineteen talks described a wide range of perspectives on elliptic genera and elliptic cohomology, including homotopy theory, vertex operator algebras, 2-vector spaces, and open string theories. The second week featured ten talks with a more specifically homotopy theoretic focus, but encompassing the higher chromatic variants of elliptic cohomology. This was the first conference on elliptic cohomology since the one organized by Peter Landweber at the Institute for Advanced Study in Princeton in 1986. The proceedings of that conference were published in [10]. The breadth of that volume is an indication of the multifaceted nature of the subject. From the start it has provided a meeting point for algebraic topology, number theory, and theoretical physics, playing in the present era a role analogous to the role of K-theory in the second half of the last century. Landweber’s introduction to that volume, together with Serge Ochanine’s contribution [13] to it, provide good introduction to the origins of this subject. The starting point was the study of genera of spin manifolds. A genus is a multiplicative bordism invariant, with values in some commutative ring. A classical result asserts that signature provides the universal example of a rational genus of oriented manifolds with the property that it is multiplicative for oriented fiber bundles with connected compact Lie structure groups. Ochinine showed that if a spin genus has this property then it is “elliptic.” In this description, Ochanine used the observation due to Quillen (understood also by Novikov in the rational case) that a genus for almost complex manifolds with values in a ring R is the same as a formal group law over R. Ochanine called a genus “elliptic” if it factors through the genus representing a formal group law used by Euler in 1756 to describe the addition law for certain elliptic integrals. Such a genus automatically factors uniquely through the map from unitary to spin bordism. The value ring can be regarded as a ring of modular forms. vii

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Ochanine conjectured that the converse is true as well, and Edward Witten provided intuitive physics rationale for this conjecture in an article [23] in the IAS conference proceedings, interpreting the elliptic genus as the charactervalued index of a certain hypothetical signature operator on the free loop space of the manifold. The conjecture was proven by Raoul Bott and Cliff Taubes [3], and later by a different method by Kefeng Liu [12]. The traditional genera occur as indexes of differential operators, and hence extend to families versions. The topological manifestation of this phenomenon is that the genus is the effect on homotopy groups of a map of ring specta. Landweber [9] had established a general criterion guaranteeing such a geometric realization, and Landweber, Bob Stong, and Doug Ravenel [11] showed that after suitable localization the Ochanine genus corresponds to a map of ring spectra MU → Ell. The target spectrum represented the first example of an elliptic cohomology theory. One of the principal attractions of elliptic cohomology is the possibility that it admits a geometric description, in analogy with the description of Ktheory via vector bundles. This dream was beautifully enunciated by Graeme Segal in his early survey [17], and expanded on in the his 1988 Oxford University lecture notes, now published as [18]. The idea, very crudely, is to regard a vector bundle with connection as associating a vector space to each point and an isomorphism to each path; the next step would then be to instead associate a Hilbert space to each loop and an operator to each bordism between loops. This is a field theory, and the hope has been that the physics of field theories would provide the ideas necessary to construct this mathematical object, which would, in turn, provide a framework for further physics. These hopes are reviewed in the paper by Segal in this volume. The torch is currently being carried by Stephan Stolz and Peter Teichner [19]. A representative of the string physics arising from the considerations which originally led Witten to his genera is provided by Emmanuel Diaconescu, Dan Freed, and Greg Moore, in their contribution. Other geometric constructions have been attempted. Po Hu and Igor Kriz [6] pursue the conformal field theory approach, while Nils Baas, Bjorn Dundas, and John Rognes [2] start from the philosophy that elliptic cohomology should be a form of the K-theory of the complex K-theory spectrum, and obtain a functor by considering the theory of 2-vector spaces. (This theory was in fact created by Mikhail Kapranov and Vladimir Voevodsky [8] with this application in mind.) Any geometric source of modular forms provides a hint for geometric sources of elliptic cohomology. One such source is vertex operator algebras. It has been globalized by Vassily Gorbunov, Fyodor

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Malikov, and Vadim Schechtman, [5], who use them identify a geometric object associated with a BU 6 structure. A useful introduction to vertex operator algebras, focusing on their modular properties, is provided by Geoffrey Mason’s article in this volume. Another approach to a geometric interpretation of elliptic cohomology starts from Witten’s interpretation of the elliptic genus and develops the notion that the corresponding cohomology theory should be a form of equivariant K-theory of the free loop space. The article of Nitu Kitchloo and Jack Morava pursues this idea. Further evidence of the geometric depth of elliptic genera is provided by the theorems of Lev Borisov, Anatoly Libgober, and Burt Totaro. Igor Krichever and Gerald Höhn had considered certain two-variable variants of the Ochanine genus, taking values in Jacobi forms and exhibiting rigidity for complex manifolds. It turns out that the same genus enjoys quite a different universal property. It is unchanged when one passes from one resolution of singularities of a given complex projective variety to another. One must either specify the process of passing from one to another (by a certain form of complex surgery, “classical flops”) or restrict to certain types of resolution of singularities. Burt Totaro reviews these theorems in his contribution. After early work of Jens Franke [4], Hopkins, Haynes Miller, Paul Goerss, and Charles Rezk have constructed an étale sheaf of commutative ring spectra over the elliptic modular stack. This construction has not yet been fully documented, though it has been the subject of courses ([16]) and a weeklong workshop in Mainz, Germany, in October, 2003. The paper of Bertrand Toën and Gabriele Vezzosi puts this construction into the general context of derived moduli spaces. The rigidity of this construction allows one to form a homotopy inverse limit, which is the spectrum tmf of “topological modular forms.” One of the hallmarks of a geometrically defined cohomology theory is the presence of equivariant versions of the theory. An analytic circle-equivariant theory associated to an elliptic curve E over C, taking values in sheaves of algebras over E, was constructed in 1994 by Ian Grojnowski. This construction has had a large impact on the subject, and his announcement appears here for the first time along with a historical introduction. Quillen [15] described the connection between even multiplicative cohomology theories and formal groups. It appears that lifting the cohomology theory to a circle equivariant form is related to extending the formal group to an algebraic group. This idea is explored in John Greenlees’s contribution. A variant of the problem of an equivariant extension is the possibility of an orbifold theory. Physicists had studied an orbifold elliptic genus, and this

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story is brought into contact with the homotopy theoretic enrichment of the Witten genus due to Ando, Hopkins, and Neil Strickland [1] in the paper of Matthew Ando and Christopher French. Elliptic spectra represent rather computable cohomology theories, which carry deep information about stable homotopy in their coefficients and in their operations. Following the approach set out by Frank Adams, these operations are most conveniently studied by means of the self homology Ell∗ Ell. Relations between this object and numerical polynomials (familiar from Adams’s study of K∗ K) are described in Keith Johnson’s article. The second part of the workshop focused on higher analogues of elliptic cohomology. From the chromatic viewpoint on stable homotopy theory, elliptic cohomology is the third in a sequence of types of theories, starting with rational cohomology and K-theory; and, locally at a prime at least, this sequence can be continued. One still wishes to realize these still higher “height” theories by some geometric data. A futuristic venture, representing joint work with the late Charles Thomas, is described by Jorge Devoto in this volume. Here the objective is to construct a variant of elliptic cohomology using moduli of K3-surfaces, and geometrically realize this theory by means of a quaternionic field theory. A step towards a different extension of the notion of elliptic cohomology is proposed in Ravenel’s article, in which he observes that the Jacobians of certain curves over Fp have one-dimensional formal factors of high height. The theory of operations of these higher height theories is the topic of the review article here by Mark Hovey. More detailed computations and homotopy theoretic constructions carried out by Goerss, Hans-Werner Henn, Mark Mahowald, and Rezk in the case of elliptic cohomology are reviewed and in part extended to higher height by Henn in his paper. Nori Minami explores a variety of analogies and connections between the various localizations of stable homotopy theory associated with these higher height theories. We dedicate this volume to the memory of our friend Professor Charles Thomas of the University of Cambridge, who died suddenly on December 16, 2005. Charles was constant in his belief in the geometric promise of elliptic cohomology. His book Elliptic Cohomology [20] and his review article [21] were important contributions to this development. His generosity and his warmth are missed by the whole community. We are pleased to publish here a paper derived from the lecture he gave at the workshop. Cambridge, MA, [email protected] Rochester, NY, [email protected] October 25, 2006

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References [1] Matthew Ando, Michael Hopkins, and Neil Strickland, Elliptic spectra, the Witten genus, and the theorem of the cube, Invent. math. 146 (2001) 595–687. [2] Nils A. Baas, Bjørn Ian Dundas, and John Rognes, Two-vector bundles and forms of elliptic cohomology. In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages 18–45. Cambridge Univ. Press, Cambridge, 2004. [3] Raoul Bott and Clifford Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989) 137–186. [4] Jens Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992) 43–65. [5] Vassily Gorbunov, Fyodor Malikov, and Vadim Schechtman, Gerbes of chiral differential operators, Math. Res. Lett. 7 (2000) 55–66. [6] P. Hu and I. Kriz, Conformal field theory and elliptic cohomology. Adv. Math., 189(2):325–412, 2004. [7] M. J. Hopkins, Algebraic topology and modular forms, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press. [8] M. Kapranov and V. A. Voevodsky, 2-categories and Zamolodchikov tetrahedra equations, in Algebraic groups and their generalizations: quantum and infinitedimensional methods (University Park, PA, 1991), Proc. Symp. Pure Math. 56 (1994) 177–259. [9] P. S. Landweber, Homological properties of comodules over MU∗ MU and BP∗ BP, Amer. J. Math 98 (1976) 591–610. [10] P. S. Landweber, editor, Elliptic curves and modular forms in algebraic topology, volume 1326 of Lecture Notes in Mathematics, Berlin, 1988. Springer-Verlag. [11] P. S. Landweber, D. C. Ravenel, and R. E. Stong, Periodic cohomology theories defined by elliptic curves. In Mila Cenkl and Haynes Miller, editors, The ˇ Cech Centennial, volume 181 of Contemporary Mathematics, pages 317–338, Providence, Rhode Island, 1995. American Mathematical Society. [12] Kefeng Liu, On SL2 Z and topology, Math. Res. Lett. 1 (1994) 53–64. [13] Serge Ochanine, Genres elliptiques equivariants, in P. S. Landweber, editor, Elliptic curves and modular forms in algebraic topology, Lecture Notes in Mathematics 1326, 1988, Springer-Verlag. [14] Serge Ochanine, Sur les genres multiplicatifs définis par des intégrales elliptiques. Topology, 26(2):143–151, 1987. [15] D. G. Quillen, On the formal group laws of oriented and unoriented cobordism theory. Bulletin of the American Mathematical Society, 75:1293–1298, 1969. [16] Charles Rezk, Supplementary notes for Math 512, Course Notes, Northwestern University, 2002. [17] G. Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others), Séminaire Bourbaki, Vol. 1987/88, Astérisque 161–162 (1989) 187–201. [18] Graeme Segal, The definition of conformal field theory. In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages 247–343. Cambridge Univ. Press, Cambridge, 2004.

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[19] Stephan Stolz and Peter Teichner, What is an elliptic object? In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages. Cambridge Univ. Press, Cambridge, 2004. [20] Charles B. Thomas, Elliptic cohomology. The University Series in Mathematics, Kluwer Academic/Plenum Publishers, New York, 1999. [21] Charles B. Thomas, Elliptic cohomology. In Survesy on surgery theory, Vol. 1 pages 409–439 Ann. of Math. Studies 145, Princeton University Press, Princeton, 2000. [22] Burt Totaro, Chern numbers for singular varieties and elliptic homology. Ann. of Math. (2), 151(2):757–791, 2000. [23] Edward Witten, The index of the Dirac operator in loop space. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math. 1326 (1988) 161–181.

Charles Thomas, 1938–2005

Charles B. Thomas, Professor of Algebraic Topology in the University of Cambridge died on 16 December 2005. Although some ill health had come his way, his death came as a shock, particularly to those who wished him well at his retirement dinner but nine days earlier. He was born on 17 August 1938 and educated at the Benedictines’ Douai School near Reading. After two years’ service in the Royal Air Force he entered Trinity College, Cambridge, in 1958. An initial intention to read physics was soon converted into a life-long commitment to the study of mathematics. He worked for his PhD initially at Trinity College under D.B.A. Epstein and then under A. Dold at Heidelberg (where he met his wife, Maria). He was a Research Associate at Cornell University (1965–1967), a Lecturer at the University of Hull (1967–1969) and a Lecturer at University College London (1969–1979). He returned to Cambridge as a University Lecturer and Fellow of Robinson College in 1979. Charles Thomas’ research concerned the interplay between algebra (in particular the intricacies of finite group theory), algebraic topology, number theory, the topology of manifolds and various structures in differential geometry. He studied finite group actions on spheres and homotopy spheres and worked on many aspects of the spherical space form problem. His study of 3-manifold groups and Poincaré complexes culminated in his book Elliptic structures on 3-manifolds. Work on characteristic classes, classifying spaces and the cohomology of finite groups led to another book; problems on the cohomology of a wide range of types of group were a continuing interest to him. In two papers in the 1970s Charles was one of the first to note the significance of contact structures on smooth manifolds. He returned to this topic with renewed vigour some twenty years later when contact and symplectic structures became fashionable. Aspects of his research led to his writing books on elliptic cohomology, on differential manifolds (written with D. Barden) and on representation theory. He edited the volumes of the proceedings of xiii

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Charles Thomas

several meetings he had organised and was particularly proud to be co-editor of a selection of the works of J.F. Adams, whom he particularly admired. Several of Charles’ former PhD students now propagate his high standards to students of their own. Charles valued academic excellence. A list of his mathematical achievements gives only a limited impression of his intellect. He spoke something of most of the main European languages and was fluent in French and German. He translated into English four mathematics books from German and one from French. He could converse on any academic subject but had a passion for medieval European and Byzantine history. Whilst giving first priority to research, Charles did his share of university administration. Recently, he was Chairman of the Faculty Board and editor of the Mathematical Proceedings of the Cambridge Philosophical Society. His careful lecturing was illuminated by his immaculate handwriting on the blackboard and on duplicated notes; an occasional stutter allowed the student a moment to catch up. He planned in his retirement to continue his work partly in Cambridge, partly in California at Santa Cruz. His wife, two sons, a daughter and a very new grandson survive him. Raymond Lickorish Slightly edited, first appearing in the Newsletter of the London Mathematical Society (No. 345 February 2006). Used by permission.

Charles Thomas, Hong Kong, 2002

DISCRETE TORSION FOR THE SUPERSINGULAR ORBIFOLD SIGMA GENUS MATTHEW ANDO AND CHRISTOPHER P. FRENCH Abstract. The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of [HKR00] to the Borel-equivariant genus associated to the sigma orientation of [AHS01] to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the same sort of formula as the orbifold “two-variable” genus of [DMVV97] and [BL02]. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable genus in the formulae of [And03] to define an analytic equivariant genus in Grojnowski’s equivariant elliptic cohomology, and we show that this gives precisely the orbifold two-variable genus. The second purpose of this paper is to study the effect of varying the BU h6i-structure in the Borelequivariant sigma orientation. We show that varying the BU h6i structure by a class in H 3 (BG; Z), where G is the orbifold group, produces discrete torsion in the sense of [Vaf85]. This result was first obtained by Sharpe [Sha], for a different orbifold genus and using different methods.

1. Introduction Let E be an even periodic, homotopy commutative ring spectrum, let C be an elliptic curve over SE = spec π0 E, and let t be an isomorphism of formal groups b∼ t:C = spf E 0 (CP ∞ ), so that C = (E, C, t) is an elliptic spectrum in the sense of [Hop95, AHS01]. In [AHS01], Hopkins, Strickland, and the first author construct a map of homotopy commutative ring spectra σ(C) : MU h6i → − E called the sigma orientation; it is conjectured in [Hop95] that this map is the restriction to MU h6i of a similar map MOh8i → E. (MU h6i is the bordism Date: Version 1.7, July 2005. We thank the participants in the BCDE seminar at UIUC, particularly David Berenstein and Eric Sharpe, for teaching us the physics which led to this paper. Ando was supported by NSF grants DMS—0071482 and DMS—0306429. Part of this work was carried out while Ando was a visitor at the Isaac Newton Institute for Mathematical Sciences. He thanks the Newton Institute for its hospitality.

1

2

Ando and French

theory of SU -manifolds with a trivialization of the second Chern class, and MOh8i is the bordism theory of spin manifolds with a trivialization of the characteristic class p21 .) The sigma orientation is natural in the elliptic spectrum, and, if KTate = (K[[q]], Tate, t) is the elliptic spectrum associated to the Tate elliptic curve, then the map of homotopy rings π∗ MU h6i → π∗ KTate

(1.1)

is the restriction from π∗ MSpin of the Witten genus. Explicitly, let M be a Riemannian spin manifold, and let D be its Dirac operator. Let T denote the tangent bundle of M. If V is a (real or complex) vector bundle over M, let V C be the complex vector bundle V C = V ⊗ C. R

If V is a complex vector bundle, let rV be the reduced bundle rV = V − rank V , and let St V =

X

tk S k V

k≥0

be the indicated formal power series in the symmetric powers of V . The operation V 7→ St V extends to an exponential operation K(X) → K(X)[[t]] because of the formula St (V ⊕ W ) = (St V )(St W ). The Witten genus of M is given by the formula w(M) = ind D ⊗

O

Sqk (rT C )

k≥1

and the diagram

!

∈ Z[[q]],

(1.2)

π∗ MU h6i

/ π∗ MSpin NNN NNN w NN π∗ σ(KTate ) NNN&

Z[[q]]

commutes [AHS01]. The Witten genus first arose in [Wit87], where Witten showed that various elliptic genera of a manifold M are essentially one-loop amplitudes of quantum field theories of closed strings moving in M. Locally on a spin manifold M, the quantum field theory associated to w(M) is a conformal field theory, and the obstruction to assembling a conformal field theory globally on M is c2 M.

Discrete torsion and the sigma genus

3

1

This gives a physical proof that, if c2 M = 0, then w(M) is the q-expansion of a modular form. Suppose that M is an SU -manifold. The formula (1.2) shows that the Witten genus is an invariant of the spin structure of M. On the other hand the sigma orientation depends on a choice of BU h6i structure, that is, a lift in the diagram BU h6i x

x

x

x

x<

c2 / BSU / K(Z, 4). M It is an interesting problem to understand how the orientation depends on this choice. The fibration sequence ι

c

2 K(Z, 3) → − BU h6i → BSU − → K(Z, 4)

shows that a lift exists precisely when c2 (M) = 0, and that the set of lifts is a quotient of H 3 (M; Z). The dependence on the choice of lift appears to have an explanation in string theory. The action for the QFT described by the Witten genus is a function on the space of maps X:Σ→M of 2-dimensional surfaces Σ to M. If the theory is anomaly-free, that is, if c2 M = 0, then one is free to add to the action a term of the form Z X ∗ B, Σ

2

where B ∈ Ω M is a differential 2-form on M (called the “B-field”), provided that H = dB is an integral three-form. It seems clear that the physics of the B-field should account for the variation in the sigma orientation from at least torsion classes in H 3 (M, Z).

This situation has already received a good deal of attention, particularly in the case of orbifolds. Eric Sharpe [Sha] showed that Vafa’s discrete torsion [Vaf85] arises from, as he put it us once, “the action of the orbifold group on the B-field.” Lupercio and Uribe [LU] have explained how discrete torsion arises from a gerbe on a total quotient orbifold M/G with finite orbifold group (and so gives rise to a class in the Borel cohomology group H 3 (MG ; Z)). The techniques used to date in the study of the sigma orientation have quite a different flavor from those used in the study of gerbes on orbifolds. 1There

And03]).

are various ways to understand this obstruction ([Wit87, BM94, GMS00,

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Ando and French

One goal of the research which led to this paper was to find using the sigma orientation some of the phenomena associated to the B-field. In this direction, we show that varying the BU h6i structure in the orbifold sigma genus of a supersingular elliptic curve indeed produces a variant of discrete torsion. More precisely, suppose that M is a complex manifold with an action by a group G, and suppose that V is a complex G-vector bundle over M. Let T denote the tangent bundle of M. If X is a space, let XG denote the Borel construction EG ×G X. If c1 (TG ) = c1 (VG ) c2 (TG ) = c2 (VG ), then there is a lift in the diagram BU h6i `

w

MG

w

w

(1.3)

w;

w

/ VG −TG

BSU,

and a choice of lift gives a Thom class U (M, `, C)G ∈ E 0 (VG − TG ). The relative zero section together with the Pontrjagin-Thom construction provide a map τ (V )G : E 0 (VG − TG ) → E 0 (−TG ) → E 0 (BG), and τ (V )G (U (M, `, C)G ) ∈ E 0 (BG) is the (Borel) equivariant sigma genus of M twisted by V (see §4). To get from it an “orbifold” genus taking its values in E 0 , we use the character theory of Hopkins, Kuhn, and Ravenel ([HKR00]; see also §5). It associates to a pair (g, h) of commuting elements of G a ring homomorphism Ξg,h : E 0 (BG) → D, where D is a complete local E-algebra which depends on the formal group of the spectrum E. It turns out that the quantity X σorb (M, `, C)G = Ξg,h τ (V )G (U (M, `, C)G ) (1.4) gh=hg

takes its values in E 0 ; we call it the orbifold sigma genus of M twisted by V (see §6).

There is already an extensive literature on the subject of “orbifold elliptic genera”, particularly the “two-variable” elliptic genus of [Kri90, EOTY89]; see for example [DMVV97, BL02]. In §6, we show that the formula (1.4) is formally analogous to the formula for the orbifold two-variable genus. More

Discrete torsion and the sigma genus

5

precisely, by applying the topological Riemann-Roch formula to the expression (1.4) and using the Thom class associated to the the two-variable elliptic genus in place of U , we obtain a formula nearly identical to that of [BL02]. Unfortunately our use of the topological Riemann-Roch formula in this context, while illuminating, is not useful for calculation, because we work with the Borel-equivariant elliptic cohomology associated to a supersingular elliptic curve, which is a highly completed situation. In order to locate the orbifold two-variable genus more precisely in the setting of equivariant elliptic cohomology, we consider in §7 the case of a finite cyclic group G = T[n] ⊂ T. We use the principle suggested by Shapiro’s Lemma to define def

EG (X) = ET (T ×G X), where ET is the uncompleted analytic equivariant elliptic cohomology of Grojnowski. We adapt the formulae in [And03], which descends from [Ros01, AB02], to write down an euler class in EG (X). The associated genus Ellan (M, G) takes its value in Γ(EG (∗)) ∼ = Γ(OC[n] ) ∼ = CG×G , and we prove the following. Theorem 1.5. Suppose that G is a finite cyclic group. Summing the analytic equivariant two-variable genus over the torsion points of the elliptic curve gives the orbifold two-variable genus: more precisely, we have 1 X Ellorb (X, G) = Ellan (M, G, g, h). |G| gh=hg We were pleased to be able to confirm that orbifold elliptic genera are (in the cyclic case) so simply obtained from equivariant elliptic genera. It would be interesting to use this observation to investigate more subtle properties of orbifold genera, such as, for example, the “McKay correspondence” of Borisov and Libgober. The rest of the paper is devoted to the study of the dependence of the orbifold sigma genus on the choice ` of BU h6i structure in (1.3). Suppose that we have chosen an element u ∈ H 3 (BG; Z), represented as a map u : BG → K(Z, 3). If π : MG → BG denotes the projection in the Borel construction, then we obtain an element π ∗ u = uπ ∈ H 3 (MG ; Z), and ` + ιuπ is another BU h6i-structure on VG − TG . In §8, we use the character theory and the sigma orientation to associate to u an alternating bilinear map δ = δ(u, C) : G2 → D × , where G2 denotes the set of pairs of commuting elements of G of p-power order. In §9 we obtain the

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Ando and French

Theorem 1.6. The orbifold sigma genus associated to the BU h6i structure `+ιuπ is related to the equivariant sigma genus associated to ` by the formula X σorb (M, ` + ιuπ, C)G = δ(g, h)Ξg,h τ (V )G (U (M, `, C)G ). gh=hg

In [Vaf85], Vafa observed that if φ=

X

φg,h

gh=hg

is an orbifold elliptic genus associated to a theory of strings on M, and if c = c(g, h) is a 2-cocycle with values in U (1), then X c(g, h)φg,h (1.7) gh=hg

is again modular; he called this phenomenon “discrete torsion”. Eric Sharpe [Sha] showed that the genus (1.7) arises from adding a B-field on the orbifold M/G. As expressed by Lupercio and Uribe, this is a U (1)-gerbe B on the orbifold M/G, whose associated cohomology class [B] ∈ H 3 (MG ; Z) satisfies [B] = [c] ∈ H 3 (MG ; Z), where [c] is the cohomology class in MG obtained from c by pulling back along MG → BG. Our result shows that varying the BU h6i-structure of MG by an element u ∈ H 3 (BG) has a similar effect on the orbifold sigma genus. When G is an abelian group of order dividing n = ps , the map δ may be viewed as a two-cocycle on G with values in D × [n] ∼ = Z/n, and as such it represents a 3 H (BG). It is not quite the cohomology cohomology class in H 2 (BG; Z/n) ∼ = class u: instead, as we shall see in §10, if c is a 2-cocycle representing u ∈ H 3 (BG; Z) ∼ = H 2 (BG; Z/n), then δ(g, h) = c(g, h) − c(h, g). 2. The sigma orientation and the sigma genus In this section we recall some results from [AHS01]. Definition 2.1. An elliptic spectrum consists of (1) an even, periodic, homotopy commutative ring spectrum E with formal group PE = spf E 0 CP ∞ over π0 E; (2) a generalized elliptic curve C over π0 E; b of PE with the formal completion of C. (3) an isomorphism t : PE → − C

Discrete torsion and the sigma genus

7

A map (f, s) of elliptic spectra E1 = (E1 , C1 , t1 ) → − E2 = (E2 , C2 , t2 ) consists of a map f : E1 → − E2 of multiplicative cohomology theories, together with an isomorphism of elliptic curves s

C2 → − (π0 f )∗ C1 , extending the induced isomorphism of formal groups. Theorem 2.2. An elliptic spectrum C = (E, C, t) determines a map σ(C) : MU h6i → E of homotopy-commutative ring spectra. The association C 7→ σ(C) is modular, in the sense that if (f, s) : C1 → C2 is a map of elliptic spectra, then the diagram σ(C )

1 /E 1 GG GG GG f G σ(C2 ) GG#

MU h6i

E2

commutes up to homotopy. If KTate = (K[[q]], Tate, t) is the elliptic spectrum associated to the Tate curve, then the diagram MU h6i

/ MSpin KK KK KK w K σ(KTate ) KK%

K[[q]] commutes, where w is the orientation associated to the Witten genus. 3. The sigma genus Definition 3.1. Let W be a virtual complex vector bundle on a space M. A BU h6i-structure on W is a map `:M → − BU h6i such that the composition `

M→ − BU h6i → − BU classifies rW . Now let M be a connected compact closed manifold with complex tangent bundle T , and let V be another complex vector bundle on M. Let d = 2 rankC T − V Let τ (V ) : S 0 −−−→ M −T → − M V −T P −T

ζ

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Ando and French

be the composition of the Pontrjagin-Thom map with the relative zero section. If ` : M → BU h6i is a BU h6i-structure on V − T , and if C = (E, C, t) is an elliptic spectrum, let U (M, `, C) ∈ E −d (M V −T ) be the class given by the map σ(C)

`

U (M, `, C) : Σd M V −T → − MU h6i −−−→ E. Definition 3.2. The sigma genus of ` in C is the element def

σ(M, `, C) = τ (V )∗ (U (M, `, C)) ∈ E −d (S 0 ) = πd E. Example 3.3. Suppose that c1 T = 0 = c2 T , so that T itself admits a BU h6istructure, say ` : M → BU h6i, and d = 2 dim M. Then we have a Thom isomorphism E 0 (M) ∼ = E −d (M −T ) 1 7→ U (M, `, C). and the usual Umkehr map π!M associated to the projection πM : M → ∗ is the composition ∼ =

π!M : E 0 (M) − → E −d (M −T ) −−−→ E −d (S 0 ) ∼ = πd E. P −T

Thus σ(M, `, C) = π!M (1) = πd (σ(C))([M]) ∈ πd E is just the genus of M with BU h6i-structure `, associated to the sigma orientation σ(C) MU h6i −−−→ E. 4. The Borel-equivariant sigma genus Now suppose that G is a compact Lie group, and, if X is a space, let XG denote the Borel construction def

XG = EG ×G X. Suppose that G acts on the compact connected manifold M, that V is an equivariant complex vector bundle, and that ` : MG → BU h6i is a BU h6i-structure on the bundle VG −TG . Since TG is the bundle of tangents along the fiber of MG → BG, we have a Pontrjagin-Thom map P −T

BG+ −−−→ (MG )−TG ,

Discrete torsion and the sigma genus

9

and so a map ζ

P −T

τ (V )G : BG+ −−−→ (MG )−TG → − (MG )VG −TG . Let U (M, `, C)G ∈ E −d (MGVG −TG ) be given by the map `

σ(C)

U (M, `, C)G : Σd (MG )VG −TG → − MU h6i −−−→ E. Definition 4.1. The (Borel) equivariant sigma genus of ` in C is the element def

σ(M, `, C)G = τ (V )G (U (M, `, C)G ) ∈ E −d (BG). 5. Character theory The equivariant sigma genus described in §4 is not so familiar, because E (BG) is not. In this section we review the character theory of [HKR00], which gives a sensible way to understand E ∗ (BG). In the next section, we apply the character theory to produce the orbifold sigma genus from the equivariant sigma genus; as we shall see, it is given by the same sort of formula as those for “orbifold elliptic genera” in for example [DMVV97, BL02] ∗

We suppose that E is an even periodic ring spectrum, and that π0 E is a complete local ring of residue characteristic p > 0. We write P for CP ∞ , so PE = spf E 0 P is the formal group of E. We assume that PE has finite height h. Let Λ∞ = (Zp )h , and for n ≥ 1, let Λn = Λ∞ /pn Λ∞ . If A is an abelian def group, let A∗ = hom(A, C× ) denote its group of complex characters, so for example Λ∗∞ ∼ = (Z[ 1p ]/Z)h . = (Qp /Zp )h ∼ Each λ ∈ Λ∗n defines a map Bλ

BΛn −→ P. Choose a coordinate x ∈ E 0 P. For each λ ∈ Λ∗n , let x(λ) = (Bλ)∗ x ∈ E 0 BΛn . Let S ⊂ E 0 BΛn be the multiplicative subset generated by {x(λ)|λ 6= 0}. Let Ln = S −1 E 0 BΛn , and let Dn be the image of E 0 BΛn in Ln . In other words, Dn is the quotient of E 0 BΛn by the ideal generated by annihilators of euler classes of non-zero characters of Λn . It is clear that Ln and Dn are independent of the choice of coordinate x. Now suppose that G is a finite group. Let α : Λn → G be a homomorphism: specifying such α is equivalent to specifying an h-tuple of commuting elements of G of order dividing pn .

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Ando and French

Definition 5.1. The character map associated to α is the ring homomorphism E 0 Bα

Ξα : E 0 BG −−−→ E 0 BΛn → − Dn . One may check directly from the definition that the map Λn+1 Λn induces maps Dn → Dn+1 Ln → Ln+1 . Let D = colim Dn n

L = colim Ln .

(5.2)

n

Since G is finite, any homomorphism α:Λ→G factors as α

n α:Λ→ − Λn −→ G for sufficiently large n, and we may unambiguously attach a character homomorphism Ξα : E 0 BG → − D such that, for sufficiently large n, the diagram

E 0 BGH

Ξαn

/D n HH HH HH Ξα HH#

D

commutes. A great deal is known about the ring Dn , because it turns out [AHS03] that spf Dn is the scheme of level Λ∗n -structures on the PE . For example, it is easy to check that the action of Aut(Λn ) on E 0 BΛn induces an action of Aut(Λn ) on Dn . Using the description of Dn in terms of level structures, one may prove the following. Proposition 5.3. The ring Dn is finite and faithfully flat over E. If PE is the universal deformation of a formal group of height h (i.e. if E is a Morava E-theory), then Dn is a complete Noetherian local domain, and in that case, and in general if p is regular in π0 E, then Ln = 1p Dn . The structural map E 0 → Dn identifies E 0 with the Aut(Λn )-invariants in Dn . Proof. [Dri74] or [Str97].

Discrete torsion and the sigma genus

11

If w ∈ Aut(Λn ) and α : Λn → G, then we have two homomorphisms from E (BG) to Dn , namely wΞα and Ξαw . 0

Lemma 5.4. The diagram E 0 BGG

Ξα

/ Dn GG GG GG w Ξαw GG#

Dn

commutes.

Corollary 5.5. The expression X

Ξα

α:Λn →G

defines an additive map E 0 BG → π0 E. In the case that the height of PE is two, the sum is over all pairs of commuting elements of G of p-power order. If G is a p-group, then we write X Ξg,h : E 0 BG → − π0 E gh=hg

for the map in the Corollary. 6. The orbifold sigma genus There has been much study of the orbifold version of the two-variable elliptic genus of [EOTY89]; see for example [DMVV97, BL02]. In this section we introduce an orbifold version of the sigma genus, in the case of a supersingular elliptic curve. Our definition is intentionally as simple as possible: we consider only total quotient orbifolds, and then extract the orbifold sigma genus from the Borel genus using the map of Corollary 5.5. Explicitly, suppose that G is a finite group acting on a manifold M with complex tangent bundle T , that V is an equivariant complex vector bundle, and that ` : MG → BU h6i is a BU h6i-structure on the bundle VG − TG . Let C be the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p > 0, and let C = (E, C, t) be the associated elliptic spectrum.

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Ando and French

Definition 6.1. The orbifold sigma genus of ` in C is the element X def σorb (M, `, C)G = Ξg,h σ(M, `, C)G ∈ π−d E.

(6.2)

gh=hg

The rest of this section is devoted to showing that the formula (6.2) is formally analogous to the formula for the orbifold two-variable genus. In section 7, we show that, in the case of a finite cyclic group, the orbifold twovariable genus is precisely the genus in Grojnowski’s circle-equivariant elliptic cohomology obtained from the characteristic series defining the two-variable genus by following the construction of [Ros01, AB02, And03]. These sections are logically independent of the discussion of discrete torsion and the proof of Theorem 1.6, and readers interested primarily in that formula may prefer to skip to section 8. Our comparison in this section is based on the analogue of the formula (6.2) in the case of a genus given by a complex orientation t : MU → − E, so that E has Thom classes and Umkehr maps for complex vector bundles. If M is a compact manifold of real dimension d, then we write π M for the projection M → ∗, M and πt for the Umkehr map πtM : E ∗ (M) → πd−∗ E. This Umkehr map is often denoted π!M ; our notation emphasizes the dependence on the orientation t. In any case, the genus associated to t is the map Φt : π∗ MU → π∗ E given by the formula Φt (M) = πtM (1). If a compact Lie group G acts on M, then we write π M,G for the projection π M,G : MG → BG, and πtM,G for the associated Umkehr map πtM,G : E 0 (MG ) → − E −d (BG). If G is a finite p-group, then the analogue of our formula (6.2) is the quantity X Φtorb (M) = Ξg,h πtM,G (1). gh=hg

If g and h are commuting elements of G, let def

M (g,h) = M g ∩ M h

13

Discrete torsion and the sigma genus

be the subset of M fixed by both g and h. Let V(g, h) be a normal bundle of M (g,h) in M. We view (g, h) as a homomorphism (g,h)

Λn −−→ G : this makes Λn act trivially on M (g,h) , and we let et (V(g, h)) ∈ E ∗ (BΛn ) ⊗ E ∗ (M (g,h) ) ∼ = E ∗ (EΛn ×Λn M (g,h) ) be the Λn -equivariant euler class of V(g, h) in the orientation t. Recall from (5.2) that L = colim Ln is the colimit of the rings Ln obtained from E ∗ BΛn by inverting the euler classes of non-trivial characters of Λ. Proposition 6.3. Suppose that (g, h) 6= (0, 0). (1) The euler class et (V(g, h)) is a unit of L ⊗E ∗ E ∗ (M (g,h) ). (2) The quantity 1⊗

(g,h) πtM

lies in the subring D ⊂ L. (3) As elements of D we have Ξg,h (πtM,G (1))

=1⊗

1 et (V(g, h))

(g,h) πtM

1 . et (V(g, h))

Thus the orbifold genus of M associated to t is given by the formula X 1 t t M (g,h) Φorb (M) = Φ (M) + 1 ⊗ πt . et (V(g, h)) gh=hg

(6.4)

(g,h)6=(0,0)

Proof. Keeping in mind that M (g,h) is a compact manifold, the first assertion follows by the argument originally due to [AS68]. Now examine the diagram j

BΛ × M (g,h) −−−→ EΛ ×(g,h) M −−−→ EG ×G M M,G (g,h) M,Λ M π y y yπ 1×π i

BΛ The right square is a pull-back, so

BΛ

B(g,h)

−−−−→

B(g, h)∗ πtM,G = πtM,Λ j ∗ . It follows that Ξ(g,h) π!M,G (1) = πtM,Λ (1),

BG

14

Ando and French

considered as an element of D. The fixed-point formula asserts that π!M,Λ (1)

=1⊗

(g,h) π!M

1 et (V(g, h))

in L: but in fact we know that the left-hand side is an element of D ⊂ L. It follows that the right hand side is too, and 1 M,G M (g,h) . Ξ(g,h) π! (1) = 1 ⊗ π! et (V(g, h)) The rest is easy.

The formula (6.4) is the analogue for the t-genus of the orbifold elliptic genera of [DMVV97, BL02]. To see this, let A be the (abelian) subgroup of G generated by g and h, and suppose that V(g, h) decomposes as a sum V(g, h) ∼ = L1 ⊕ · · · ⊕ L r of complex line bundles, with A acting on Li by the character χi . Let e(χi ) ∈ E(BA) be the euler class of the character χi , using the orientation t, and let yi ∈ E(M (g,h) ) be the (non-equivariant) euler class of the line bundle Li . Then Y et (V(g, h)) = yi +F e(χi ). i

We can be even more explicit. Let F (x, y) ∈ E[[x, y]] be the formal group law over E induced by the orientation t. If R is a complete local E-algebra, let us write F (R) for the maximal ideal of R, considered as an abelian group using the power series F to perform addition. Associating to a character λ ∈ Λ∗n its first chern class in E-theory using the orientation t defines a group homomorphism Λ∗n → − F (E(BΛn )), which gives rise to a homomorphism Λ∗n → − F (Dn ); in fact, this is the “level structure” referred to in §5. The dual of the epimorphism Λ → Λn → A

Discrete torsion and the sigma genus

15

is a monomorphism A∗ → Λ∗n which composes with the level structure to give a homomorphism φ : A∗ → Λ∗n → − F (Dn ). By construction, φ(χi ) = e(χi ). If e1 , e2 are a basis for Λ∗n , then we can write χi = a i e 1 + b i e 2 in Λ∗n , where ai , bi ∈ Z/n. If vi = `(ei ) for i = 1, 2, then φ(χi ) = [ai ](v1 ) +F [bi ](v2 ). Our typical summand in the formula (6.4) for the orbifold genus becomes ! Y 1 (g,h) 1 ⊗ π!M . (6.5) y + [a ](v ) + [b ](v ) i F i 1 F i 2 i It is customary to calculate expressions like (6.5) by using the topological Riemann-Roch formula to pass to ordinary cohomology. In fact this approach is not available in our situation. To do so, one introduces the exponential ba → F exp : G

b a (L) such that of the group law F , and finds xi ∈ L ⊗ E(M (g,h) ) and wi ∈ G yi = f (xi )

v1 = f (w1 )

v2 = f (w2 ). However, if v1 = f (w1 ) then 0 = [n]F (v1 ) = f (nw1 ), which implies that nw1 = 0, and, as L is torsion free, we conclude that w1 must be zero! Nevertheless, we shall proceed formally in order to compare our formula with those of [DMVV97, BL02]. We have Y et (V(g, h)) = f (xi + ai w1 + bi w2 ). i

Let uj , j = 1, . . . , r be the roots of the total Chern class of the tangent bundle of M (g,h) : Y c(M (g,h) ) = (1 − uj ). j

Then the Riemann-Roch formula gives ! Z Y uj Y 1 1 M (g,h) πt = . et (V(g, h)) f (uj ) f (xi + ai w1 + bi w2 ) M (g,h) j i

(6.6)

16

Ando and French

In [BL02], Borisov and Libgober use the two-variable elliptic whose exponential is θ(x, τ ) f (x) = , (6.7) θ(x − z, τ ) where 1

x

x

θ(x, τ ) = −iq 8 (e 2 − e− 2 )

Y

(1 − q n )(1 − q n ex )(1 − q n e−x ),

(6.8)

n≥1

τ is a complex number with positive imaginary part, and q = e2πiτ . (We have adopted slightly different conventions regarding factors of 2π. The simplest way to compare is to say that we work with the elliptic curve C/(2πiZ + 2πiτ Z), while they work with the elliptic curve C/(Z + τ Z)) Their expression for the orbifold two-variable genus is Ellorb (M, G) =

1 X Φg,h , |G| gh=hg

where, with our conventions, ! Z Y uj Y 1 Φg,h = ezBi /n . (6.9) f (uj ) f (xi + Ai (1/n) − Bi (τ /n)) M (g,h) j i and the Ai and Bi are integer representatives of ai and bi . This differs from the formal expression (6.6) by only the factors ezbi /n . These factors are familiar from the study of equivariant genera; for example 1/k they are analogous to the factors νs in (11.26) of [BT89], S(c1 (V 1/n )T ) in k ˆ ¯ in (5.16) of [And03]. Their role is to make the (6.17) of [AB02], or u n I(m) expression (6.9) independent of the choice of representatives Ai and Bi . It is necessary to introduce these factors because f is not doubly periodic; instead we have f (x + 2πi` + 2πikτ ) = y −k f (x),

(6.10)

where y = ez , as one checks easily using (6.8). So the expression doesn’t depend on the choice of Ai . If Bi0 = Bi + nδi , then Y Y P P 0 f (xi + Ai /n − Bi0 τ /n)ezBi /n = f (xi + Ai /n − Bi τ /n)y − δi ezBi /n y δi i

i

=

Y

f (xi + Ai /n − Bi τ /n)ezBi /n .

i

This is not an issue in our expression (6.5), and so the factors ezBi /n have no role in our genus.

Discrete torsion and the sigma genus

17

7. Comparison with the analytic equivariant genus In fact, we can use the expression (6.7) for the two-variable elliptic genus in terms of theta functions to construct a Thom class in Grojnowski’s equivariant cohomology, following [Ros01, AB02, And03]. When G = T[n] is a cyclic group of order n acting on a compact manifold M, we can write down a formula for a T-equivariant genus on T ×G M, which by Shapiro’s lemma is a sensible notion of G-equivariant genus on M. When we do so, we obtain the formula of [DMVV97, BL02]. Let Λ be the lattice (2πiZ + 2πiτ Z), let C be the elliptic curve C/Λ, and let ET be Grojnowski’s equivariant elliptic cohomology associated to C. For convenience we identify T∼ = R/Z so that 1 G∼ = Z[ ]/Z. n We identify T×T∼ =C by the formula (r + Z, s + Z) 7→ 2πir + 2πisτ + Λ. (7.1) Let M be a G-manifold with an equivariant complex structure on its tangent bundle T . We define def

EG (M) = ET (T ×G M). For a ∈ C, (T ×G M)a = 0 unless a ∈ C[n], so ET (T ×G M) is a metropolitan sheaf (collection of skyscraper sheaves) supported at C[n]. The stalk at a point a of order dividing n is H(EG ×G X a ). Let Tˇ be the lattice of cocharacters in Spin(2d). In [And03], the first author constructed orientations for theta functions Θ = Θ( − , τ ) : Tˇ ⊗ C → C satisfying Θ(x + 2πi` + 2πikτ, τ ) = exp(−2πiI(k, x)) exp(−φ(k))Θ(x, τ ) for x ∈ Tˇ ⊗ C and k, ` ∈ Tˇ , where φ :Tˇ → Z I :Tˇ × Tˇ → Z

(7.2)

18

Ando and French

are respectively quadratic and bilinear functions related by φ(` + `0 ) = φ(`) + I(`, `0 ) + φ(`0 ). The building block of the orientation is a family of functions F which we now describe. For simplicity we have supposed that T M is a complex vector bundle, and so our structure group is U (d) instead of Spin(2d). We let T be the maximal torus of diagonal matrices; our choices so far identify the lattice Tˇ = hom(T, T ) of cocharacters with Zd in the usual way. Let (g, h) ∈ G2 ; let a be the corresponding point of C[n], and let A ⊆ G be the subgroup generated by g and h. The action of A on T M|M A is described by characters m = (m1 , . . . , md ) ∈ hom(A, T ) ∼ = Tˇ /|A|Tˇ ∼ = (Z/|A|)d . Choose integer lifts

m ¯ = (m ¯ 1, . . . , m ¯ d ) ∈ Tˇ ∼ = Zd .

Choose a¯ = 2πi

` k + 2πi τ n n

so that a=a ¯ + Λ. In terms of these choices, the stalk of the orientation at (g, h) is built from the function k k ¯ exp(2πi φ(m)τ ¯ )Θ(x + m ¯ ⊗a ¯) F (x) = exp(2πi I(x, m)) n n The essential feature of F is that the functional equation (7.2) satisfied by Θ implies that F is Weyl invariant and independent of the choice of preimage m, ¯ and its dependence on a ¯ is under control. Now consider the exponential f (6.7) associated to the two-variable elliptic genus. Comparison of its functional equation (6.10) with the functional equation (7.2) for Θ suggests that we should build the orientation for the two-variable genus from the simpler function Y k X F (x) = exp(2πi z m ¯ j) f (xj + 2πim ¯ j `/n + 2πiτ m ¯ j k/n). (7.3) n j

The argument at the end of §6 shows that indeed, this F is independent of the choice of lift m. ¯ In fact, the simple transformation rule (6.10) for f implies that F is also independent of the choice of representative a¯ for a.

Now use this F to write down a class Ellan (M, G) ∈ EG (M) following the instructions in [And03]. (In general one gets a section of the cohomology of the Thom space, but the Thom isomorphism in ordinary cohomology defined by f identifies this with EG (M)). More precisely, the formula for the value in the stalk at a ∈ C[n] is the one for γa before Lemma 8.12, taking V 0 to be

Discrete torsion and the sigma genus

19

trivial and θ 0 to be 1. That formula refers to an expression R which is defined in terms of F in Lemma 5.28. (The expression for R also includes a product of σ functions, which should be replaced with the corresponding product of f ’s). Theorem 7.4. With these substitutions, the value of Ell an (M, G) in the stalk at (g, h) is the class in H(EG ×G M (g,h) ) whose restriction to H(BA × M (g,h) ) is the integrand in the summand Φg,−h of the orbifold two-variable elliptic genus (see (6.9)). With the remarks so far, the proof is straightforward. We omit the details, except note that if (g, h) = (`/n + Z, k/n + Z), then in (6.9), (Ai , Bi ) can be taken to run over the set (`m ¯ i , km ¯ i ). Thus the typical factor in (6.9) can easily be seen to identify with the typical factor in (7.3). The genus associated to Ellan (M, G) is the global section of OC[n] whose value at (g, h) is Z Ellan (M, G)g,h . M (g,h)

Summing over (g, h) ∈ G, we get exactly |G| times the orbifold two-variable elliptic genus of [DMVV97, BL02]. 8. The cocycle We now return to the orbifold sigma genus, and study the effect of varying the BU h6i structure. The fibration of infinite loop spaces K(Z, 3) → BU h6i → BSU gives a map of E∞ ring spectra i

Σ∞ K(Z, 3)+ → − MU h6i. If C = (E, C, t) is an elliptic spectrum, then the sigma orientation σ(C) : MU h6i → E gives rise to a map of ring spectra def

w(C) = σ(C)i : Σ∞ K(Z, 3)+ → − E. In particular, w(C) is a Thom class for the trivial bundle over K(Z, 3), and so it is a unit of E 0 K(Z, 3). If α = (g, h) : Λ → G, then we define def

δn (α) = δn (u, C, α) = Ξg,h (u∗ w(C)) ∈ D × .

(8.1)

20

Ando and French

Lemma 8.2. If α

α0 = Λn+1 → Λn − → G, then δn+1 (α0 ) = δn (α),

(8.3)

and so we have a well-defined unit δ(g, h) ∈ D × . It satisfies δ(g, h)n = 1 δ(h, g) = δ(g, h)−1 δ(g + g 0 , h) = δ(g, h)δ(g 0, h) δ(g, h + h0 ) = δ(g, h)δ(g 0, h) for any g, g 0 , h, and h0 in G for which these equations make sense. Proof. The arguments for the various claims are similar to each other; as an illustration we show that δ is exponential in the first variable, and for simplicity we suppose that G is abelian. The proof in the general case will be given in §10. Let C be the cyclic group of order n. The universal example of an abelian group with three elements (g1 , g2 , h) of order n is C 3 . Let (g1 ,h)

C 2 −−−→ C 3 (g2 ,h)

C 2 −−−→ C 3 (g1 +g2 ,h)

C 2 −−−−−→ C 3 be the maps which represent the selection of the indicated pairs elements formed from the triple (g1 , g2 , h). It suffices to show that for every homotopy class u : BC 3 → − K(Z, 3), the outside rectangle of the diagram BC 2 ∆y

(g1 +g2 ,h)

−−−−−→

(g1 ,h)×(g2 ,h)

BC 3

u

−−−→

u×u

K(Z, 3) x

σ(C)

−−−→

σ(C)×σ(C)

E x

BC 2 × BC 2 −−−−−−−→ BC 3 × BC 3 −−−→ K(Z, 3) × K(Z, 3) −−−−−−→ E × E (8.4)

21

Discrete torsion and the sigma genus

commutes. The right-side rectangle commutes, because σ(C) is a map of ring spectra. To show that the left-side rectangle commutes, consider the diagram BC 2 ∆y

(g1 +g2 ,h)

−−−−−→

BC 3

(g1 ,h)×(g2 ,h)

v

−−−→

v×v

K(C, 2) x

β

−−−→

β×β

K(Z, 3) x

BC 2 × BC 2 −−−−−−−→ BC 3 × BC 3 −−−→ K(C, 2) × K(C, 2) −−−→ K(Z, 3) × K(Z, 3). (8.5) Once again, the right square commutes, this time because the Bockstein is an additive group homomorphism. An easy calculation shows that the Bockstein β : H 2 (BC 3 ; C) → H 3 (BC 3 ; Z) is surjective, so there is a v such that βv = u. Moreover H 2 (BC 3 ; C) ∼ = C{µ12 , µ13 , µ23 }, where µij : BC 3 ∼ = K(C, 1)3 → K(C, 2) is the map which represents the natural transformation µij : H 1 (X; C)3 → H 2 (C; C) given by µij (x1 , x2 , x3 ) = xi ∪ xj . If v = aµ12 + bµ13 + cµ23 , then the top row of the diagram represents the natural transformation H 1 (X; C)2 → − H 3 (X; Z) given by (x, y) 7→ β(ax ∪ x + bx ∪ y + cx ∪ y), while the other (i.e. counterclockwise) composition represents the natural transformation (x, y) 7→ β(bx ∪ y + cx ∪ y). These coincide since β(x ∪ x) = 0. In other words, the diagram (8.5) commutes for any v, and so the diagram (8.4) does too, as required.

22

Ando and French

8.1. The Weil pairing. As an example, let’s consider the case that G = Λ, and u : BG → K(Z/N, 2) is the map representing the cup product: indeed u is a generator of E 0 BG. A homomorphism α:Λ→G gives rise to a homomorphism G∗ → − Λ∗ → C[N ]. Thus we may view the homomorphism α = (g, h) as a pair of N -torsion points of C[N ]. The argument of [AS01] then shows Lemma 8.6. δ(u) is the Weil pairing of the elliptic curve C.

9. Discrete torsion We are now ready to state our basic formula. Theorem 9.1. If α = (g, h) : Λn → G, then Ξα σ(M, ` + ιuπ, C)G = δ(u, C, α)Ξα σ(M, `, C)G , and so abbreviating δ(g, h) = δ(u, C, α), we have X σorb (M, ` + ιuπ, C)G = δ(g, h)Ξg,h σ(M, `, C)G ∈ π−d E. gh=hg

Proof. Since σ(C) : MU h6i → E is a map of ring spectra, and since the multiplication on MU h6i arises from the addition on BU h6i, we have U (M, ` + ιuπ, C)G = U (M, `, C)G U (M, ιuπ, C)G = U (M, `, C)G π ∗ u∗ w(C). Recall that E 0 (VG − TG ) is an E 0 (MG )-module, and so an E 0 (BG)-module via E 0 (π). As such the Pontrjagin-Thom map τ (V )G : E 0 (VG − TG ) → E 0 (BG) is a homomorphism of E 0 (BG)-modules. It follows that σ(M, ` + ιuπ, C)G = τ (V )G (U (M, ` + ιuπ, C)G ) = τ (V )G (U (M, `, C)G π ∗ u∗ w(C)) = u∗ w(C)σ(M, `, C)G in E ∗ (BG). If α : Λn → G,

Discrete torsion and the sigma genus

23

then applying Ξα gives Ξα σ(M, ` + ιuπ, C)G = Ξα u∗ w(C)Ξα σ(M, `, C)G = δn (u, C, α)Ξα σ(M, `, C)G . as required.

10. The non-abelian Case

In this section, we prove Lemma 8.2 in the case that G is non-abelian. We fix an n sufficiently large that |G| divides n. We first construct an isomorphism H 3 (BΛn ) ∼ = Z/n. Consider the following commutative diagram, where the columns are universal coefficient exact sequences. Ext1 (H1 (BΛn ); Z) /

Ext1 (H1 (BΛn ); Z/n)

H 2 (BΛn ) /

H 2 (BΛn ; Z/n)

Hom(H2 (BΛn ); Z) /

/ H 3 (BΛ ) n l l l l l∼ l = ul

Hom(H2 (BΛn ); Z/n).

Here, the middle row is part of Bockstein long exact sequence. Note that this is a short exact sequence since multiplication by n kills group cohomology of Λn . Since H2 (BΛn ) is torsion, Hom(H2 (BΛn ); Z) = 0. We therefore obtain the dotted arrow. Since Ext1 (H1 (BΛn ); −) is right exact,the top map is a surjection. It follows that the dotted arrow is an isomorphism. Now it is not hard to check that e1 ⊗ e2 − e2 ⊗ e1 is a generator for H2 (BΛn ) ∼ = Z/n. Composing the dotted arrow above with evaluation on this generator yields an isomorphism H 3 (BΛn ) ∼ = Z/n. Definition 10.1. If u is an element in H 3 (BG) and α = (g, h) : Λn → G is any map, then we obtain a class α∗ u ∈ H 3 (BΛn ). Let (g, h) = nu (g, h) be the image of this class in Z/n under the isomorphism above. Remark 10.2. It is easy to check that (g, h) = u˜(g, h) − u ˜(h, g) where u ˜ : G × G → Z/n is a 2-cocycle whose cohomology class maps to u under the Bockstein. Lemma 10.3. Whenever the expressions are defined, the following properties hold. (g, h) = −(h, g)

24

Ando and French (h, j) − (gh, j) + (g, hj) − (g, h) = 0 (gg 0 , h) = (g, h) + (g 0 , h) (g, hh0 ) = (g, h) + (g, h0 )

Proof. The first and second properties follow easily from the remark. For the third property, it suffices to show that u ˜(gg 0 , h) − u ˜(h, gg 0) − u ˜(g, h) + u ˜(h, g) − u ˜(g 0 , h) + u ˜(h, g 0 ) is zero. Since u ˜ is a cocycle, we may rewrite the expression using the following equations: u ˜(gg 0 , h) − u˜(g 0 , h) = u ˜(g, g 0 h) − u ˜(g, g 0), −˜ u(h, gg 0 ) + u ˜(h, g) = u ˜(g, g 0 ) − u ˜(hg, g 0 ), −˜ u(g, h) + u ˜(h, g 0 ) = u ˜(gh, g 0 ) − u˜(g, hg 0 ). Then, canceling terms, we get u ˜(g, g 0 h) + u ˜(gh, g 0) − u ˜(hg, g 0) − u ˜(g, hg 0 ). Since gh = hg and g 0 h = hg 0 , this is zero as needed. The last property follows similarly, or directly from the first and third. Proof of Lemma 8.2. Let F : BΛn → K(Z, 3) represent the element 1 ∈ Z/n ∼ = H 3 (BΛn ) under the isomorphism above. Let x ∈ D be the image of F ∗ w(C) ∈ E 0 BΛn under the map tautological map E 0 BΛn → − Dn → − D. It is easy to check from the definitions (8.1) and (10.1) of δ and that δ(g, h) = xg,h . The result now follows from Lemma 10.3.

References [AB02] [AHS01]

[AHS03] [And03]

Matthew Ando and Maria Basterra. The Witten genus and equivariant elliptic cohomology. Mathematische Zeitschrift, 240(4):787–822, 2002. Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. Elliptic spectra, the Witten genus, and the theorem of the cube. Inventiones Mathematicae, 146:595–687, 2001. DOI 10.1007/s002220100175. Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. The sigma orientation is an H∞ map. Amer. J. Math., To appear. Matthew Ando. The sigma orientation for analytic circle-equivariant elliptic cohomology. Geometry and Topology, 7:91–153, 2003. math.AT/0201092.

Discrete torsion and the sigma genus

25

[AS68]

M. F. Atiyah and G. B. Segal. The index of elliptic operators. II. Ann. of Math. (2), 87:531–545, 1968. [AS01] M. Ando and N. P. Strickland. Weil pairings and Morava K-theory. Topology, 40(1):127–156, 2001. [BL02] Lev A. Borisov and Anatoly Libgober. Elliptic genera of singular varieties, orbifold elliptic genus and chiral de Rham complex. In Mirror symmetry, IV (Montreal, QC, 2000), volume 33 of AMS/IP Stud. Adv. Math., pages 325–342. Amer. Math. Soc., Providence, RI, 2002. [BM94] J.-L. Brylinski and D. A. McLaughlin. The geometry of degree-four characteristic classes and of line bundles on loop spaces. I. Duke Math. J., 75(3):603–638, 1994. [BT89] Raoul Bott and Clifford Taubes. On the rigidity theorems of Witten. J. of the Amer. Math. Soc., 2, 1989. [DMVV97] Robbert Dijkgraaf, Gregory Moore, Erik Verlinde, and Herman Verlinde. Elliptic genera of symmetric products and second quantized strings. Comm. Math. Phys., 185(1):197–209, 1997. [Dri74] V. G. Drinfeld. Elliptic modules. Math. USSR-Sb., 23(4):561–592, 1974. [EOTY89] H. Eguchi, H. Ooguri, A. Taormina, and S.-K. Yang. Superconformal algebras and string compactification on manifolds with SU (N ) holonomy. Nucl. Phys. B, 315, 1989. [GMS00] Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman. Gerbes of chiral differential operators. Math. Res. Lett., 7(1):55–66, 2000. math.AG/9906117. [HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. Generalized group characters and complex oriented cohomology theories. J. Amer. Math. Soc., 13(3):553–594 (electronic), 2000. [Hop95] Michael J. Hopkins. Topological modular forms, the Witten genus, and the theorem of the cube. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), pages 554–565, Basel, 1995. Birkh¨ auser. [Kri90] Igor Moiseevich Krichever. Generalized elliptic genera and Baker-Akhiezer functions. Mat. Zametki, 47(2):34–45, 158, 1990. [LU] Ernesto Lupercio and Bernardo Uribe. Deligne cohomology for orbifolds, discrete torsion, and B-fields. hep-th/0201184. [Ros01] Ioanid Rosu. Equivariant elliptic cohomology and rigidity. Amer. J. Math., 123(4):647–677, 2001. [Sha] Eric Sharpe. Recent developments in discrete torsion. hep-th/0008191. [Str97] Neil P. Strickland. Finite subgroups of formal groups. J. Pure and Applied Algebra, 121:161–208, 1997. [Vaf85] C. Vafa. Modular invariance and discrete torsion. Nuclear Physics B, 261:678– 686, 1985. [Wit87] Edward Witten. Elliptic genera and quantum field theory. Comm. Math. Phys., 109, 1987. E-mail address: [email protected] E-mail address: [email protected] Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana IL 61801, USA

QUATERNIONIC ELLIPTIC OBJECTS AND K3-COHOMOLOGY JORGE A. DEVOTO Abstract. We shall outline a research program which aims to provide a geometric approach to K3-cohomology. We define for this a quaternionic analogue of Segal’s elliptic objects.

1. Introduction The relation between one dimensional formal group laws and complex oriented generalised cohomology theories, which is a consequence of Quillen’s theorem [19, 20] which relates Lazard’s ring – the ring over which is defined the universal formal group law – and the coefficient ring of complex cobordism, has been a very fruitful source of research in algebraic topology. Among one dimensional formal group laws three have a direct geometric interpretation. They are obtained from the additive group, the multiplicative group and elliptic curves respectively by taking completion at the origin. The corresponding cohomology theories, Ordinary cohomology, K-theory and Elliptic cohomology also have geometric interpretations 1. These interpretations have extended their use, and helped to find applications of them, to other fields of mathematics and physics besides algebraic topology. No other one dimensional commutative formal group law is associated to a group in this direct way. However M. Artin and B. Mazur [2] showed that one can associate formal groups to certain algebraic varieties – among them K3 surfaces – in a way which generalises the case of elliptic curves. M. Hopkins used their work to define an associated generalised cohomology theory: K3cohomology. The geometric origin of the formal group laws associated to K3 surfaces gives a reason to hope that there exists a geometric definition of K3 cohomology. There is some evidence that the K3-cohomology of a manifold M should be related to some form of K-theory on the double loop space L (2) M = C ∞ (S 1 × S 1 , M) of M. This relation should be analogous to the relation between the elliptic cohomology of M and K-theory of the loop space of M. If this belief is true, then it is possible that one can find some interesting Date: 1st September 2003. 1991 Mathematics Subject Classification. 55N35, 14J81. Key words and phrases. K3 cohomology, K3 surfaces, Elliptic objects, Quaternionic Geometry. 1 Conjectural in the case of elliptic cohomology

26

Quaternionic elliptic objects and K3-cohomology

27

applications of K3 cohomology to other areas of mathematics and physics, as in the case of K-theory, Ordinary cohomology and Elliptic cohomology. The aim of the present article is to outline a research program whose purpose is to give a geometric interpretation of K3 cohomology. This interpretation is based on an appropriate modification of G. Segal’s elliptic objects. The main difference is that in the case of K3-cohomology the relevant geometry is defined over the quaternions rather than the complex numbers, as in the case of elliptic objects. We expect that the geometric interpretation will be related to: • Dirac operators on double loop spaces. • Representation theory of double loop groups. • Representation theory of DiffΩ (S 1 ×S 1 ) where Ω is a fixed volume form and DiffΩ (S 1 × S 1 ) is the group of diffeomorphisms which preserve Ω. • Siegel modular forms. • Classical and quantum gravity. • D-Branes. There are few published references for K3-cohomology. We refer the reader to [27, 28] for an exposition of different aspects of K3 cohomology. This project is a joint project with Charles Thomas and is a pleasure to express my gratitude to him for sharing with me his insight and for his continuous support over the years. I want also to express my gratitude to the funders of the Newton Institute for the invitation to be in the conference. Finally I want to thank Haynes Miller and the referee for their suggestions which greatly helped to improve the present manuscript.

2. Definitions of K3-cohomology 2.1. Formal groups and algebraic varieties. In this section we will describe the work of Artin and Mazur [2] which is at the basis of the definition of K3 cohomology. Let Gm denote the multiplicative group scheme. If X is an algebraic curve defined over a field k, then applying Gm to the sheaf O of regular functions on X and taking identity component of the first cohomology group H 1 (X, Gm ) one obtains Pic(X), the Picard group of X. If X is an elliptic curve and j = 1 the identity component of Pic(X) is isomorphic to X. One can ask if it is possible to obtain in this way the formal completion of the Picard group, which in the case of elliptic curves is the relevant formal group for elliptic cohomology. A positive answer, and a generalisation of this construction, was obtained by Artin and Mazur [2]. Let X be a variety defined over a field k. If A is a local Artin k-algebra with spectrum Spec A, then XA = X ×Spec k Spec A can be thought as an infinitesimal thickening of X which can be used to study deformation theory. The inclusion Spec k → Spec A induces a morphism X → XA .

28

Jorge A. Devoto

Consider now, for each natural number j, the functor from the category of Artin local algebras over k to the category of groups defined by: (2.1)

A → ker{H j (XA , Gm ) → H j (X, Gm )}.

Cohomology in this case means etale cohomology. One can ask if this functor is prorepresentable by some formal object. If X is a X and j = 1, then the answer is yes and the representing object is the formal completion of Pic(X). Artin and Mazur showed that the answer is also yes in other cases. 2.2. K3-Surfaces. Let us consider Artin and Mazur’s construction for the case j = 2. They showed that if H 1,0 (X) = H 0,1 (X) = 0, then the functor (2.1) is prorepresented by a smooth formal group, defined over Spec k, of dimension dim H 2,0 (X). The representing object is called the formal Brauer group. This group has a natural coordinatisation – see Theorem 4.9 and Corollary 4.10 of [25] – and hence it has an associated formal group law, which is the real object of interest in K3-cohomology. We shall call this formal group c law the formal Brauer group law and we shall denote it by Br(X).

Definition 2.1. A K3-surface X is a compact surface with h1,0 (X) = 0 and trivial canonical bundle. If X is a K3-surface, then the Hodge diamond has the following form: 1 0 0 1 20 1 0 0 1. From this it follows that the formal Brauer group has dimension one. 2.2.1. Examples of K3 surfaces. Example 2.2. The Fermat quartic S = {[z0 , z1 , z2 , z3 ] ∈ CP3 : z04 + z14 + z24 + z34 = 0} Brauer’s formal group law is defined over Z and has as logarithm X (4m)! x4m+1 log(x) = . 4 4m + 1 (m!) m≥0 Example 2.3. Any nonsingular quartic in CP3

Example 2.4. A complete intersection of a cubic and a quadric in CP4 Example 2.5. Let Λ be a lattice in C2 . Then the quotient T = C2 /Λ is a 4-torus. Define a map σ : T → T by σ : (z1 , z2 )+Λ → (−z1 , −z2 )+Λ. Then σ defines an action of Z2 which fixes the 16 points {(z1 , z2 ) + Λ | (z1 , z2 ) ∈ 12 Λ}. The quotient space by this action T/σ is a singular complex orbifold with 16 singular points. Let K be the blow-up of T/σ at the singular points. Then

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K is a K3 surface called a Kummer surface. The map T → K is called the Kummer map. If one is not working with smooth surfaces the surface T/σ is also called a Kummer surface. For more references about Kummer surfaces see [12]. 2.3. K3 cohomology. Let p be a prime. Then in characteristic p the formal c group law Br(X) is either the additive group law, of height h = ∞, or a formal group law of height h less or equal than 10.

Definition 2.6. Let X be a K3-surface defined over a field of characteristic p 6= 0. Then if h = ∞ we will say that X is supersingular.

Example 2.7. The formal Brauer group associated to the Fermat quartic is supersingular for the primes p ≡ 3 mod 4. The filtration by height induces in characteristic p a stratification M1 ⊃ M2 ⊃ · · · ⊃ M10 ⊃ M11 = M∞ of the moduli space of K3 surfaces. In this case M∞ corresponds to supersingular surfaces. The strata M(i) corresponding to K3 surfaces of height h = i are, for i < 11 regular and locally complete intersection of codimension h − 1 [29, Thm 15.1]. From Theorem 8.21A and Proposition 8.23 of [11] it follows that if X is a K3 surface which is not supersingular, then the associated formal group law satisfies Landweber exactness criterion reduced modulo p. One could study different theories associated to this construction. The first one is defined in the spirit of elliptic cohomology, trying to consider c a universal example.The construction X → Br(X) induces a map b from the moduli space of K3 surfaces to the space F of formal group laws. If MU (X) denotes the complex cobordism of a space X, then one has a sheaf over F associated to the MU -module structure of MU (X). Here MU is identified with Lazard’s ring. One can use b to pull-back this sheaf of modules associated to the moduli space of K3 surfaces. This corresponds to consider a cohomology theory associated to a generic K3 surface. A second approach is to study theories associated to certain families of K3 surfaces, like Kummer surfaces. We hope that this approach will produce a theory with a geometric model. There is a second approach to K3 cohomology – see [28] – inspired by the h8i work of M. Kreck and S. Stoltz in elliptic cohomology [16]. Let Ω∗ denote the bordism ring of manifolds with w1 (M) = w2 (M) = 12 p1 (M) = 0. Let h8i I ⊂ Ω∗ be the ideal generated by elements of the form [E] − [OP2 ][M], where E → M is a fibration. The fibre OP2 , is the Cayley projective plane, defined for example in [5, 28]. The structural group is the exceptional Lie group F4 which is the isometry group of OP2 . Define then the functor M → h8i h8i Ω∗ (M)/IΩ∗ (M). Is this functor, may be after inverting some primes, a generalised homology theory? How is it related to the previously mentioned construction?

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Jorge A. Devoto 3. Elliptic Objects

The physically motivated heuristic relation between elliptic cohomology and Dirac operators on loop spaces – see for example [31, 32] – provided evidence that there exists a geometric definition of elliptic cohomology related to some form of Diff+ (S 1 ) equivariant K-theory on free loop spaces. The group Diff+ (S 1 ) is the group of orientation preserving diffeomorphisms of the circle and the action on the loop spaces is by reparametrisation of loops. G Segal proposed a definition of elliptic objects and conjectured in [22] that they might be used to give the sought for geometric construction of elliptic cohomology. Elliptic objects are closely related to Segal’s axiomatic formulation of conformal field theory [23]. Conformal field theories are defined as functors from a cobordism category C to suitable topological vector spaces. We shall describe now Segal’s definitions. Then in the next section we shall generalise the category C and later we shall consider some generalisations of the functors involved in conformal field theory. Let C be the category which has an object set {C0 , C1 , . . .}. The set C0 = {∅} and for each n ≥ 0 the set {Cn } contains all the 1-dimensional manifolds formed by the disjoint union of n parametrised circles. A morphism Σ : Cm → Cn is a Riemann surface Σ with boundary ∂Σ together with an orientation preserving identification Cn t Cm → ∂Σ. Here Cm means Cm with the orientation reversed. The circles in Cm are called incoming and the e are equivalent if there is circles in Cn are called outgoing. Two surfaces Σ, Σ e which is analytic in the interiors of the surfaces a diffeomorphism f : Σ → Σ and respects the identifications of the boundary. Composition of morphisms e is defined by sewing surfaces together along the boundary. The set Σ◦Σ Cm,n of morphisms Σ : Cn → Cm is a topological space with one connected component for each topological type of surface Σ. Segal showed that one can define a structure of holomorphic manifold on Cm,n and that composition Ck,m × Cm,n → Ck,n is a holomorphic map [23, Section 4]. There are a number of natural operations which can be performed on C . (1) The symmetric groups Sm and Sn act on Cm,n by permuting the numbering of the circles. (2) If Σ ∈ Cm,n , then the complex conjugated surface Σ ∈ Cn,m and Σ → Σ is an antiholomorphic map. (3) By reversing the orientation of various boundary circles one can obtain several holomorphic crossing maps Cm,n → Cp,q , with m + n = p + q. (4) By sewing k incoming circles to k outgoing one obtains a holomorphic map Cm,n → Cm−k.n−k which is referred to as collapsing. Definition 3.1. Let H be a topological vector space with a hermitian form and a given real structure induced by an anti-involution H → H . A conformal field theory based on H is a continuous functor U from C to the category of topological vector spaces with the following properties:

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(1) For U (∅) = C and U (Cn ) = H ⊗n for n ≥ 1. ⊗m ⊗ H ⊗n . The image (2) For each Σ ∈ Cm,n the element U (Σ) ∈ H of the natural homomorphism H ⊗ H → End(H ) consists of trace class operators [24]. Therefore the element U (Σ) can be identified with a trace class operator. (3) The map Cm,n × H m → H n is compatible with the action of the symmetric groups Sm and Sn . (4) U (Σ) = U (Σ)∗ for all morphisms Σ. This property is termed reflection positivity. ⊗m ⊗ (5) The crossing maps Cm,n → Cp,q are compatible with the maps H ⊗p ⊗n ⊗q H →H ⊗H induced by the real structure of H . (6) The collapsing maps Cm,n → Cm−k.n−k are compatible with the maps ⊗m ⊗(m−k) H ⊗H ⊗n → H ⊗H ⊗(n−k) induced by the hermitian structure. Remark 3.2. The set A of equivalence classes of morphisms Σ : C1 → C1 which are topologically equivalent to an annulus form a semigroup. Segal shows that A plays the role of a complexification of Diff+ (S 1 ). As a consequence of the definition of a conformal field theory it follows that the space H carries a representation of Diff+ (S 1 ) which can be extended to a representation of A . A first step in the construction of a conformal field theory is to understand which representations of Diff+ (S 1 ) can be extended to representations of A . These representations are called positive energy representations. ∂ be the tangent vector field on S 1 which generates the rigid rotations. Let ∂θ ∂ Then ∂θ defines an operator on H . A representation of Diff+ (S 1 ) is a positive ∂ energy representation if H = −i ∂θ is a self-adjoint operator with spectrum bounded from below. Proposition 3.3. [23, Proposition 3.1] There is a one to one correspondence between positive energy representations of Diff+ (S 1 ) and holomorphic (contraction) representations of A . Remark 3.4. let Σ ∈ A and let U be a conformal field theory based on the space H . Let Σ be the complex torus obtained by sewing the incoming and outgoing circles. This torus can be seen as a morphism Σ : ∅ → ∅. In physical terms a vacuum to vacuum transition. The last condition in the definition of conformal field theory implies that the number U (Σ) is the trace of the operator U (Σ). The torus Σ has a distinguished point given by the image of 1 ⊂ S 1 which induces the structure of elliptic curve on it. The assignment Σ → U (Σ) for all Σ ∈ A defines a modular function. In this way the theory of elliptic objects gives a geometric explanation of the appearances of elliptic curves and modular forms in elliptic cohomology. Let M be a manifold. An elliptic object with target M is a modification of the definition of a conformal field theory which reflects some of the properties

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of a sigma model with target M. One considers the category C (M) with objects pairs {Cn , f }, where Cn is an object of C and f : Cn → M is a smooth map. A morphism (Σ, H) : (Cm , f ) → (Cn , g) is a pair formed by a morphism Σ : Cm → Cn and a smooth function H : Σ → M which restricts on the boundary, via the corresponding identifications, to f and g respectively. An elliptic object is a functor U from C (M) to topological vector spaces satisfying some conditions similar to Definition 3.1. One can relate elliptic objects to vector bundles on L M in the following way: The vector space U (S 1 , f ) can be seen as the fibre of a vector bundle E → L M over the element f ∈ L M. The linear transformations U (Σ, H) associated to morphisms (Σ, H), where Σ ∈ A , can be seen as the holonomy of a connection on E over the path H(Σ) in L M. 4. The categories T

2

and T 3

If one tries to mimic Segal’s definition of elliptic objects in order to define objects related to vector bundles over double loop spaces, then the natural thing to do is to consider a cobordism category T 2 whose objects are a finite disjoint union of oriented manifolds diffeomorphic to the standard two torus T = S 1 × S 1 . The next logical step would be to understand which are the morphisms in T 2 . They should be 3-dimensional manifolds Σ with some extra structure. If this construction is going to be related to K3 cohomology, then the geometry of the morphisms should give natural explanations of the role played by K3-surfaces in a way analogous to Remark 3.4. There is a clear problem with this approach. Morphisms in T 2 are manifolds of real dimension 3 instead of 4, which is the real dimension of K3-surfaces. We shall consider for this reason two different categories at the same time. One will be T 2 . A notion of elliptic objects based on this category will be related to bundles on the double loop spaces. The other, called T 3 will be explain the appearance of K3 surfaces. Both categories are related by a functor which we call the change of dimensions functor. 4.1. The category T 3 . Let T3 = S 1 × S 1 × S 1 be the standard torus equipped with the volume form Ω = dθ1 ∧ dθ2 ∧ dθ3 . A parametrised torus is a pair (M, ΩM ) formed by a manifold M with a fixed diffeomorphism f : T → M and a volume form ΩM such that f ∗ ΩM = Ω. For each n ≥ 0 the category T 3 has a space of objects Tn3 . For n > 0 the objects are pairs (M, Ω), where M is the disjoint union of n copies of parametrised tori. In the case n = 0 it has only one object which is the empty set. The morphisms Σ : (M1 , Ω1 ) → (M2 , Ω2 ) are equivalence classes of 4-manifolds with a hyperk¨ ahler structure smooth up to the boundary together with a volume preserving diffeomorphism ∂Σ ' M 1 t M2 . Two hyperk¨ahler manifolds Σ1 and Σ2 representing morphisms (M1 , Ω1 ) → (M2 , Ω2 ) are equivalent if there exists a diffeomorphism f : Σ1 → Σ2 analytic in the interior of the manifolds ,

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which preserves the hyperk¨ahler structure. Composition of morphisms is obtained by gluing manifolds along the boundary. The reasons for this choices are explained in the next sections. 4.2. Hyperk¨ ahler Geometry. Let X be a 4n dimensional manifold. An almost hyperk¨ ahler structure on X is a quadruple (J1 , J2 , J3 , g) where Ji are anticommuting almost complex structures on X with J1 J2 = J3 and g is a Riemannian metric on X which is hermitian with respect to J1 , J2 and J3 . The family of complex structures Ji defines a H-module structure in each of the tangent spaces Tx X. The form g is called a hyperhermitian form. An almost hyperk¨ahler structure on X is called hyperk¨ ahler if in addition ∇Ji = 0, i = 1, 2, 3, where ∇ is the Levi-Civita connection of g. Hyperk¨ahler structures are a quaternionic analogue of K¨ahler structures. The basic examples of hyperk¨ahler manifolds are the quaternionic n-dimensional spaces Hn . Identify Hn = R4n using coordinates (q1 , . . . qn ) with qj = x0j + x1j J1 + x2j J2 + x3j J3 , where J1 = i, J2 = j, J3 = k are the usual quaternionic units. Introduce complex coordinates (z1 , . . . , z2n ) by z2j−1 = x0j + ix1j and P z2j = x2j + ix3j then the form qj q j defines a metric g and three symplectic forms ωi , i = 1, 2, 3 by g=

2n X j=1

2n

2

|dzj | ,

iX ω1 = dzj ∧ dz j , 2 j=1

ω2 + iω3 =

n X

dz2j−1 ∧ dz2j .

j=1

The forms g and ωi , i = 1, 2, 3 can be defined in any almost hyperk¨ahler manifold. The next result gives different characterisations of hyperk¨ahler manifolds. Proposition 4.1. [13, Proposition 7.1.2] Suppose that X is a 4n manifold and that (J1 , J2 , J3 , g) is an almost hyperk¨ ahler structure on X. Let ω1 , ω2 , ω3 be the hermitian forms of J1 , J2 , J3 . Then the following conditions are equivalent (1) (J1 , J2 , J3 , g) is a hyperk¨ ahler structure, (2) dω1 = dω2 = dω3 = 0, (3) ∇ω1 = ∇ω2 = ∇ω3 = 0, where ∇ is the Levi-Civita connection of g. (4) The holonomy of ∇ is contained in Sp(n) and J1 , J2 , J3 are the complex structures induced by the holonomy. In the category T 3 closed hyperk¨ahler manifolds X appear as a vacuum to vacuum transition X : ∅ → ∅ and one can expect that they will be related to the partition function in any quantum field theory defined on T 3 . In this way, due to the next result, one can see that smooth K3-surfaces play a natural and essential part of the theory. Theorem 4.2. [13, Thm 7.3.13] Let (X, J) be a K3 surface. Then each K¨ ahler class on X has a unique metric with holonomy SU (2) = Sp(1). Conversely any compact Riemannian 4 manifold (X, g) with holonomy SU (2)

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admits a covariantly constant complex structure J such that (X, J) is a K3 surface. 4.3. The semigroup of tubes. Let M be the set of morphisms Σ : (T3 , Ω) → (T3 , Ω) which are topologically tubes T3 × [a, b]. This set forms a semigroup which is an analogue of the semigroup A considered by Segal. We shall presents some arguments that show that M could be considered as a quaternionification of the volume preserving diffeomorphisms of T3 . In conformal field theory the diffeomorphisms of a circle arise in relation to Lorentz invariance in a two dimensional Mikowskian space S 1 × R where the space is the circle [23, Section 1]. One of the reasons for the choice of the volume preserving diffeomorphisms of a three dimensional manifold T3 as the relevant group of symmetry is that we want to make contact with the work of Ashtekar [3] which relates this group to general relativity, in particular to self dual solutions of Einstein equations on T3 × R 2. The arguments are based on a work of S. Donaldson which we will briefly describe now. Let M be a three dimensional compact closed manifold with a fixed volume form Ω and let DiffΩ (M) be the Lie group of diffeomorphisms of M preserving the form Ω. Donaldson defined in [8] a set of manifolds equipped with extra geometric structures which could be used as a the elements of a semigroup SM . He suggested that the semigroup SM plays the role of a complexification of DiffΩ (M). In particular he showed that the natural adjoint representation of DiffΩ (M) on its Lie algebra VectΩ (M), consisting of divergence free vector fields, has a natural extension to an action of SM on VectΩ (M)⊗C. The basis of his construction is the following definition. Let G be a real Lie group, with Lie algebra g, which acts on a real vector space V . We shall say that two elements U + , U − of V ⊗ C are GC -related if there are: (1) An element g ∈ G. (2) Smooth one parameter families U1 (t), U2 (t) of elements of V parametrised by [−a, a] such that U1 (−a) + iU2 (−a) = U − , U1 (a) + iU2 (a) = gU + . (3) A smooth one parameter family α(t) of elements of g such that d (4.1) (U1 + iU2 ) = iα(t)(U1 + iU2 ). dt Remark 4.3. When G has a complexification GC , then two elements U + and U − of a complex representation V of G are GC related if and only if they are in the same GC orbit of the natural extension of the action of G to GC . Remark 4.4. If the case G = Diff+ (S 1 ) then two elements U + and U − are GC related if and only if there is an element A in Segal’s semigroup of annuli 2There

are also some other contexts in physics where this group appears as the natural generalisation of Diff+ (S 1 )

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35

such that AU − = U + . What makes Segal’s construction special is that in this case there is an intrinsic representation of this construction in terms of complex structures on the annuli. 4.3.1. Nahm’s equations. Nahm’s equations are a system of ordinary differential equations for three smooth one parameter families Ti (t) with i = 1, 2, 3 of elements of a real Lie algebra g. The equations are: dT1 (4.2) = [T2 , T3 ], dt dT2 = [T3 , T1 ], dt dT3 = [T1 , T2 ], dt where t ∈ [−a, a] for some a ∈ R. One can combine the first two equations into a single equation d (4.3) (T1 + iT2 ) = [iT3 , T1 + iT2 ], dt defined for elements of the complexified Lie algebra g ⊗ C. Definition 4.5. Two elements U + , U − of elements of g ⊗ C are related by a Nahm’s solution if U + = T1+ + iT2+ , U − = T1− + iT2− , where Ti± = Ti (±a) and Ti , i = 1, 2, 3 is a solution of Nahm’s equations. A solution of Nahm’s equation defines a pair of GC -related elements in the adjoint representation. Take U1 = T1 , U2 = T2 and α = T3 . When G is compact the converse holds. Two elements of g ⊗ C are GC related if and only if they are related by a Nahm’s solution, see for example [7] for the case of the unitary groups. The result has been extended to other groups by Saskida [21]. 4.3.2. The geometry behind Nahm’s equations. The essential idea in Donaldson’s complexification of DiffΩ (M) is to find an intrinsic geometric characterisation of solutions of Nahm equations. Recall that a symplectic form in a 2n dimensional manifold M is a nondegenerate closed two form ω. That ω is non-degenerate means that ω n is a volume form. Definition 4.6. A complex symplectic surface Σ is a smooth four dimensional manifold with a complex structure and a holomorphic symplectic form w ∈ Ω2,0 (Σ) . In the complex symplectic context the non-degeneracy condition means that w ∧ w is a complex valued volume form. If we write w = ω1 + iω2 , then the algebraic constraints imposed by the non-degeneracy condition are (4.4)

ω1 ∧ ω1 6= 0,

ω2 ∧ ω2 6= 0,

at each point of Σ. Both forms are closed.

and ω1 ∧ ω2 = 0,

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Remark 4.7. Donaldson showed that if one have two real closed 2-forms ω1 and ω2 on a 4 manifold Σ which satisfy conditions (4.4) at each point, then they define a complex symplectic structure on Σ. Example 4.8. If X is a hyperk¨ahler manifold, then the form ω2 + iω3 defines a complex symplectic structure on X. Let M be a three dimensional manifold with a fixed volume form Ω and suppose that vi (t), i = 1, 2, 3 are three families of vector fields parametrised by t ∈ [−a, a] such that for each t the fields are divergence free and linearly independent. Suppose that also d (4.5) (v1 + iv2 ) = [iv3 , v1 + iv2 ]. dt We shall now define a complex symplectic structure on Σ = M × [−a, a]. For each t let 1 , 2 , 3 be the one forms dual to the vector fields v1 (t), v2 (t), v3 (t). Then there exists a smooth function f : Σ → R such that Ω = f 1 ∧ 2 ∧ 3 . Define then (4.6)

ω1 = f (dt ∧ 1 + 2 ∧ 3 ), ω2 = f (dt ∧ 2 + 3 ∧ 1 ).

Let w = ω1 + iω2 . An elementary algebraic computation shows that the forms ω1 and ω2 satisfy the algebraic conditions (4.4). The forms ω1 and ω2 are closed if and only if the vector fields vi (t) satisfy equation (4.5). Thus by Donaldson’s construction quoted in Remark 4.7 w defines a complex symplectic structure on M. Remark 4.9. It is an open problem to know if any complex symplectic structure on a 4 manifold Σ ' M × [−a, a] comes from a solution of Nahm’s equations. See the discussion in Donaldson’s paper. 4.4. The semigroup M as a quaternionification of DiffΩ (T3 ). We shall use some work of Atiyah and Bielawski [4] to argue that Nahm’s equations really define a quaternionification of a Lie Algebra. Then we shall show that the set of manifolds that appears in Donaldson’s work are elements of the semigroup M . 4.4.1. Algebraic Point of View. Let g be a real Lie algebra. Then in Lie theory one considers the usual complexified Lie algebra gC = g ⊗ C ' g ⊗ R2 . The elements of g ⊗ C can be identified to pairs g1 , g2 of elements of g. In the case of Segal’s theory the elements v1 + iv2 , where vi are vector fields on the circle can be seen as infinitesimal annuli so, quoting Segal, A has the right size to be a complexification of Diff+ (S 1 ). In the case of Nahm’s equations one has to work with a family of three elements T1 , T2 , T3 of g, which can be naturally identified to an element of g ⊗ R3 . Atiyah and Bielawski argued in [4] that the one should identify R3 with the imaginary quaternions I and use time t as the real coordinate so

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37

H = {t + ix1 + jx2 + kx3 }. Introduce a fourth function T0 : [−a, a] → g and consider the one form (4.7)

ω = T0 dt + T1 dx1 + T2 dx2 + T3 dx3 .

as an R3 invariant connection on H ⊗ g → H. Nahm’s equations have a natural interpretation in terms of the curvature of the connection. When g is the Lie algebra of a compact Lie group G, the natural gauge group to consider consists of G valued functions of t. Then one can set T0 = 0 using a convenient gauge transformation. In this way we can argue that Nahm’s equations are related to the quaternionification g ⊗ H of g. 4.4.2. Geometric Point of View. In Section 4.3.2 we used the vector fields v1 and v2 to define an complex symplectic form w1 , but the cyclic symmetry of Nahm’s equations implies that we can define another two complex symplectic forms w2 and w3 . Donaldson showed that these forms are compatible with a Riemannian metric in the sense that g + iw1 + jw2 + kw3 define a hyperher∂ mitian metric which is a hyperk¨ahler structure. In this case ∂t becomes the gradient flow for a harmonic time function. From this observation if follows that the manifolds defined by solutions of Nahm’s equations define elements of M . 4.5. The Category T 2 . The category T 3 seems to have many of the desired properties for a geometric model of K3 cohomology, however it has no clear relation with double loop spaces. This relation will reappear if we consider the category T 2 . Let us now define the morphisms in T 2 . Definition 4.10. A contact structure φ on a three dimensional manifold M is a one form φ with the property that at each point x ∈ M the form φ ∧ dφ defines a volume form. Definition 4.11. A 3 dimensional manifold M has a taut contact circle if there is a pair φ1 , φ2 of contact structures with the property that for each pair of real numbers λ1 , λ2 such that λ21 + λ22 = 1 the contact one form λ1 φ1 + λ2 φ2 defines the same volume form Ω = (λ1 φ1 + λ2 φ2 ) ∧ d(λ1 φ1 + λ2 φ2 ). The category T 2 has objects T which are two dimensional manifolds with fixed volume forms. Each of the connected components Tc of T is parametrised by an area preserving diffeomorphism φ : S 1 × S 1 → Tc . The relevant group is the group of area preserving diffeomorphisms of S 1 × S 1 . This group is the natural generalisation of Diff+ (S 1 ) in many physical situations, particularly 2d-gravity, W-gravity, matrix models, see [15] and references therein. A morphism Σ : T1 → T2 is a equivalence class of oriented 3-manifolds with a taut contact circle, together with an identification ∂Σ = T1 t T2 . Two manifolds Σ1 and Σ2 are equivalent if there is a diffeomorphism φ : Σ1 → Σ2 which preserves the taut contact circles and the identifications on the boundary.

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4.6. Change of Dimensions. Manifolds with a taut contact circle have been classified by Gonzalo and Geiges in [10]. They used the following result and the classification of complex surfaces. Theorem 4.12. [10] There is a one to one correspondence between 3-manifolds with a taut contact circle and four dimensional manifolds with a complex symplectic structure and a distinguished copy of S 1 . The correspondence is defined by M → M × S 1 . This results suggests that there is a functor, given by multiplication by S 1 between T 2 and T 3 . We can use this functor to extend and relate constructions in T 2 and T 3 . 5. Positive energy representations of DiffΩ (M) In the definition of quaternionic elliptic objects one would need to consider functors from T 3 to an appropriate category of topological vector spaces satisfying axioms similar to the axioms of conformal field theory. The first of these axioms implies that these spaces should carry a representation of DiffΩ (T 3 ) and the representation should extend to a representation of M . The notion of positive energy representations can be extended to DiffΩ (T), where T = S 1 × S 1 × S 1 in the following way. Begin with R3 , then the space of spinors can be identified with the quaternions H using the identification Spin(3) ' Sp(1) and the Dirac operator on R3 is (5.1)

D=i

∂ ∂ ∂ +j +k ∂x1 ∂x2 ∂x3

acting on the space of H-valued Schwartz functions S (R3 , H). If one takes the standard L2 metric it is easy to show that D is a self-adjoint operator. This results can be easily extended to spinors on T; just take periodic functions f : R3 → H. Then again D is a self adjoint operator with a discrete spectrum which acts on the Hilbert space H = L2 (T, H). Using this operator we can define a polarisation of H = H+ ⊕ H− into the spaces generated by eigenvectors corresponding to the positive and negative eigenvalues of D and e−tD restricted to H+ is a contraction operator for positive t. Definition 5.1. A representation H of DiffΩ (T) will be termed admissible if (1) H is a H-module. (2) the metric on H is hyperhermitian. (3) VectΩ (T) acts by skew-adjoint operators Let H be an admissible representation of DiffΩ (T). Then identify the partial derivatives ∂/∂θi with the vector fields which generate rigid rotations in each of the copies of S 1 in T. In this way we obtain three elements of the Lie algebra of DiffΩ (T). If we think of these elements as operators on H , then there is a natural operator D on H defined by the formula (5.1).

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Definition 5.2. An admissible representation H of DiffΩ (T) is a positive energy representation if D has a discrete spectrum bounded from bellow. If H is a positive energy representation we can associate to any manifold T × [0, t] equipped with the natural hyperhermitian metric a contraction operator e−tD . If Σ ∈ M is a morphism defined by a solution of Nahm’s equations, then let t : Σ → R be the time function and let v1 (t), v2 (t), v3 (t) be the vector fields on Σ which are the solutions of Nahm’s equations. Assume that t has no critical points, then t induces a foliation on Σ by manifolds Tt diffeomorphic to T. For each t we have an operator Dt = iv1 (t) + jv2 (t) + kv3 (t) defined on H . Then we can use the one parameter family e−tDt to define an operator associated to Σ. So in principle any positive energy representation of DiffΩ (T3 ) can be extended to a representation of M . The next step is to construct positive energy representations. In the case of conformal field theory the two basic representations, free bosons and free fermions, are related to sections of holomorphic bundles over Riemann surfaces. We shall argue that something similar should be true in the present case. If v ∈ H is a vector and vt = e−tD v, where D is the operator defined in (5.1), then the “evolution equation” for vt is ∂vt ∂vt ∂vt ∂vt = −Dvt = −i −j −k ∂t ∂θ1 ∂θ2 ∂θ3 which is the Cauchy-Riemann-Fueter equation. This equation is a quaternionic analogue of the Cauchy-Riemann equations. The use of the CauchyRiemann-Fueter equation allowed Fueter to extend the theory of holomorphic functions of one variable to the case of functions of a quaternionic variable, descriptions of Fueter’s work can be found in [26]. So the final idea is that the holomorphic structure in Segal’s elliptic objects has to be replaced by some form of quaternionic manifolds and quaternionic holomorphic bundles over them. (5.2)

6. Quaternionic algebra In this section we shall summarise some of the ideas of D. Joyce, see [14, 30], about quaternionic algebra and quaternionic geometry and see how they lead to natural definitions of quaternionic Fock spaces. The basic construction in Joyce’s theory is the quaternionic tensor product. This product is also necessary in principle to extend condition one of the definition of conformal field theory. An alternative approach to quaternionic quantum mechanics and quaternionic quantum field theory is given in [1]. Let I = {ai + bj + ck ∈ H} denote the imaginary quaternions. If U is a (left) H-module U , we shall write U ∗ for the set (6.1)

U ∗ = {φ : U → H, | φ(qu) = qφ(u), ∀ q ∈ H, u ∈ U }.

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Jorge A. Devoto

Let U be a (left) H-module U with a distinguished (real) real subspace U 0 and define U † = {φ ∈ U ∗ such that φ(u) ∈ I, ∀u ∈ U 0 } An AH-module is a generalisation of the pair (H, I); it consists of a (left) H-module U with a distinguished real subspace U 0 such that if u ∈ U 0 and φ(u) = 0, ∀φ ∈ U † , then u = 0. Usually we shall not write U 0 which is ∗ understood. If U is a H-module, then define a map iU : U → H ⊗R (U † ) by iU (u).α = α(U ). It is not difficult to show that iU is a quaternionic linear ∗ monomorphism, where H acts on H ⊗R (U † ) in the obvious way. Definition 6.1. Let U and V be two AH-modules and consider the H-module ∗ ∗ ∗ H ⊗R (U † ) ⊗R (V † ) . Exchanging the factors we may regard (U † ) ⊗R iV (V ) ∗ ∗ ∗ and iU (U ) ⊗R (V † ) as submodules of H ⊗R (U † ) ⊗R (V † ) . The quaternionic tensor product U ⊗H V is the H-module n o n o ∗ ∗ (6.2) U ⊗H V = iU (U ) ⊗R (V † ) ∩ (U † ) ⊗R iV (V ) . The module U ⊗H V has a distinguished real subspace ∗

∗

(U ⊗H V )0 = (U ⊗H V ) ∩ (I ⊗ (U † ) ⊗ (V † ) ), and the pair (U ⊗H V, (U ⊗H V )0 ) defines a AH-module. Lemma 6.2. Let U, V, W be AH-modules. Then there are canonical AHhomomorphisms H ⊗H U ∼ = (U ⊗H V ) ⊗H W. = V ⊗H U and U ⊗H (V ⊗H W ) ∼ = U, U ⊗H V ∼ Two important consequences of this lemma are that firstly one can form symmetric and antisymmetric powers of an AH-module U denoted by S k U and ∧k U and secondly one can define AH-algebras, examples of the last construction are the symmetric algebra S ∗ U and the exterior algebra ∧∗ U . An appropriate choice of U gives a candidate for a quaternionic bosonic Fock space of a particle 3. Recall that using holomorphic quantisation – see for example the first section of Fadeev Lectures in [6] or chapter two of [9] one can identify the space of quantum states of a harmonic oscillator in terms of the holomorphic functions on the plane. In this representation the symmetric power S n (C) appears both as the space of homogeneous polynomials of degree n (in the power series expansion of holomorphic functions) or as the n-particle space. Quoting Fadeev [9, Section 2.2]: This representation is very useful for the field theory because in this case the free Hamiltonian can be represented by the sum of Hamiltonians of an infinite set of oscillators. Let U be the set of linear functions f : H → H of the form f (x0 + ix1 + jx2 + kx3 ) = q0 x0 + q1 x1 + q2 x2 + q3 x3 3We

ignore if there is a real physical content in these definitions.

Quaternionic elliptic objects and K3-cohomology

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for certain quaternions qi with i = 0, . . . , q3 . The function f is q-holomorphic if and only if q0 + q1 j + q2 j + q3 k = 0. Thus the set U of q-holomorphic linear functions is isomorphic to H3 . Let U 0 = {(q1 , q2 , q3 ) ∈ H3 : qj ∈ I for j = 1, 2, 3 and q1 i + q2 j + q3 k ∈ I}. Then (U, U 0 ) defines an AH module. The k-th symmetric power SHk U has a natural identification with the q-holomorphic functions which are homogeneous polynomials of degree k. So the sum ⊕k SHk U could serve as a quaternionic analogue of the state space of a harmonic oscillator. One might hope that this Fock space could be the basis for the definition of a Fock space for free bosonic fields 4. 7. Other Aspects of the Program It is our aim in this section to briefly discuss some other aspects which need to be developed. Everything is highly speculative. 7.1. The coefficient ring of K3 cohomology. In this section we shall conjecture about the possible nature of the coefficient ring of K3 cohomology. This ring should be related to sections of some bundle over the moduli space of K3-surfaces. One question if is possible to relate this ring to modular forms. We will follow closely the exposition of [28] where we refer the reader for details. 7.1.1. Siegel Modular Forms. One of the interesting thing about elliptic cohomology is the relation with modular forms. Can we find a relation between K3 cohomology and modular forms? Definition 7.1. The Siegel upper half plane h2 is the space of 2 × 2 complex matrices τ which are symmetric and have a positive definite imaginary part. The symplectic group Sp4 (R) acts on h2 by A B (7.1) × τ → (Aτ + B)(Cτ + D)−1 C D Definition 7.2. A Siegel modular from of degree 2 and weight k is a holoA B morphic function f : h2 → C such that for each matrix C D ∈ Sp4 (R) and τ ∈ h2 (7.2)

f (τ ) = det (Cτ + D)−k f ((Aτ + B)(Cτ + D)−1 )

Siegel modular forms can be seen as sections over the moduli space of principally polarised Abelian surfaces. They are related to the moduli space of K3-surfaces via the Kummer map. One might speculate that if one considers only Kummer surfaces one can show the coefficient ring of K3 cohomology is some subset of the set of Siegel Modular forms. 4This

definition has be extended to consider free boson fields. This is work in progress

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7.2. Singular K3-surfaces. Besides the smooth K3-surfaces there are also important examples of singular ones. Which role they play in the theory? Can one extend the definition of morphisms in M 3 to include them? 7.3. Projective representations of DiffΩ (M). How can one build other positive energy representations of DiffΩ (M)? Are they related to the representations of double or triple loop groups? One aspect to consider is that representations coming from quantum physics are projective and therefore they correspond to genuine representations of central extensions of groups. Pressley and Segal showed in [18] that in a certain sense there are no interesting central extensions of double and triple loop groups. Since we are working with the quaternions it is reasonable to conjecture that one might need to consider some non-central extensions. Projective quaternionic representations are studied in [1]. 7.4. Index theory on L (2) X. The most important question if there is some formal operator on the double loop space related to the genus associated to a K3 surface. References 1. S. Adler, Quaternionic quantum mechanics and quantum fields, Oxford University Press, 1995. ´ 2. M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. Ecole Norm. Sup. 10 (1977), 87–131. 3. A. Ashtekar, T. Jacobsen and L. Smolin, A new characterization of half-flat solutions to Einstein’s equations, Commun. Math. Phys. 115 (1988), 631-648. 4. M. Atiyah and R. Bielawski, Nahm’s equations, configuration spaces and flag manifolds, xxx.lanl.gov – math.RT/0110112v1, 2001. 5. J. Baez, The octonions, Bull. of the A.M.S. 39 (2002), no. 2, 145–206. 6. P [et all] Deligne (ed.), Quantum fields and strings: A course for mathematicians, vol. I, A.M.S., 1999. 7. S. Donaldson, Nahm’s equations and the classification of monopoles, Commun. Math. Phys. 96 (1984), 387–407. , Complex cobordisms, astekar’s equations and diffeomorphisms, London Math. 8. Soc. Lecture Notes (1993), no. 192, 45–55. 9. L. Faddeev and A. Slavnov, Gauge fields: An introduction to quantum theory, second edition ed., Frontiers in Physics, Perseus Books, 1991. 10. H Geiges and J. Gonzalo, Contact geometry and complex surfaces, Inv. Mathematicae 121 (1995), no. 1, 147–211. 11. R. Hartshorne, Algebraic geometry, Springer-Verlag (1983). 12. R. W. H. T. Hudson, Kummer’s quartic surfaces, Cambridge Univ. Press, 1990. 13. D. Joyce, Compact manifolds with special holonomy, Oxford University Press, 2000. 14. , A theory of quaternionic algebra with applications to hypercomplex geometry, Available from arXiv.math.DG/0010079, October 2001. 15. D. Karachanyan , R. Mavelyan and R. Mkrtchyan, Area preserving structure of 2d gravity, Available from arXiv.hep-th/9401031. 16. M. Kreck and S. Stolz, HP2 bundles and elliptic cohomology, Acta Math. 171 (1993), 232–261.

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17. L. J. Mason and E. T. Newman, A connection between the Einstein equations and Yang-Mill equations, Commun. Math. Phys. 121 (1989), 659–668. 18. A. Pressley and G. B. Segal, Loop groups and their representations, Oxford University Press, 1986. 19. D. Quillen, On the formal group laws of unoriented and complex cobordism, Bull. of the A. M. S. 75 (1969), 1293–1298. 20. , Elementary proof of some results of cobordism theory using Steenrod operations, Adv. in Math. 7 (1971), 29–56. 21. P Saskida, Unpublished dissertation. 22. G. B. Segal, Elliptic cohomology, S´eminaire Bourbaki 1987/88, Ast´erisque, vol. 161-162, Soc. Mathematique de France, 1988, pp. 187–201. , The definition of conformal field theory, Unpublished Notes Oxford, 1989. 23. 24. Simon, B, Trace ideals and their applications London Mathematical Society. Lecture Notes vol. 35, (1979). 25. J. Stienstra, Formal group laws arising from algebraic varieties, Am. J. of Math. 109 (1987), no. 5, 907–926. 26. A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Phyl. Soc. 85 (1979), 199–225. 27. C. Thomas, Elliptic cohomology, Surveys on Surgery Theory (S Capell, A Ranicki, and Rosenberg J., eds.), Annals of Mathematics Studies, Princeton University Press, 1999, pp. 409–439. 28. C. Thomas, Elliptic cohomology, Plenum, 1999. 29. G. van der Geer and T. Katsura, On a stratification of the moduli of K3 surfaces, Preprint (1999), Available from math.AG/9910061. 30. D. Widdows, Quaternion algebraic geometry, Ph.D. thesis, Univ. of Oxford, 2000. 31. E. Witten, Elliptic genera and quantum field theory, Communications in Mathematical Physics 109 (1987), 525–536. 32. , The index of the Dirac operator on the loop space, Elliptic curves and Modular Forms in Algebraic Topology (P. Landweber, ed.), Lecture Notes in Math, vol. 1326, Springer-Verlag, 1988, pp. 161–181.

Dept. of Math. ITBA Av. Madero 399, Buenos Aires, Argentina Dept. of Math. FCECN, University of Buenos Aires, Ciudad Universitaria, Buenos Aires, Argentina [email protected]

THE M -THEORY 3-FORM AND E8 GAUGE THEORY EMANUEL DIACONESCU, GREGORY MOORE, AND DANIEL S. FREED

Abstract. We give a precise formulation of the M -theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8 gauge field. This leads to a precise version of the Chern-Simons interaction of 11-dimensional supergravity on manifolds with and without boundary. As an application of the formalism we give a formula for the electric C-field charge, as an integral cohomology class, induced by self-interactions of the 3-form and by gravity. As further applications, we identify the M theory Chern-Simons term as a cubic refinement of a trilinear form, we clarify the physical nature of Witten’s global anomaly for 5-brane partition functions, we clarify the relation of M -theory flux quantization to K-theoretic quantization of RR charge, and we indicate how the formalism can be applied to heterotic M -theory.

1. Introduction This paper summarizes a talk given at the conference on Elliptic Cohomology at the Isaac Newton Institute, in December, 2002 [1] In this paper we will discuss the relation of M-theory to E8 gauge theory in 10, 11, and 12 dimensions. Our basic philosophy is that formulating Mtheory in a mathematically precise way, in the presence of nontrivial topology, challenges our understanding of the fundamental formulation of the theory, and therefore might lead to a deeper understanding of how one should express the unified theory of which 11-dimensional supergravity and the five 10-dimensional string theories are distinct limits. To be more specific, let us formulate three motivating problems for the formalism we will develop. The first problem concerns 11-dimensional supergravity. We will be considering physics on an 11-dimensional, oriented, spin manifold, Y . When it has a boundary we will denote ∂Y = X. The basic fields of 11-dimensional supergravity are a metric g, (Lorentzian or Euclidean) a “C-field,” and a gravitino ψ ∈ Γ(S ⊗ T ∗ Y ) where S is the spin bundle on Y . Our main concern in this paper is with the mathematical nature of the C-field. In the standard formulation of supergravity the C-field is regarded as a 3-form gauge potential (1.1)

C ∈ Ω3 (Y ).

Date: Nov. 30, 2003; Revised March 22, 2004.

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The M-theory 3-form and E8 gauge theory

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This generalizes the Maxwell potential A ∈ Ω1 (M) of electromagnetism on a manifold M. The fieldstrength is defined to be (1.2)

G = dC ∈ Ω4 (Y ).

Whereas in classical electromagnetism the gauge potential A is a global 1form, in the quantum theory Dirac’s law of charge quantization changes the geometric nature of A: it is now a connection on a U (1) line bundle over M. (See [2] (sec. 3) and [3] (sec.2) for expository accounts of how charge quantization leads to a U (1) connection and its generalizations in differential cohomology theory.) One of our goals is to give a similar geometric description of the C-field. In standard supergravity the path integral measure is a canonical formal measure on the space of fields, defined by the metric g and the Faddeev-Popov procedure, weighted by an action. Schematically, the exponentiated action is given by " # Z 1 1 / Φ(C) (1.3) exp −2π vol(g)R(g) + 3 G ∧ ∗G + ψ¯Dψ 9 2` Y ` plus 4-fermion terms, where ` is the 11-dimensional Planck length and, roughly speaking, Z 1 2 (1.4) Φ(C) ∼ exp 2πi CG − CI8 (g) Y 6 Here I8 (g) is a quartic polynomial in the curvature tensor. We will give precise normalizations for the C-field and I8 in sections 3.2 and 4.1 below. Now let us suppose we wish to formulate the action in the presence of nontrivial topology. This means, among other things, that we wish to allow the fieldstrength G in (3.37) to define a nontrivial class in the DeRham cohomology of Y . Evidentally, we cannot use (1.4). One might be tempted to introduce a 12-manifold bounding Y and use Stokes formula. This procedure works (after accounting for several subtleties) when Y is closed but fails when ∂Y 6= ∅. Thus our first problem is: Find a mathematically precise definition of Φ(C) when G is cohomologically nontrivial, and ∂Y = X is nonempty. We will give a complete answer to this question in section 5 below. Having formulated the measure we can next turn to applications. When ∂Y = X is nonempty, the path integral for the C-field on a manifold with boundary Y defines a wavefunction of the boundary values CX of C. We may denote this wavefunction as Ψ(CX ). Now, there is a group of gauge transformations G of the C-field and the wavefunction must be suitably gauge invariant. Our second problem is If Ψ(CX ) is a nonvanishing gauge invariant wavefunction, what conditions are imposed on the values of CX ? Put more simply: What is the Gauss law for the C-field? We will find nontrivial conditions on CX in section 7.1 and will interpret them as the condition that

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the induced electric C-field charge associated to a C-field configuration and gravity must vanish. An analogy might be useful at this point. Because of the Chern-Simons phase Φ(C), the path integral is a generalization of the familiar 3-dimensional massive gauge theory: Z Z Z 1 k 2 3 (1.5) Ψ(AX ) = [dAY ] exp − TrF ∧ ∗F + i Tr AdA + A 2e2 4π Y 3

where AY is a connection on a bundle PY over the three-manifold Y and AX is the fixed boundary value. The “Gauss law” is the statement that this “function” is suitably gauge invariant. The Gauss law implies, among other things, that one can only define a nonvanishing gauge invariant wavefunction when c1 (PX ) = 0, where PX is the restriction of PY . In the case e2 = ∞ we have pure Chern-Simons theory. The Gauss law then implies F (AX ) = 0 and leads directly to the mathematical interpretation of Ψ as a section of a complex line bundle over the moduli space of flat connections on a Riemann surface. The analogy to Chern-Simons gauge theory cannot be pushed too far. In (1.5) the integer parameter k appears. In (1.4) the analogous parameter is, roughly speaking, k = 1/6. It is precisely this fractional value which makes the proper definition of Φ(C) subtle. Finally, and most importantly, let us turn to the third motivating problem, namely an understanding of the emergence of E8 gauge symmetry in Mtheory. There are several hints that there is a fundamental role for the group E8 in M-theory. First, there are the famous U -duality global symmetries that arise when M-theory is compactified on tori. These symmetries involve exceptional groups. Next the duality relating M theory on K3 to heterotic string theory on T 3 implies that there are enhanced gauge symmetries when the K3-surface develops A − D − E singularities. These gauge symmetries certainly include E8 × E8 . Next, a construction of Horava-Witten shows that a quotient of M-theory on X × S 1 by an orientation reversing isometry of S 1 leads to the E8 × E8 heterotic string, with E8 gauge fields propagating on the boundary [4, 5]. Furthermore, in [6] Witten gave a definition of Φ(C) that used E8 gauge theory in 12 dimensions. This definition was then used in [7] to establish a connection to the K-theoretic classification of RR fluxes in the limit that M theory reduces to type II string theory. All this suggests a hidden E8 structure in M theory which might point the way to a useful reformulation of the theory. Thus we have our third problem: What is the precise relation of the C-field of 11-dimensional supergravity to an E 8 gauge field? We will propose an answer to this question in section 3. The remainder of the paper endeavors to demonstrate that this answer can be useful. Finally, we would like to close this introduction with a general remark. In the physics literature gauge fields are traditionally modeled as belonging to a space with a group action, or simply as an element of the quotient space. For

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example, a 1-form gauge field is typically taken to be a connection on a fixed principal bundle, and the group of gauge transformations acts on the space of all such connections. It is technically sounder to use a different model. Namely, we consider instead the groupoid of all connections on all principal bundles; morphisms are maps of principal bundles which preserve the connections. There is still a quotient space of equivalence classes of connections, and this is the space over which one writes the functional integral. This model is local, whereas fixing a particular principal bundle is not. Also, we can replace the groupoid by an equivalent groupoid without changing the physics. This allows us the convenience of using different models adapted to different purposes. In section 3 we introduce a few different models for the C field. One (described in secs. 3.1-3.3) is closer to the physics tradition, while the other two (described in secs. 3.4-3.5) involves a groupoid model. In this paper we mostly rely on the first model, which emphasizes the connection to E8 . There is an alternative approach to some of the issues in this paper [8] which uses yet other models, following the ideas developed in [9]. That point of view leans more heavily on the (differential) algebraic topology to construct the cubic form which appears in the M-theory action. In particular, the model for the C-field does not involve E8 gauge fields. Two very recent papers on subjects closely related to the present work are [10, 11]. 2. The gauge equivalence class of a C-field Our first task is to give a precise answer to the question: “What is a Cfield?” In this section we will give a partial answer, by describing the gauge equivalence class (gec) of a C-field. The answer will be that the gec of a C-field is a (shifted) differential character. To motivate this description let us consider the description of the gauge equivalence class of a U (1) gauge field on a manifold M. The key to answering this problem turns out to be to consider the holonomy around 1-cycles γ. This is certainly gauge-invariant information, and it turns out to be all the gauge invariant information. More precisely, given a connection A on a line bundle over M we may regard the holonomy as a map on closed 1-cycles χ ˇA : Z1 (M) → U (1) given by I (2.1) χˇA (γ) = exp i A γ

It is then natural to ask how χ ˇA differs from an arbitrary such map. The answer is that there exists a closed 2-form, the fieldstrength of χ, ˇ such that if γ = ∂B2 is a boundary then Z (2.2) χ ˇA (γ) = exp i F B2

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Note that it follows that F has 2π ZZ-periods, denoted F ∈ Ω22πZZ (M), and that F is closed. Note also that it follows from (2.2) that χ ˇ is a homomorphism of abelian groups. It is smooth in an appropriate sense. Such maps χ ˇ such that a fieldstrength exist are in 1-1 correspondence with the gec of U (1) bundles with connection. In supergravity theories one often encounters p-form gauge potentials. One natural mathematically precise formulation of such gauge potentials is in terms of differential characters, which generalize the above description of U (1) gauge connections. By definition a Cheeger-Simons character, or differential ˇ p+1 (M) is a homomorphism character χ ˇ∈H (2.3)

χˇ : Zp (M) → U (1)

where Zp (M) is the group of p-cycles on M, such that there is a fieldstrength ω(χ) ˇ ∈ Ωp+1 ˇ ZZ (M). That is, there exists a closed differential (p + 1)-form ω(χ) with ZZ-periods such that Z (2.4) Σp = ∂Bp+1 ⇒ χ(Σ ˇ p ) = exp 2πi ω(χ) ˇ . Bp+1

We will identify the space of gec’s of a p-form gaugefield with the space of ˇ p+1 (M). In order to establish some notation let differential characters χˇ ∈ H us recall the basic facts about differential characters. (See [9] and references therein for further details about differential characters and cohomology). The gauge invariant information in a Cheeger-Simons character can be expressed in two distinct ways, each of which is summarized by an exact sequence. The first sequence is related to the space of flat characters, that is, characters with ω(χ) ˇ = 0: (2.5)

ˇ p+1 (M) → Ωp+1 (M) → 0 0 → H p (M, U (1)) → H ZZ

This is clear because a flat character defines a homomorphism of abelian groups Hp (M, ZZ) → U (1), which, by Poincar´e duality is H p (M, U (1)). In order to define the second sequence we begin with the topologically trivial characters. If A ∈ Ωp (M) is globally defined, then we may define a differential character: Z (2.6) χˇA (Σp ) := exp[2πi A]. Σp

Note that, first of all, χˇA only depends on A modulo ΩpZZ , and secondly, the fieldstrength ω(χˇA ) = dA is trivial in cohomology. Thus, the cohomology class of the fieldstrength is an obstruction to writing a character as a trivial character (2.6). In fact we have the second sequence: (2.7)

ˇ p+1 (M) → H p+1 (M, ZZ) → 0 0 → Ωp /ΩpZZ → H

The projection map in this sequence defines the characteristic class of χˇ which we will denote as a(χ) ˇ ∈ H p+1 (M, ZZ). Note that this is a class in integral

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cohomology, not DeRham cohomology. The compatibility relation between the two sequences states that (2.8)

a(χ) ˇ IR = [ω(χ)] ˇ DR

where a(χ) ˇ IR denotes the image of a(χ) ˇ in DeRham cohomology. It turns out that the gec of a C-field is not quite a differential character, but is rather a “shifted” differential character [6]. The reason for this is best explained in terms of the coupling to the “membrane.” In the formulation of M-theory - as it is presently understood - one posits the existence of fundamental “electrically charged” membranes with 3-dimensional worldvolumes. In Maxwell theory, a charged particle, of charge e, moving along a worldline γ, couples to the background gauge potential A via the holonomy: Z (2.9) exp[i eA]. γ

In M theory the membrane couples to the C-field in an analogous way. The standard coupling of supergravity fields to the membrane wrapping a 3-cycle Σ is usually written: Z (2.10) ∼ exp[2πi C]. Σ

More accurately, because of the worldvolume fermions on the membrane, the topologically interesting part of the membrane amplitude is Z q / S(N ) exp 2πi C DetD (2.11) Σ

where N is the rank 8 normal bundle of the embedding ι : Σ ,→ Y and S(N ) is an associated chiral spinor bundle. The squareroot has an ambiguous sign when one considers (2.11) on the space of all 3-cycles Z3 (Y ) and metrics, and the holonomy (2.10) has a compensating sign such that the product is welldefined [6]. The net effect is that it is the gec of the difference [C1 − C2 ] of ˇ 4 (Y ). We can be slightly C-fields which is an honest differential character in H more precise about the nature of the shift. As shown in [6] the requirement that (2.11) is well-defined implies quantization condition on the fieldstrength: 1 [G]DR = aIR − λIR 2 4 where a ∈ H (Y, ZZ) is an integral class and λ is the canonical integral class of the spin bundle of Y . 1 In conclusion the we have a partial answer to the question “What is a C-field?” The gauge equivalence class of a C-field is a shifted differential character. We have not yet defined precisely what is meant by “shifted.”

(2.12)

1Put

differently, only twice the characteristic class of the twisted differential character is well-defined, and it is constrained to be equal to w4 modulo two.

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

This will be explained in the section 3.4 (equation 3.25). The space of shifted ˇ 41 (Y ). characters, with shift 12 λ will be denoted H λ 2

3. Models for the C-field In the previous section we have explained what the gauge equivalence class of a C-field is, but have not answered the question: “ What is a C-field?” An analogous situation would be to have in hand a formulation of nonabelian gauge theory in terms of gauge invariant quantities without having introduced connections on principal bundles. In physical theories the requirement of locality, that we be able to formulate the theory in terms of local fields, forces us to introduce redundant variables, such as gauge potentials. Similarly, in the standard formulation of supergravity one takes the C field to be an ordinary 3-form C ∈ Ω3 (Y ) subject to a gauge invariance C → C + ω where ω is a closed 3-form. When ω is exact such gauge transformations are referred to as small gauge transformations. When ω is closed but not exact the gauge transformation is a large gauge transformation. In M-theory, quantization of membrane charge requires that ω have integral periods. In this view, the gauge equivalence class of a C-field is (3.1)

[C] ∈ Ω3 (Y )/Ω3ZZ (Y ).

According to (2.7) such fields define topologically trivial differential characters. However, many interesting nontrivial phenomena in string/M-theory involve topologically nontrivial characters and hence we must modify the geometric description of the C-field. In this paper we will focus on the “E8 model for the C-field,” which seems well-suited to describing the M-theory action. As we will discuss below, there are other models of the C-field which can be considered to be equivalent to the E8 model. 3.1. The E8 model for the C-field. The E8 model is motivated by Witten’s definition [6] of the M-theory action as an integral in 12-dimensions. (We will review Witten’s definition in section 4.1 below.) This model is based on the topological fact that there is a homotopy equivalence (3.2)

E8 ∼ K(ZZ, 3)

up to the 14-skeleton. Equivalently, the homotopy groups πi (E8 ) of E8 vanish for 4 ≤ i ≤ 14. It follows from (3.2) that BE8 ∼ K(ZZ, 4) and therefore, for dim M ≤ 15, there is a one-one correspondence between integral classes (3.3)

a ∈ H 4 (M, ZZ)

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51

and isomorphism classes of principal E8 bundles over M. 2 For each a we pick a specific bundle P (a) → M. 3 We now come to a central definition; we will say that a “C-field on Y with characteristic class a ” is an element of (3.4)

EP (Y ) := A(P (a)) × Ω3 (Y )

where A(P ) is the space of smooth connections on the principal bundle P . Thus, our “gauge potentials,” or “C-fields,” will be pairs (A, c) ∈ EP (Y ). We will often denote C-fields by Cˇ = (A, c), and we will call c the “little c-field.” ˇ Given the “gauge potential” Cˇ = 3.2. The gauge equivalence class of C. (A, c) ∈ EP (Y ) we will now describe its gauge equivalence class. The principle we use is that the holonomy, or coupling of the C-field to an elementary membrane contains all the gauge invariant information in C. The holonomy of Cˇ = (A, c) ∈ EP (Y ) around Σ is defined to be " Z # 1 (3.5) χˇA,c (Σ) := exp 2πi CS(A) − CS(g) + c 2 Σ Here CS(A) and CS(g) are Chern-Simons invariants associated to the gauge field A and the metric g, normalized by 4 1 F2 (3.6) dCS(A) = trF 2 := Tr248 60 8π 2 1 dCS(g) = trR2 := − (3.7) Tr11 R2 16π 2 Note that 1 (3.8) [trF 2 ]DR = aIR [trR2 ]DR = (p1 (T Y ))IR 2 It follows immediately that the fieldstrength of the character is 1 (3.9) ω(χˇA,c) = G = trF 2 − trR2 + dc 2 R Note that the normalizations are chosen such that exp[2πi CS(A)] is Σ R well-defined. However, exp[πi Σ CS(g)] has a sign ambiguity when regarded

2A more elementary way to understand this is to use the obstruction theory arguments of [12]. 3This is somewhat unnatural and will lead to problems with gluing of manifolds and hence with locality. One can easily modify the definition below to include triples (P, A, c) where P is an E8 bundle in the isomorphism class determined by a. The discussion that follows is not changed in any essential way, except that one must account for bundle isomorphisms when discussing equivalences of (P, A, c) with (P 0 , A0 , c0 ). We have suppressed this refinement here in the interest of simplicity and brevity. R 4In general CS(A) is not well defined as a differential form, and only exp[2πi Σ CS(A)] is well-defined. See [13]. For a recent account see [14], sec.1.2. Below we will have occasion R to use the relative Chern-Simons invariant CS(A, A0 ) := [0,1] trF 2 defined by integrating along a straightline between the two connections. This is well-defined as a differential form.

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as a function on Z3 (Y ) × Met(Y ). This sign ambiguity is cancelled by the sign ambiguity of the worldvolume fermion determinant: q / S(N ) ) (3.10) Det(D

It is the metric dependence required to cancel the sign ambiguity of (3.10) which leads to χˇA,c being a shifted character. Previous authors have proposed that there should be an identification of the C field with an E8 Chern-Simons form. See, for examples, [15], and [16]. However, a formula such as C = CS(A) is unsatisfactory for many reasons. A similar idea based on OSp(1|32) was proposed in [17]. Related ideas, in which the 3-form gauge potential of 11-dimensional supergravity should be considered as a composite field, have been explored in [18]. The model we have just described, ( which was inspired by the IR/ZZ index theory of Lott [19], and was first announced in [20]) fits into this circle of ideas, but we emphasize that the E8 connection plays a purely auxiliary role, at least on manifolds without (spatial) boundaries. This becomes more clear when one considers alternative formulations which do not involve E8 at all. 3.3. The C-field gauge group. Now that we have defined “gauge potentials” we seek a gauge group G so that ˇ 41 (Y ). (3.11) EP (Y )/G = H 2

λ

The fiber of the group orbit should be defined by (3.12)

(A, c) ∼ (A0 , c0 )

⇔

χˇA,c = χ ˇA0 ,c0

This condition is easily solved: (A, c) ∼ (A0 , c0 ) iff there exists α ∈ Ω1 (adP ) and ω ∈ Ω3ZZ (Y ) such that : (3.13) (3.14)

A0 = A + α

α ∈ Ω1 (adP )

c0 = c − CS(A, A + α) + ω

ω ∈ Ω3ZZ (Y )

To prove this, note that equality of fieldstrengths implies c0 −c = CS(A0 , A)+ ω for some closed 3-form ω. Then equality of the holonomies shows that ω must have integral periods. ¿From (3.12,3.13,3.14) we might conclude that the “C-field gauge group” is: (3.15)

?

G = Ω1 (adP ) × Ω3ZZ (Y ).

The right hand side of (3.15) is indeed a group, with group law (3.16)

(α1 , ω1 )(α2 , ω2 ) = (α1 + α2 , ω1 + ω2 + d(trα2 ∧ α1 ))

and does satisfy (3.11). Nevertheless, it is not precisely the gauge group we need. This can be understood in two ways, one physical and one mathematical.

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53

Let us first explain the physical point of view. In electromagnetism, if ∂Y = X, the worldline of a charge particle can end on a point P ∈ X. The coupling of the charged particle to the background gauge potential Z (3.17) exp[i eA] γ

is not gauge invariant. Now note that gauge transformations in Maxwell ˇ 1 (Y ), since H ˇ 1 (Y ) is just the theory can be thought of as elements χˇ ∈ H group of U (1)-valued functions on Y . In this view, the gauge transformation by χˇ of the “open wilson line” (3.17) is: Z Z (3.18) exp[i eA] → χ(P ˇ ) exp[i eA]. γ

γ

In M theory, the worldvolume of a membrane Σ can end on a 2-cycle σ. By analogy with electromagnetism we should define C-field gauge transformations to act on such “open membrane Wilson lines” as: (3.19)

exp[2πi

Z

Σ

C] → χ(σ) ˇ exp[2πi

Z

C]

Σ

R where exp[2πi Σ C] is short for (3.5), and χ(σ) ˇ is a U (1)-valued function on Z2 (X). Moreover, the existence of a fieldstrength for C shows that there must exist a fieldstrength for χ. ˇ Indeed, suppose σ = ∂Σ, and C → C + x with x ∈R Ω3ZZ (Y ). Then, if Σ ⊂ X, consistency with (3.19) demands χ(σ) ˇ = 3 ˇ exp[2πi Σ ω]. But this is the defining property of χˇ ∈ H (X) ! Since we could choose any 10-dimensional subspace X in Y in this discussion, we conclude ˇ 3 (Y ). that the gauge transformation parameter should be regarded as χˇ ∈ H In this way we conclude that the proper definition of the gauge group should be ˇ 3 (Y ) (3.20) G := Ω1 (adP ) × H Recall that ˇ 3 (Y ) → Ω3 (Y ) → 0 0 → H 2 (Y, U (1)) → H ZZ

(3.21)

and thus we have a nontrivial extension of the naive gauge group (3.15). The gauge group (3.20) acts on EP (Y ) via (3.22)

(α, χ) ˇ · (A, c) = (A + α, c − CS(A, A + α) + ω(χ)) ˇ

The group law is (3.23)

(α1 , χ ˇ1 )(α2 , χ ˇ2 ) = (α1 + α2 , χ ˇ1 + χˇ2 + χˇb )

where χˇb is a topologically trivial character with b = Tr(α2 α1 ). It is convenient to introduce some terminology. We will refer to gauge transformations in the connected component of the identity in (3.20) as small gauge transformations. Otherwise, the gauge transformation is referred to as a large gauge transformation. We will refer to gauge transformations in H 2 (Y, U (1))

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

as micro gauge transformations since they leave (A, c) unchanged. 5 As we shall see, they nevertheless have a crucial physical effect. This formalism makes clear the physical role of the E8 gauge field. It is a kind of topological field theory since we can shift A to any other connection A0 ∈ A(P (a)), and hence A is only constrained by topology. 6 3.4. C-fields and categories. Let us now turn to a more mathematical justification of the definition (3.20). Quite generally, bosonic fields with internal symmetry should be viewed as objects of a groupoid —a category with all arrows invertible—and equivalent groupoids provide alternative models for the same physical object. In this section we demonstrate an equivalence of our model with another possible model for the C-field. To begin, since we have a G-action on a space EP (Y ) we can form an associated groupoid. We regard this groupoid as a category. The objects of this category are elements of EP (Y ). The morphisms Mor(Cˇ1 , Cˇ2 ) are the group elements taking Cˇ1 to Cˇ2 . We denote this category by EP (Y )//G. Note from (3.22) that the objects Cˇ in the category have automorphism group given by the flat characters of degree three: H 2 (Y, U (1)). Now, in [9] M. Hopkins and I. Singer have formulated a theory of “differential cochains” and “differential cocycles” which refines the theory of differential cohomology. Differential cocycles may be regarded as one definition of what is meant by gauge potentials for abelian p-form gauge fields.7 In the framework of [9] a “shifted differential character” is the equivalence class of a differential cocycle which trivializes a specific differential 5-cocycle related to W5 (Y ). The Hopkins-Singer theory can therefore be applied to the C-field of M-theory. To be more specific, 8 the cohomology class w4 (Y ) ∈ H 4 (Y, ZZ2 ) defines a differential cohomology class, w ˇ4 , because we can include H 4 (Y, ZZ2 ) ,→ ˇ 4 (Y ). The characteristic class of the flat character w H 4 (Y, IR/ZZ) ,→ H ˇ4 is the integral class W5 (Y ) ∈ H 5 (Y, ZZ), given by the Bockstein homomorphism applied to w4 (Y ). This class is interpreted as the background magnetic charge induced by the topology of Y , and this class must vanish in order to be able to formulate any (electric) C-field at all. On a spin manifold, W5 (Y ) = 0, since the class λ is an integral lift of w4 (Y ). When W5 (Y ) = 0 we may refine 5Note

that some micro gauge transformations can also be large gauge transformations! The characteristic class a(χ) ˇ for such gauge transformations will be torsion. 6In Donaldson-Witten theory there is also a symmetry under arbitrary shifts of the gauge potential, δA = ψ. In that case ψ is nilpotent. 7For some abelian p-form gauge fields, such as the Ramond-Ramond fields in Type II superstring theory, the quantization law is not in terms of ordinary cohomology so gauge fields are defined in terms of “differential functions” and “generalized differential cohomology theories.” 8This paragraph assumes some familiarity with [9].

The M-theory 3-form and E8 gauge theory

55

the differential cohomology class w ˇ4 to a differential cocycle by defining (3.24)

ˇ 5 = (0, 1 λ(g), 0) ∈ Zˇ 5 (Y ) ⊂ C 5 (Y, Z) × C 4 (Y, R) × Ω5 (Y ) W 2

1 2 where λ(g) = − 16π 2 TrR is functorially attached to the metric g. We then define a C-field to be a differential cochain Cˇ = (¯ a, h, ω) ∈ C 4 (Y, ZZ)×C 3 (Y, IR)× 4 ˇ 5: Ω (Y ) trivializing W

(3.25)

ˇ 5. δ Cˇ = W

Written out explicitly this means that

(3.26)

δ¯ a=0 1 δh = ω − a¯R + λ(g), 2 dω = 0.

We refer to these as shifted differential cocycles, and denote the space of such cochains by Zˇ 41 λ(g) . This is a principal homogeneous space for Zˇ 4 (Y ) and 2 hence may be considered as the set of objects in a category. Now, finally, we may explain the mathematical motivation for the choice of gauge group (3.20). It is only with this choice that we have the crucial theorem Theorem. There is an equivalence of the categories EP (Y )//G and Zˇ 41 λ(g) . 2

Proof: To prove this it suffices to establish the existence of a fully faithful functor F : EP (Y )//G → Zˇ 41 λ(g) such that any object in Zˇ 41 λ(g) is isomor2 2 ˇ for some C. ˇ Since both categories are groupoids, the morphic to F (C) phism spaces are principal homogeneous spaces for the automorphism group. Since the automorphism group is independent of object and category, namely H 2 (Y, U (1)), and since set of isomorphism classes of objects is the same, ˇ 41 (Y ) it simply suffices to establish the existence of a functor. namely H λ 2 Begin by choosing a Hopkins-Singer 4-cocycle (¯ c0 , h0 , ω0 ) on the classifying 4 space BE8 where [¯ c0 ] is a generator of H (BE8 ; ZZ), and ω0 is determined from the universal connection Auniv on EE8 by ω0 = trF (Auniv )2 . Now, there exists a map γ : P → EE8 which classifies the connection: γ ∗ (Auniv ) = A. 9 Let γ¯ : Y → BE8 be the induced classifying map on the base space Then we 9For

a nice proof, see [21], section 2. One needn’t rely on this, however. Instead, we introduce an equivalent category which includes a classifying map; see [14], section 3.1.

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Emanuel Diaconescu, Gregory Moore and Daniel S. Freed

define our functor on objects by: F (A, c) = (¯ cHS , hHS , ωHS ) where c¯HS = γ¯ ∗ (c0 ) (3.27)

hHS = γ¯ ∗ (h0 ) + c 1 ωHS = γ¯ ∗ (ω0 ) + dc = trF 2 (A) + dc − λ(g) 2

and one checks that (¯ cHS , hHS , ωHS ) is a shifted differential cocycle. ♠ In particular, and crucially, the automorphism group of a C-field (analogous to the constant gauge transformations in nonabelian gauge theory) is the same in both categories, namely H 2 (Y, U (1)). Finally, given the E8 model for the C-field a natural question one may ask is how E8 gauge transformations, that is, bundle automorphisms of P , are related to the C-field gauge transformations. The answer involves a construction which will prove useful later. Suppose (3.28)

(A, c) → (Ag , c)

g ∈ Aut(P )

is an E8 gauge transformation. Every transformation of this type is equivalent to a C-field gauge transformation (α, χ) ˇ acting on (A, c). We first find α, trivially it is (3.29)

α = Ag − A := g −1 DA g.

It follows that ω(χ) ˇ = CS(A, A+α) = CS(A, Ag ). A natural way to construct a character with this fieldstrength is to use g to construct the twisted bundle over X × S 1 (3.30)

Pg := (P × [0, 1])/(p, 0) ∼ (pg, 1)

and take the vertical connection dt∂t + dX + A where (3.31)

A = (1 − t)A + tAg

We can then set χ ˇ = χˇ(g,A) where (3.32)

"

χ ˇ(g,A) (σ) := exp 2πi

Z

σ×S 1

CS(A)

#

It is straightforward to check that (3.33)

1 ω(χˇ(g,A) ) = CS(A, Ag ) = − tr(g −1 DA g)3 + db(g, A) 3

where b(g, A) is globally well-defined. It is rather interesting to note that Aut(P ) is not a subgroup of G, since (α, χ) ˇ depends on A. Of course, since it is a group acting on EP (X) it does define a sub-groupoid. We will come back to this point in section 12 below.

The M-theory 3-form and E8 gauge theory

57

3.5. A third model for the C-field. There is a different approach to differential cohomology theory based on differential function spaces [9]. This motivates a different model for the C-field which will be described in some detail in the following. Of particular importance is the filtration on differential function spaces. While this approach will be less familiar to physicists, because gauge transformations are replaced by morphisms in a category, the approach has an appealing flexibility. As motivation for the construction, let us begin with a simpler example, ˇ 2 (Y ), which is isomorphic to namely the group of differential 2-characters H the group of equivalence classes of line bundles with connection on Y . One can construct a cocycle category for this group as follows. We take the objects to be pairs (L, A) consisting of a line bundle with connection on Y . The space of morphisms between two objects (L, A), (L0 , A0 ) consists of equivalence classes of pairs (L, A) on Y × ∆1 so that10 (3.34)

(L, A)|Y ×{0} = (L, A)

(L, A)|Y ×{1} = (L0 , A0 ).

Here ∆1 is the standard 1-simplex. Two pairs (L0 , A0 ), (L1 , A1 ) satisfying the same boundary conditions (3.34) are said to be equivalent if they are homotopy equivalent relative to the boundary conditions. This means that there exists a pair (L, A) on Y × ∆1 × I which restricts to (Li , Ai ) on Y × ∆1 × {i}, for i = 0, 1. Moreover, (L, A) should also restrict to (L, A), (L0 , A0 ) on the two boundary components Y × (∂∆1 ) × I. Composition of morphisms is defined by concatenation of paths. One can check that the construction sketched above yields a groupoid, but ˇ 2 (Y ). Two objects it is easy to see that this is not a cocycle category for H 0 0 (L, A), (L , A ) are equivalent if they are connected by a morphism (L, A). Then the parallel transport associated to the connection A determines an isomorphism φ : L → L0 . For a general connection A, φ is not compatible with the connections A, A0 . Therefore the equivalence classes of objects are in one to one correspondence to line bundles on Y up to isomorphism, which is not the answer we want. ˇ 2 (Y ) in this manner we have to refine our groupoid In order to obtain H structure so that the resulting equivalence relations are compatible with connections. This is a special case of a general construction described in detail in [9]. The main idea is to impose an extra condition on morphisms by keeping only pairs (L, A) so that (3.35)

F 1,1 (A) = 0

where F 1,1 (A) denotes the component of the curvature with one leg along ∆1 . The effect of this condition is that the parallel transport along ∆1 defined by A 10More

precisely, a morphism is a pair (L, A) together with isomorphisms (3.32) all up to the equivalence relation below. Also, note that the equivalence relation on morphisms and the composition of morphisms can be defined by working on Y × ∆2 , where ∆2 is the standard 2-simplex.

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becomes horizontal. Then the isomorphism φ : L → L0 preserves connections φ∗ A0 = A, and we obtain the desired set of equivalence classes. Note that for consistency we have to impose a similar filtering condition on the pairs (L, A) introduced below (3.34). Namely, we require all components with legs along I × ∆1 to vanish (3.36)

F 1,1 (A) = F 0,2 (A) = 0.

Note that this construction is valid even if Y is a manifold with boundary since we did not have to invoke integration by parts at any stage. The same will be true for the category of C-fields constructed below. (In physical applications one might wish to impose boundary conditions.) In the following we will produce a cocycle category E for C-fields on a manifold Y proceeding in a similar way. The objects are triples (P, A, c) as in section 3.1. Hence P is a principal E8 bundle on Y , A is a connection on P and c ∈ Ω3 (Y ). The “curvature” of a triple is the closed 4-form G defined by 1 G = trF 2 − trR2 + dc 2 The space of morphisms between two objects (P, A, c) and (P 0 , A0 , c0 ) consists of equivalence classes of triples (P, A, γ) on Y × ∆1 so that (3.37)

(3.38)

(P, A, γ)|Y ×{0} = (P, A, c)

(P, A, γ)|Y ×{1} = (P 0 , A0 , c0 ).

Denoting the curvature of (P, A, γ) by G, we impose a filtering condition G 3,1 = 0, where G (4−k,k) , k = 0, 1 denotes the component of G of degree k along ∆1 . Since dG = 0, this implies that G is pulled back from Y . (This means that G is “gauge invariant,” as it should be.) Two triples (P0 , A0 , γ0 ), (P1 , A1 , γ1 ) satisfying the same boundary conditions (3.38) are said to be equivalent if they are homotopy equivalent relative to the boundary. This means that there exists a triple (P, A, c) on Y ×∆1 ×I with obvious restriction properties, as explained below (3.34). For consistency we have to impose a filtering condition G3,1 = G2,2 = 0 analogous to (3.36). One can check that this construction defines a groupoid E. We claim this is a correct cocycle category for the C-field. In the remaining part of this section we will verify this claim by showing that i) the equivalence classes of objects are in one to one correspondence with shifted differential characters, and ii) the automorphism group of any object (P, A, c) is isomorphic to H 2 (Y, IR/ZZ). Let us start with (i). Two objects (P, A, c) and (P 0 , A,0 , c0 ) are equivalent if they are connected by a morphism. Suppose (P, A, γ) represents such a morphism. Since G is pulled back from Y , using the boundary conditions (3.38) we find that G = G = G0 . Moreover, the parallel transport associated to the connection A determines an isomorphism φ0 : P → P 0 . Therefore P, P 0 have the same characteristic class a ∈ H 4 (Y, ZZ). We conclude that we have a well

The M-theory 3-form and E8 gauge theory

59

defined map Ee → A41 λ (Y )

(3.39)

2

where Ee denotes the set of equivalence classes of objects of E and A41 λ (Y ) 2

denotes the set of pairs (a, G) ∈ H 4 (Y, ZZ) × Ω4ZZ+ λ (Y ) subject to the com2

patibility condition aIR −

λIR 2

= [G]DR . Note that A41 λ (Y ) is a principal 2

homogeneous space over the group A4 (Y ) of pairs (a, G) subject to the unshifted compatibility condition aIR = [G]DR . This map is clearly surjective. This is an encouraging sign since the group of shifted differential characters ˇ 1 λ is expected to surject onto this space. In order to finish the identificaH 2 tion, we should show that the fiber of this map is isomorphic to the torus H 3 (Y, IR)/H 3 (Y, ZZ). The fiber over a point (a, G) ∈ A41 λ (Y ) consists of isomorphism classes of 2 triples (P, A, c) with fixed (a, G). Up to isomorphism, a triple satisfying this condition can always be taken of the form (P0 , A0 , c) for some fixed (P0 , A0 ) with a(P0 ) = a. Therefore it suffices to determine the set of isomorphism classes of triples of the form (P0 , A0 , c) with fixed (a, G). Since only the threeform c is allowed to vary, it is straightforward to check that these triples are parameterized by the space of closed three-forms z ∈ Ω3cl (Y ). Two triples (P0 , A0 , c), (P0 , A0 , c + z) are isomorphic if they are connected by a morphism (P, A, γ) in the groupoid. Now we make use of the filtering condition G 3,1 = 0, which yields (3.40)

dt γ (3,0) + dY γ (2,1) + (trF (A)2 )(3,1) = 0.

Integrating this relation along ∆1 , and using the boundary conditions (3.38), we find Z Z (3.41) z = −dY β − η β= γ η= (trF (A)2 ). ∆1

∆1

At this point note that the bundle with connection (P, A) on Y × ∆1 satisfies identical boundary conditions along the two boundary components Y × {0} and Y × {1}. By a standard gluing argument, we can construct a bundle with e A) e over Y × S 1 . Then we have connection (P, Z e 2 ), (3.42) η= (trF (A) S1

which is a closed form with integral periods. In conclusion, the three-forms c and c+z parameterizing isomorphic triples (P0 , A0 , c), (P0 , A0 , c+z) with fixed (a, G) differ by closed forms with integral periods. However Ω3cl (Y )/Ω3ZZ (Y ) ' H 3 (Y, IR)/H 3 (Y, ZZ), hence we obtain the expected result. This shows that the gauge equivalence classes of C-fields in this model are indeed shifted differential characters.

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To conclude this section, let us compute the automorphism group of an arbitrary object (P, A, c) of E. Using the boundary conditions (3.38) and a standard gluing argument, we can easily show that Aut(P, A, c) consists of equivalence classes of triples (P, A, γ) on Y × S 1 subject to the condition G 3,1 = 0. Moreover, S 1 is equipped with a base point ∗ ∈ S 1 , and (P, A, γ) restricts to (P, A, c) on Y × {∗}. As noted below (3.38), since G is closed, it follows that it is a pull back from Y , G = π ∗ G, where π : Y × S 1 → Y is the canonical projection, and G is the curvature of (P, A, c). An important observation is that triples (P, A, γ) on Y × S 1 satisfying G 3,1 = 0 form themselves a groupoid M. The morphism space between two such triples (P, A, γ) and (P 0 , A0 , γ 0 ) consists of triples (P, A, c) on Y ×S 1 ×∆1 which restrict to (P, A, γ) and respectively (P 0 , A0 , γ 0 ) on the two components of the boundary. Two triples (P0 , A0 , c0 ) and (P1 , A1 , c1 ) are said to be equivalent if they are homotopy equivalent relative to boundary conditions. Furthermore, we have to impose a filtering condition of the form G3,1 = G2,2 = 0, where Gk,4−k denotes the component of G of degree k along Y . Note that M is in fact a cocycle category for C-fields on Y × S 1 subject to a filtering condition on the curvature. The automorphisms of (P, A, c) are classified by equivalence classes of objects of M so that (P, A, γ)|Y ×{∗} = (P, A, c). Proceeding by analogy with E, the equivalence classes of objects of M are in one to one correspondence ˇ 4 (Y × S 1 ) so that ω(χ) to differential characters χˇ ∈ H ˇ 3,1 = 0. In particular, this implies that ω(χ) ˇ is pulled back from Y . Adding the extra condition ˇ 4 (Y ) is the charac(P, A, γ)|Y ×{∗} = (P, A, c) fixes χ| ˇ Y ×{∗} = ρˇ, where ρˇ ∈ H ter determined by (P, A, c). Using the exact sequence (3.43)

ˇ 4 (Y × S 1 ) → Ω4 (Y × S 1 ) → 0 0 → H 3 (Y × S 1 , IR/ZZ) → H ZZ

and the K¨ unneth formula H 3 (Y × S 1 , IR/ZZ) ' H 3 (Y, IR/ZZ) ⊕ H 2 (Y, IR/ZZ), we see that ρˇ fixes the component in the first summand, but not the second. It follows that the characters (P, A, γ) are parametrized by H 2 (Y, IR/ZZ). In conclusion, Aut(P, A, c) ' H 2 (Y, IR/ZZ) for any object (P, A, c). This is in agreement with our previous discussion in section 3.3. We have so far given two different constructions of cocycle groupoids for C-fields which have identical equivalence classes of objects and automorphism groups. Therefore the two groupoids must be equivalent. 4. The definition of the C-field measure for Y without boundary 4.1. Witten’s definition. In [6] Witten gave a definition of the phase factor Φ(C) in (1.3) using E8 gauge theory in 12-dimensions. We will review his definition, and then show how to recast it in intrinsically 11-dimensional terms. The 11-dimensional formulation will then be in a form suitable to generalization to the case when Y has a boundary.

The M-theory 3-form and E8 gauge theory

61

Suppose P (a) → Y admits an extension (4.1)

PZ (aZ ) → Z

where Z is a bounding spin manifold (and hence ∂Y = ∅). The existence of such an extension follows from Stong’s theorem [22]. Suppose, moreover, that CˇZ = (AZ , cZ ) extends CˇY = (AY , cY ) ∈ EP (Y ) to EPZ (aZ ) (Z) (such extensions always exist if PZ (aZ ) exists). Then Witten’s definition is ) ( Z 1 3 GZ − GZ I8 (gZ ) (4.2) ΦW (CˇY ; Y ) = exp 2πi Z 6

where

1 GZ = trFZ2 − trRZ2 + dcZ 2 and I8 (g) is defined on a manifold M by 1 2 2 4 TrR − 4TrR . (4.4) I8 (g) = (4π)4

(4.3)

I8 (g) is a closed form, depending on a metric, which represents 1 (p2 (T M) − λ(T M)2 ) 48 in H 8 (M, IR), where we recall that λ = p1 /2. One needs to check that (4.2) is independent of the extensions aZ , PZ , CˇZ . This is almost true thanks to some remarkable identities involving E8 index theory. To check independence of extensions it suffices to check that the right hand side of (4.2) is equal to 1 on a closed 12-manifold Z. 11 / V (aZ ) coupled to the bundle V (aZ ) associated Consider the Dirac operator D to PZ (aZ ) via the 248 of E8 , and endowed with connection AZ . The index density is formed from (4.5)

(4.6)

I8 :=

ˆ Z) / V (aZ ) ) := Tr248 exp[FZ /2π]A(g i(D

ˆ Z ) is the usual Dirac index where FZ is the fieldstrength of AZ , while A(g density formed from the curvature 2-form R, computed using gZ . Using properties of E8 we find that 1 (4.7) Tr248 exp (FZ /2π) = 248 + 60αZ + 6αZ2 + αZ3 3 1 2 2 where αZ = 60 Tr248 F /8π represents (aZ )IR in DeRham cohomology. Now, ˆ eR/2π − / RS ) denote the 12-dimensional gravitino index density 12 ATr let i(D 11It

is quite crucial that we extend the entire triple (AY , cY , PY ), and not just the fieldstrength GZ . Otherwise Φ can vary continuously with a choice of extension. 12This is not the gravitino density of a 12-dimensional theory, but rather that appropriate to an 11-dimensional theory.

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ˆ Then, extracting the 12-form part, we have the crucial identity 1 + 8A. (12) 1 1 1 / A ) + i(D / RS ) i(D = G3Z − GZ I8 (gZ ) 2 4 6 # " (4.8) 1 1 1 2 − d cZ ( GZ − I8 (gZ )) − cZ dcZ GZ + cZ (dcZ )2 2 2 6 and thus, on a closed 12-manifold Z the RHS of (4.2) is 1 1 / V (aZ ) ) + I(D / RS ) (4.9) exp 2πi I(D 2 4

where I is the index. In 12-dimensions the index is always even, and hence (4.9) / RS )/2 . Thus, (4.2) is independent of extension, up simplifies to a sign (−1)I( D / RS )/2 . to the sign (−1)I( D 4.2. Intrinsically 11-dimensional definition. We would now like to recast the definition (4.2) into intrinsically 11-dimensional terms. This is readily done using the APS index theorem: Z /E) = / E ) − ξ(D /E) (4.10) I(D i(D Z

for the Dirac operator coupled to a bundle with connection E. Here 1 / E ) := (η(D / E ) + h(D / E )) (4.11) ξ(D 2 This is the standard invariant involving the η invariant and the number of / E ). zeromodes h(D Motivated by (4.2,4.8, and 4.10) we define iπ ˇ / RS ) + 2πiIlocal / V (a) ) + ξ(D (4.12) Φ(C; Y ) := exp iπξ(D 2

The total derivative in (4.8) leads to the elementary local factor Z 1 1 1 2 2 (4.13) Ilocal = c G − I8 (g) − cdcG + c(dc) 2 2 6 Y / RS ) is short for The expression ξ(D

(4.14)

/ T ∗ Y ) − 3ξ(D) / ξ(D

As a function of the C-field, (4.12) is gauge invariant, that is, Φ(A, c; Y ) = Φ(A0 , c0 ; Y ) for (A0 , c0 ) = (α, χ) ˇ · (A, c). Thus it is a U (1)-valued function on 4 ˇ H 1 λ (Y ). 2 Let us now comment briefly on Φ as a function of the metric g. Note that we have dropped a sign-factor in passing from (4.2) to (4.12): (4.15)

/ RS )/2 Φ ΦW = (−1)I( D

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/ T ∗ Y and D / have zero modes, and Let D be the locus of metrics g such that D suppose g ∈ Met(Y ) − D. On Met(Y ) − D , Φ is well-defined as a number. However, it does not extend continuously to the full space of metrics Met(Y ). This difficulty was essentially solved in [6]. One should consider Φ as a continuous section of a complex line bundle associated to a principal ZZ2 bundle T . Then the gravitino path integral, which we will denote schematically as / RS ) is a section of a real line bundle associated to a canonically isoPfaff(D morphic principal ZZ2 bundle, and will have (in general) a first order zero along / RS ) · Φ becomes a well-defined the locus of zeromodes. 13 The product Pfaff(D smooth diffeomorphism-invariant function on the gauge equivalence classes of metrics and C-fields. ˇ for 3-form gauge potentials deThere is a standard formal measure µ(C) fined by the the metric g on Y together with the Faddeev-Popov procedure applied to small C-field gauge transformations. Taking this into account, the well-defined measure for the C-field path integral, after integrating out the gravitino, is R ˇ − 2`13 Y G∧∗G Pfaff(D ˇ Y ). / RS ) · Φ(C; (4.16) µ(C)e 5. The C-field measure when Y has a boundary Now we are ready to reap some of the benefits of the E8 model for the C-field. Using this formalism we will completely solve problem 1 of the introduction. When we try to formulate the C-field measure in the case where ∂Y = X is nonempty the same formula (4.12) holds. In particular we continue to take iπ ˇ Y ) = exp iπξ(D / A ) + ξ(D / RS ) + 2πiIlocal (5.1) Φ(C; 2

/ A )] However, there is now a conceptually important distinction: the factor exp[iπξ(D is a section of a U (1) bundle with connection over the space of C-fields on / RS )] is X. As in the case when Y has no boundary, the factor exp[ 12 iπξ(D more subtle and must be combined with the gravitino determinant. We hope to discuss this elsewhere [23]. In this paper we consider a fixed metric g ∈ Met(Y ) − D. It is well-known in Chern-Simons theory that in the presence of a boundary / E )] should be viewed as a section of a the exponentiated invariant exp[iπξ(D line bundle. In our case Φ is valued in a principal U (1) bundle (5.2)

Q → EP (X).

This bundle can be defined by the following property: Each extension CˇY of CˇX ∈ EP (X), defined for some Y with ∂Y = X, produces an element: (5.3) Φ(CˇY ; Y ) ∈ Q ˇ CX

13The

definition of this principal ZZ2 bundle and the canonical isomorphism will be explained in detail in [23].

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and two such extensions satisfy the “gluing law” Φ(CˇY ; Y ) (5.4) = Φ(CˇY − CˇY 0 ; Y Y¯ 0 ) 0 ˇ Φ(CY 0 ; Y ) where Y¯ denotes orientation reversal. Alternatively, we may define QCˇX more directly as follows. Suppose P → X is a fixed E8 bundle of characteristic class aX . For any extension PY of P define EPY (CˇX ) to be the set of all extensions CˇY of CˇX . 14 We may then define the fiber QCˇX to be the set of all U (1)-valued functions F on EPY (CˇX ), for some Y , such that the ratio F (CˇY )/F (CˇY 0 ) is the right hand side of (5.4). This space of functions is a principal homogenous space for U (1) and varies smoothly with CˇX . Thus the fibers define a smooth U (1) bundle Q → EP (X). The circle bundle Q we have just defined carries a canonical connection. Recall that a connection is simply a law for lifting of paths in the base space to the total space satisfying the natural composition laws. A path p(t) = (AX (t), cX (t)), 0 ≤ t ≤ T in EP (X) defines a C-field on the cylinder Cˇp ∈ EP (X × [0, T ]). So (5.5)

Φ(Cˇp ; X × [0, T ]) ∈ Q(A,c)(T ) ⊗ Q∗(A,c)(0) = Hom(Q(A,c)(0) , Q(A,c)(T ) )

defines the appropriate parallel transport. Given the explicit formula (5.1) we can give an explicit formula for the connection. Choose an extension (Y, PY ) of (X, P ). Since every cotangent vector (δA, δc) to EP (X) has an extension to a cotangent vector of EPY it suffices to evaluate the covariant derivative on a section provided by an infinitesimal family of extensions CˇY . The main variational formula is: Z ∇Φ(CˇY ; Y ) 1 2 G − I8 2tr(δAF ) + δc (5.6) = 2πi Φ(CˇY ; Y ) Y 2

where 2tr(δAF ) + δc is a form of type (1, 3) on EP (Y ) × Y . It is now a simple matter to compute the curvature. Identifying the tangent space T EP (X) = Ω1 (adP ) ⊕ Ω3 (X) evaluation on tangent vectors (αi , χi ) gives the curvature 2-form Ω of Q: Z (5.7) Ω((α1 , ξ1 ); (α2 , ξ2 )) = 2πi G ∧ 2tr(α1 F ) + ξ1 ∧ 2tr(α2 F ) + ξ2 X

The theory of determinant line bundles shows that the associated line bundle L to Q may be regarded as a determinant line bundle. More precisely, / adP ), the Pfaffian line bundle for adjoint E8 fermions on the L = PFAFF(D boundary X. But, we emphasize, we are not introducing physical E8 fermions on the boundary.

˜ spin (K(ZZ, 4)) = ZZ2 ⊕ ZZ2 in general we cannot Ωspin Z2 ⊕ ZZ2 ⊕ ZZ2 and Ω 10 (pt) = Z 10 extend P → X. However, in the physical situations of interest there will always be such an extension since X is the boundary of the theory defined on the spacetime Y . 14Since

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The line bundle we have defined can also be defined by considering families of Dirac operators with boundary conditions along the lines of [24]. This definition fits in more naturally with the inclusion of the Pfaffian of the gravitino operator and will be discussed in [23]. It is also possible to formulate the line, together with its connection, in the categorical approach of sections 3.4 and 3.5. 6. The action of the gauge group on the physical wavefunction and the Gauss law In this section we will study the action of the C-field gauge group on the wavefunctions defined by the C-field path integral. Fix an extension PY of P to Y . For CˇX ∈ EP (X) recall that EPY (CˇX ) is the set of all extensions CˇY of CˇX . Then we set, formally, Z Z 1 ˇ ˇ (6.1) Ψ(CX ) = µ(CY ) exp[− 3 G ∧ ∗G] Φ(CˇY ; Y ) 2` ˇ ˇ EPY (CX )/G(CX ) Here G(CˇX ) is the group of C-field gauge transformations on Y which fixes CˇX . That is, we integrate over all isomorphism classes of extensions CˇY of CˇX . As we have seen, Φ(CˇY ; Y ) is valued in QCˇX and hence Ψ(CˇX ) is valued in the line associated to QCˇX . We will denote this line LCˇX . 6.1. The Gauss Law. Physically meaningful wavefunctions must be gauge invariant. In our case, the wavefunction is a section of a line bundle L → EP (X). In order to formulate gauge invariance we will need to define a lift of the group action: L (6.2)

G

→

↓ EP (X)

L ↓

G

→

EP (X)

Then the condition of gauge invariance is simply (6.3) g · Ψ(CˇX ) = Ψ(g · CˇX ) for all g ∈ G, CˇX ∈ EP (X). This condition is known as the Gauss law. Formally, any wavefunction defined by a path integral with gauge invariant measure, such as (6.1) automatically defines a gauge invariant wavefunction. This is essentially because the path integral integrates over gauge copies of fields and thus projects onto gauge invariant quantities. Of course, this projection might vanish, so the Gauss law should be considered, in part, as a condition on CˇX such that nonvanishing gauge invariant wavefunctions can be supported on CˇX .

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We will now give a precise definition of the group lift (6.2). We will then use that to derive an important consequence of the Gauss law. 6.2. Definition of the G action on L. The main result of this section is the construction of a well-defined lift (6.2). The full construction is rather lengthy. We confine ourselves to sketching the most important part, namely defining the action of gauge transformations of the form (0, χ) ˇ on L. We will construct the group lift using the natural connection on L defined above. To do this, we introduce some standard paths in EP (X). For any ξ ∈ Ω3 (X) define the linear path (6.4) pC,ξ ˇ (t) = A, c + tξ connecting Cˇ to Cˇ + ξ. (It is convenient to write Cˇ + ξ := (A, c + ξ) for a 3-form ξ. ) ˇ 3 (X) then we will define Now, if Ψ ∈ LCˇ and χ ˇ∈H (6.5)

ˇ χ) (0, χ) ˇ · Ψ := ϕ(C, ˇ ∗ U (pC,ω( ˇ χ) ˇ )·Ψ

where ϕ is a phase. The reason we must introduce the phase factor ϕ in (6.5) is that the connection on L has curvature. Indeed, an elementary computation shows that we have: (6.6)

iπ U (pC,ω( ˇ χ ˇ χ ˇ χ ˇ1 +χ ˇ2 ) ) = U (pχ ˇ1 C,ω( ˇ2 ) )U (pC,ω( ˇ1 ) )e

R

X

G∧ω(χ ˇ1 )∧ω(χ ˇ2 )

where χˇ1 Cˇ denotes the gauge transform of Cˇ by χˇ1 . It follows that to define a group theory lift we will need to find functions ˇ χ) ϕ(C, ˇ such that: (6.7)

ˇ χ ˇ χˇ2 )ϕ(C, ˇ χˇ1 )e+iπ ϕ(C, ˇ1 + χˇ2 ) = ϕ(χˇ1 C,

R

X

G∧ω(χ ˇ1 )∧ω(χ ˇ2 )

.

We now construct a cocycle satisfying the relation (6.7). ( In the following section we will provide some physical motivation for this choice of ϕ. ) We use the data CˇX and χˇ to construct the character on the closed 11-fold Y = X ×S 1 defined by (6.8) π ∗ [CˇX ] + π ∗ (tˇ) · π ∗ (χ) ˇ 1

2

1

ˇ 1 (S 1 ) is the canonical Here πi are projections (henceforth dropped) and tˇ ∈ H character associated with S 1 ∼ = U (1). This character (6.8) has fieldstrength G + dt ∧ ω(χ), ˇ characteristic class a + [dt]a(χ) ˇ and holonomy (6.9)

χ ˇ(A,c) (Σ3 )e

2πit

R

Σ3

ω(χ) ˇ

on cycles of type Σ3 × {t} and holonomy χ(σ ˇ 2 ) on cycles of type σ2 × S 1 . We will refer to the character (6.8) as the twist of [CˇX ] by χ. ˇ Choosing 1 the Neveu-Schwarz, or bounding spin structure, S− , we define the correction factor in terms of the the M-theory phase (4.12): (6.10)

ϕ(CˇX , χ) ˇ := Φ([CˇX ] + tˇ · χ; ˇ X × S−1 )

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ˇ 41 (Y ) for closed spin 11Note that since Φ is defined as a function on H λ 2 manifolds Y equation (6.10) makes sense. Also note that, since we are working / RS )/2] must be with a fixed metric a definite choice of the sign of exp[iπξ(D made here. The idea behind (6.10) is essentially stated in [25], footnote 19. An easy argument shows that (6.10) indeed satisfies the requisite cocycle law. The cocycle (6.10) can be expressed in terms of η invariants of Dirac operators by using the construction of equations (3.28 - 3.33) above. A similar construction allows one to define the group lift for the entire group G. We need to check that (6.10) really satisfies (6.7). The proof is a standard cobordism argument. We consider a differential character on the spin 11ˇ + tˇ · χˇ1 , [C] ˇ + tˇ · manifold (X × S−1 ) ∪ (X × S−1 ) ∪ (X × S−1 ) restricting to [C] ˇ − tˇ· (χˇ1 + χ χ ˇ2 , [C] ˇ2 ) on the three components. We then choose the extending spin 12 manifold to be Z = X × ∆ where ∆ is a pair of pants bounding the three circles with spin structure restricting to S−1 on the three components. To be explicit we can choose ∆ to be the simplex {(t1 , t2 ) : 0 ≤ t1 ≤ t2 ≤ 1} with identifications ti ∼ ti + 1. Now we extend the differential character as (6.11)

ˇ + tˇ1 · χ [C] ˇ1 + tˇ2 · χ ˇ2

which clearly restricts to the required character on the boundary. The field strength is GZ = GX + dt1 ∧ ω1 + dt2 ∧ ω2 . For such extensions, ΦWitten = Φ because the gravitino index is trivial. We can therefore use (4.2) to show that the product of phases around the boundary is Z Z 1 ˜3 ˜ 1 (6.12) exp 2πi G − GI8 = exp −2πi G ∧ ω 1 ∧ ω2 2 X ∆×X 6 where 21 is the area of ∆. This proves the desired cocycle relation (6.7). ˇ χ) The following property of the cocycle ϕ(C, ˇ will be useful in what follows. If χˇb is a topologically trivial character then

(6.13)

ˇ χˇ + χˇb ) = ϕ(C, ˇ χ)e ϕ(C, ˇ −2πi

R

X

b( 12 G2 −I8 )

This can be proved using the variational formula (5.6), or by using a cobordism argument, since for a topologically trivial χˇb we may fill in S−1 with a disk and continue the character as χˇrb where r is the radius on the disk. 6.3. A physical motivation for formula (6.10). The choice of cocycles defining the group lift (6.2) is not unique. Therefore we would like to justify the choice (6.10). This definition of the group lift is necessary if we want to represent path integrals in certain twisted topological sectors as traces over Hilbert space. Suppose Ψ(Cˇ1 , Cˇ2 ) is the path integral on the cylinder X × [0, 1]. Then, it is the kernel of the operator e−βH where H is the Hamiltonian and β is the ˇ C) ˇ length of the cylinder. By integrating over the diagonal of the kernel Ψ(C,

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we represent a trace on Hilbert space as a path integral over X × S 1 : (6.14)

Za (X × S 1 ) = TrHa (X) e−βH

On both the left and right hand side we are working within a topological sector defined by a ∈ H 4 (X, ZZ). On the right hand side we have the trace over the Hilbert space of wavefunctions with support on C-fields of characteristic class a; on the left-hand side the path integral is the integral over isomorphism classes of C-fields with Cˇ ∈ EP (a) (X × S 1 ), where P (a) is simply pulled back from X to X × S 1 . We would now like to twist this construction using the group G(X). That is, we would like a path integral representation of (6.15)

TrHa (X) U (g)e−βH

where U (g) is the unitary operator implementing the action of g on the Hilbert space. Therefore, we would like to glue together the ends of a cylinder with boundary conditions related by an element g ∈ G(X). Thus we consider ˇ C) ˇ for Cˇ ∈ EP (a) (X). This is an element of a line L∗ˇ ⊗ L ˇ . Now, to Ψ(g C, gC C give a lift of the group action to L is to give a specific isomorphism (6.16)

L∗Cˇ ⊗ LgCˇ ∼ =C

Such an isomorphism is necessary if we are to integrate over the boundary values Cˇ to produce a path integral over C-fields on X × S 1 in the topological sector a + [dt]a(χ). ˇ The path integral is, after all, just a complex number. If we consider simple field configurations, such as those with isomorphism class [CˇX ] + tˇ · χˇ on the cylinder then there is no phase factor in the evolution on the cylinder, since the evolution is through a path of gauge transformations. On the other hand, the phase factor in the path integral is (6.10) and this phase can only enter through the choice of isomorphism (6.16). Here again the E8 formalism proves to be very useful since it allows one to make precise the notation of a “loop L in the space of C fields on R” used in [25]. To do this one relates a group transformation (α, χ) ˇ to an E8 bundle automorphism (3.28) and uses this to define a twisted bundle with connection on X × S 1 as in (3.30,3.31). 7. The definition of C-field electric charge In electromagnetism the electric current satisfies (7.1)

d ∗ F = je .

If je has compact support (or falls off sufficiently rapidly) one then defines n−1 [je ] ∈ HDR,cpt (X) as the electric charge of the source. On a closed manifold X, this must be zero, since ∗F provides an explicit trivialization.

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ˇ is cubic and hence the theory In M-theory, the Chern-Simons term Φ(C) is nonlinear. The equation of motion (in Lorentzian signature) is: 1 1 d ∗ G = G2 − I8 (g) 3 ` 2 Thus, background metrics and C-fields induce electric charge. As an element of DeRham cohomology 1 8 (7.3) [ G2 − I8 ] ∈ HDR (X) 2 is the induced charge. Nevertheless, (7.3) is not a suitable expression for the electric charge induced by a C-field and metric. The quantization law (2.12) on the C-field implies, via Dirac quantization, that the C-field electric charge is valued in integral cohomology H 8 (X, ZZ). 15 Thus, in order to describe the electric charge induced by the self-interactions of C (and gravity) we need to define a precise integral lift of (7.3). We will do this in the next section by considering the micro gauge transformations of section 3.3. (7.2)

7.1. The action of automorphisms on L. In section 6.2 we have defined a group action (0, χ) ˇ : LA,c → LA,c+ω(χ) ˇ is flat, i.e., χˇ ∈ H 2 (X, U (1)) ˇ . When χ then ω(χ) ˇ = 0 and hence (0, χ) ˇ · (A, c) = (A, c). These are the automorphisms of the objects (A, c) in the categorical approach. If Ψ ∈ LA,c then (7.4)

(0, χ) ˇ · Ψ = ϕ(CˇX , χ) ˇ∗Ψ

and ϕ(CˇX , χ) ˇ can be a nontrivial phase. If Ψ is a gauge invariant wavefunction nonvanishing at CˇX then the invariance under the automorphisms of CˇX requires ϕ(CˇX , χ) ˇ = 1 for all flat χ. ˇ We will now regard flat characters as 2 elements χ ˇ ∈ H (X, IR/ZZ). The cocycle law (6.7) then implies that on flat characters, ϕ(CˇX , ·) is a homomorphism (7.5)

ϕ(CˇX , ·) : H 2 (X, IR/ZZ) → U (1)

Poincare duality now implies the existence of an integral class, ΘX (CˇX ) ∈ H 8 (X, ZZ) such that (7.6)

ϕ(CˇX , χ) ˇ = exp[2πihΘX (CˇX ), χi] ˇ

Equation (7.6) defines, mathematically, an important integral class ΘX (CˇX ). Note that if a gauge invariant wavefunction is nonvanishing at CˇX then (7.7) ΘX (CˇX ) = 0. This is the first nontrivial consequence of the Gauss law. 15Moreover,

recall that the quantization law on G follows from the existence of elementary membranes. Indeed, the elementary membrane has C-field electric charge given by the isomorphism H2 (X, ZZ) ∼ = H 8 (X, ZZ).

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Since ΘX (CˇX ) is an integral class it only depends on the characteristic class a of CˇX (and the spin structure of X). Therefore, we will sometimes denote it as ΘX (a). ˇ is C-field electric charge. To interpret ΘX (a) let us “insert” 7.2. ΘX (C) a membrane wrapping a cycle σ ∈ Z2 (X) in the boundary X. Let us conˇ Relative to the sider the gauge invariance of the new wavefunction Ψσ (C). wavefunction without the insertion of σ the flat gauge transformations χˇ with χ ˇ ∈ H 2 (X, IR/ZZ) act on the wavefunction with an extra phase (7.8) χ(σ) ˇ = exp 2πihP D[σ], χi ˇ

Membranes carry C-field electric charge. Comparison of (7.8) with (7.6) shows that we should interpret ΘX as the C-field charge induced by the background metric and C-field. An important consistency check on our argument is the following. Using (6.13) and comparing with the definition (7.6) shows that 1 (7.9) [ΘX ]IR = [ G2 − I8 ]DR 2 The above arguments lead to a simple extension of the Gauss law: In the presence of membranes wrapping a spatial cycle σ ∈ Z2 (X), a gauge invariant wavefunction can have support at Cˇ only if ˇ + P D([σ]) = 0 (7.10) ΘX (C)

We have discussed a necessary condition for the existence of a nonvanishing gauge invariant wavefunction. The E8 formalism is very useful for giving the full statement of the Gauss law, and provides an interesting interpretation of the quantization of “Page charges.” This will be discussed elsewhere [23]. ˇ 8. Mathematical Properties of ΘX (C) ˇ is a subtle integral cohomology class defined by E8 ηThe class ΘX (C) invariants, and not much is known about it. Here we collect a few known facts. As we have explained above, restriction to topologically trivial and flat Cfield gauge transformations χˇb , for db = 0 shows that in DeRham cohomology we have (7.9). Since [G]IR = aIR − 21 λIR , we learn that 16 1 (8.1) [ΘX (a)]IR = aIR (aIR − λIR ) + 30Aˆ8 2 2 (To derive this note that 30Aˆ8 = 18 λ2 − I8 = 7λ 48−p2 .) There are three basic facts about the integral class ΘX (a). First, hΘX (a), χi ˇ is a spin cobordism invariant of (X, a, α), where a ∈ H 4 (X) and α ∈ H 2 (X, IR/ZZ). incidentally, that this implies that there is a natural integral lift of 30Aˆ8 defined on spin 10-folds. One easily checks that on CP 5 , 30Aˆ8 = (xIR )4 , where x generates H 2 (CP 5 ; ZZ), so 30 is the smallest multiple of Aˆ8 that has an integral lift. 16Note,

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Second, from this and a computation of spin bordism groups one can prove that ΘX is a quadratic refinement of the cup product: (8.2)

ΘX (a1 + a2 ) + ΘX (0) = ΘX (a1 ) + ΘX (a2 ) + a1 a2

Third, Θ is “parity invariant,” i.e. it satisfies the identity (8.3)

ΘX (a) = ΘX (λ − a).

ˇ 41 via In order to prove (8.3) we define a parity transformation on H λ 2

(8.4)

χˇP (Σ) = (χ(Σ)) ˇ

∗

The parity transform takes (8.5)

a→λ−a G → −G.

Under this transformation the M-theory phase transforms as (8.6)

ˇ P ; Y ) = (ΦW ([C]; ˇ Y )∗ ΦW ([C]

ˇ then [CˇZ ]P extends [C] ˇ P , and G([CˇZ ]P ) = This follows since, if [CˇZ ] extends [C] −G([CˇZ ]). Now we simply note that the integral in (4.2) is odd in GZ . Applying this observation to the case Y = X × S 1 we see that ∗ (8.7) ΦW ([CˇX ]P − tˇ · χ; ˇ X × S−1 ) = ΦW ([CˇX ] + tˇ · χ; ˇ X × S−1 )

where we have made a parity transformation t → −t on the circle. Now, (8.3) follows from (8.7). The cobordism invariance of hΘX (a), χi ˇ and the bilinear identity (8.2) allows us to compute ΘX (a) in several simple examples. To choose but one example take X = L(3, p1 )×L(3, p2 )×S 4 , where L(3, p) is any three-dimensional Lens space S 3 /ZZp . Let v be a generator of H 4 (S 4 , ZZ). If bi ∈ H 2 (L(3, pi )) = ZZpi (we omit pullback symbols) then (8.8)

˜ + b1 b2 ) = Θ(v) ˜ ˜ 1 b2 ) + vb1 b2 Θ(v + Θ(b

˜ X (a) := ΘX (a) − ΘX (0). Now Θ(b ˜ 1 b2 ) = 0 because we can fill in where Θ 4 S = ∂B5 and extend the Cheeger-Simons character over this manifold. At ˜ the same time we can extend the class α ∈ H 2 (X, U (1)). Similarly, Θ(v) = 0. Here we fill in one of the two Lens spaces (any oriented 3-manifold has a bounding oriented 4-manifold). So, we conclude: (8.9)

˜ + b1 b2 ) = vb1 b2 Θ(v

where we used that the spin bordism group is trivial in 3 dimensions. Equation (8.9) is potentially relevant to G2 compactifications and 5-brane physics. It would be helpful for topological investigations of M-theory if further methods were developed to compute the subtle integral class ΘX (a).

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9. Φ as a cubic refinement, with applications to integration over flat C-fields 9.1. The sum over flat C-fields as a subintegration in the path integral for Ψ(CˇX ). In this section we make some comments on the nature of the wavefunction Ψ(CˇX ) associated to a manifold X with boundary Y . The sum over the gauge copies of CˇY in the path integral automatically projects onto gauge invariant wavefunctions. Therefore, the wavefunction will vanish unless ΘX (CˇX ) = 0. We restrict attention to these components. The wavefunction will be a sum over topological sectors. These are labelled by extensions a ˜ of the characteristic class of CˇX . Let ι : X ,→ Y , then we ∗ sum over ker ι ⊂ H 4 (Y, ZZ). By the long exact sequence this is equivalent to a sum over H 4 (Y, X; ZZ)/δH 3 (X; ZZ)

(9.1)

where δ is the connecting homomorphism. For each extension a ˜ of a ∈ H 4 (X; ZZ) we choose an extending bundle PY of characteristic class a ˜. We can fix the Ω1 (adP ) gauge symmetry by choosing an extension (A˜0 , c˜0 ) of (AX , cX ). All other extensions are given by (A˜0 , c˜0 + cY ) with cY ∈ ker ι∗ ⊂ Ω3 (Y ). Let us denote this space by Ω3 (Y, X). The integral (6.1) reduces to an integral over Ω3 (Y, X)/Ω3 (Y, X)ZZ. In the path integral we integrate over the compact space of harmonic forms H3 (Y, X)/H3 (Y, X)ZZ

(9.2)

where H3 (Y, X) := ker ι∗ restricted to H3 (Y ), and the addition of such fields yields no cost in action. That is, the real part of the Euclidean action is unchanged by the addition of such fields. In the sum over (9.1) one can (noncanonically) split the integration over the torsion subgroup and a lattice. The sum over the torsion subgroup can be combined with the integral over (9.2) to produce an integral over ker ι∗ applied to H 3 (Y, U (1)). Using the topological considerations of this paper one can make some exact statements about the nature of this subintegration. In the next section we spell out these statements. 9.2. The cubic refinement law. In order to study the sum over flat C-fields we will need a result which is interesting and important in its own right,17 namely the interpretation of the M-theory phase as a cubic refinement of a ˇ 4 (Y ). trilinear form on H The product of differential characters on Y together with evaluation on Y leads to a trilinear form (ˇ a1 a ˇ2 aˇ3 )(Y ) ∈ U (1). 18 The cubic refinement law 17Indeed, 18The

this is the starting point for [8]. product of differential characters is described in [9].

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states that (9.3) ˇ +a ˇ +a ˇ + aˇ2 ; Y )Φ([C] ˇ +a Φ([C] ˇ1 + a ˇ2 + a ˇ3 ; Y )Φ([C] ˇ1 ; Y )Φ([C] ˇ3 ; Y ) = (ˇ a1 a ˇ2 a ˇ3 )(Y ) ˇ ˇ ˇ ˇ Y) Φ([C] + a ˇ1 + a ˇ2 ; Y )Φ([C] + a ˇ1 + a ˇ3 ; Y )Φ([C] + aˇ2 + a ˇ3 ; Y )Φ([C]; Recall that Φ descends to shifted differential characters, so it makes sense ˇ 4 (Y ), while [C] ˇ is a shifted character. In order to prove (9.3) to take aˇi ∈ H ˇ note that if C, a ˇi all extend simultaneously on the same spin 12-fold then we may use Witten’s definition (4.2) as an integral in 12 dimensions. The cubic refinement law follows from the simple algebraic identity 1 1 1 1 (a + x + y + z)3 − (a + x + y)3 − (a + y + z)3 − (a + x + z)3 6 6 6 (9.4) 6 1 1 1 1 + (a + x)3 + (a + y)3 + (a + z)3 − a3 = xyz 6 6 6 6 It thus follows that if we can simultaneously extend the differential characters ˇ and a [C], ˇi then we have (9.3). It follows from the E8 model for the C-field that, if we can extend the characteristic classes of the characters then we can extend the entire character. Therefore we consider the purely topological ˇ together with the classes ai . The extenproblem of extending the class a(C) sions of the individual classes exist by Stong’s theorem. Next, consider the obstruction to the existence of an extension of an 11-manifold together with a pair of classes (Y, a1 , a2 ). The obstruction to finding an extension of (9.5)

(Y, a1 , a2 ) + (Y, 0, 0) − (Y, a1 , 0) − (Y, 0, a2 )

is measured by the group Ωspin 11 (K(Z, 4) ∧ K(Z, 4)). Similarly, the obstruction for triples (Y, a1 , a2 , a3 ) lies in Ωspin 11 (K(Z, 4) ∧ K(Z, 4) ∧ K(Z, 4)) and that for spin 4 quartets lies in Ω11 (∧ K(Z, 4)). All of these groups can be shown to vanish by a simple application of the Atiyah-Hirzebruch spectral sequence. 9.3. Summing over the flat C-fields. We can now apply the cubic refinement law to learn some facts about the sum over flat C-fields on Y . For simplicity we will consider the case of ∂Y = ∅. As we discussed, evaluating the sum over flat C-fields reduces to evaluation of the integral Z (9.6) [dCˇf ]Φ(Cˇ + Cˇf ; Y ) H 3 (Y,U (1))

where [dCˇf ] is the natural measure given by the Riemannian metric. Recall that H 3 (Y, U (1)) is a disjoint union of connected tori. The connected component of the identity is H 3 (Y, ZZ) ⊗ U (1), and the quotient by this subgroup is isomorphic to the torsion group HT4 (Y ). 19 To evaluate this we first integrate 19We

will denote the torsion subgroup of H p (Y, ZZ) by HTp (Y ).

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over the subgroup of topologically trivial flat C-fields, writing the sum (9.6) as: X Z (9.7) [dCˇf ]Φ(Cˇ + Cˇf ; Y ) HT4 (Y )

H 3 (Y )⊗U (1)

Using the variational formula for the topologically trivial flat fields we learn that the integral over the topologically trivial flat characters H 3 (Y ) ⊗ U (1) vanishes unless 1 ˜2 (9.8) [ G − I8 ]DR = 0 ∈ H 8 (Y ; IR) 2 Thus, the sum over flat C-fields projects onto fields allowing a solution to the equation of motion. When (9.8) is satisfied the function of flat characters Cˇf → Φ(Cˇ + Cˇf ; Y ) descends to an interesting U (1)-valued function (9.9)

ˇ + Cˇf ) φ¯[C] ˇ (aT ) := Φ(C

of torsion classes aT ∈ HT4 (Y ). The way Φ(Cˇ + Cˇf ; Y ) depends on aT is not obvious since changing aT changes the isomorphism class of the E8 bundle. It is here that the cubic refinement law (9.3) becomes quite useful, since it follows that the function φ¯[C] ˇ is a cubic refinement of the U (1)-valued trilinear form (9.10)

exp[2πiha1 a2 , a3 i]

on HT4 (Y ). Here we have used the torsion pairing HT8 (Y ) × HT4 (Y ) → Q/ZZ. In conclusion the sum over flat fields projects onto fields satisfying (9.8) and for such fields we have a further projection onto topological sectors such that X (9.11) φ¯[C] ˇ (a) a∈HT4 (Y )

is nonzero. The sum in (9.11) is a sum of exponentiated cubic forms on a finite abelian group, and is hence a kind of “Airy function” generalization of Gauss sums. If we specialize further to the case Y = X × S+1 with Ramond, or nonbounding, spin structure on S 1 we can go further, and make contact with the results of [7, 15]. In this case HT4 (Y ) ∼ = HT3 (X) ⊕ HT4 (X) and hence the sum over torsion classes may be arranged as a sum X X (9.12) φ¯[C] ˇ (hdt + a) h∈HT3 (X) a∈HT4 (X)

and it is fruitful to study the sum over HT4 (X) for fixed values of h. ˇ is also pulled back from X. Then, as noted in [7] Let us assume that [C] ˇ 4 (X) be a differential character we have a dramatic simplification. Let a ˇ∈H

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with characteristic class a (not necessarily torsion). Then Φ(Cˇ + a ˇ; Y ) ˇ ˇ (9.13) = eiπ[f (a(C)+a)−f (a(C ))] ˇ Φ(C; Y ) / V (a) on X. Moreover, where f (a) is the mod 2 index of the Dirac operator D f (a) is a quadratic refinement of a bilinear form: Z (9.14) f (a1 + a2 ) = f (a1 ) + f (a2 ) + r2 (a1 )Sq 2 r2 (a2 ). X

where r2 denote reduction modulo two. It now follows that φ¯[C] ˇ (hdt + a) 4 defines a quadratic refinement as a function of a ∈ HT (X). Indeed, the cubic refinement law reduces to the quadratic refinement law: ¯ ˇ (hdt) φ¯[C] ˇ (hdt + a1 + a2 )φ [C] 3 (9.15) = exp 2πiha , (Sq + h)a i 1 2 ¯ ˇ (hdt + a2 ) φ¯[C] ˇ (hdt + a1 )φ[C] where we have used the bilinear identity for the E8 mod 2 index proved in [7], and again have used the torsion pairing HT4 (X) × HT7 (X) → Q/ZZ. Thus, for fixed values of h we are averaging a quadratic form over a finite abelian group in (9.12). We now need a little lemma from group theory. Let A, B be finite abelian groups, with a perfect pairing (9.16)

A × B → Q/ZZ

which we shall write as ha, bi. Suppose we have a homomorphism ϕ : A → B such that (9.17)

Q(a1 , a2 ) = ha1 , ϕ(a2 )i

is a symmetric bilinear form. Let fQ : A → Q/ZZ be a quadratic refinement of this bilinear form, that is: (9.18)

fQ (a1 + a2 ) = fQ (a1 ) + fQ (a2 ) + ha1 , ϕ(a2 )i

Note first that, when restricted to ker ϕ, fQ is a character, and hence there is P ∈ B/Imϕ such that the restriction of fQ to ker ϕ is given by (9.19)

fQ (a) = ha, P i

a ∈ ker ϕ

Moreover, P 6= 0 implies fQ (a) 6= 0 for some a. Now, we have the key statement: The sum X (9.20) S(fQ ) := e2πifQ (a) a∈A

is nonzero if and only if P = 0. We now apply the above discussion with A = HT4 (X), B = HT7 (X) and ϕ = Sq 3 + h and Φ(Cˇ + hdt + a) ˇ (a) = e2πifC,h (9.21) ˇ Φ(C + hdt)

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3 where fC,h ˇ (a) is a quadratic refinement of the quadratic form ha1 , (Sq +h)a2 i on HT4 (X). When restricted to a ∈ ker(Sq 3 + h) we have fC,h ˇ (a) = ha, PC,h ˇ i. It follows that the path integral vanishes due to the sum over flat fields unless there exists some C-field Cˇ0 , such that

(9.22)

fCˇ0 ,h (a) = ha, P0 i = 0

for all a ∈ ker(Sq 3 + h)|HT4 . Suppose such a C-field exists. We may ask what other C-fields contribute. Applying once more the cubic refinement law we find (9.23)

3 ˇ − a(Cˇ0 ))i fC,h ˇ0 ,h (a) + ha, (Sq + h)(a(C) ˇ (a) = fC

ˇ and a(Cˇ0 ) need not be torsion. Thus we conclude where a is torsion, but a(C) ˇ which contribute to the path integral must that the topological sectors a(C) satisfy: (9.24)

ˇ − a(Cˇ0 )) = 0 (Sq 3 + h)(a(C)

mod(Sq 3 + h)HT4 (X)

This simplifies (considerably) the discussion of the “Gauss law” in [7] and generalizes it to arbitrary torsion h-fields. Using a similar strategy one can easily derive an SL(2, ZZ) “equation of motion” for the torsion components of the C-field on 11-manifolds of the type Y = X9 × T 2 where T 2 carries the RR (a.k.a. odd, or nonbounding) spin structure. (The reader should compare our discussion with sec. 11 of [7].) Choose coordinates t1 , t2 on T 2 with ti ∼ ti + 1 and consider C-fields ˇ · tˇ2 where Cˇ0 , gˇ, h ˇ are all pulled back from of the type Cˇ = Cˇ0 + gˇ · tˇ1 + h ˇ can be X9 and, for clarity, we will drop all pullback symbols. The fields gˇ, h interpreted in type IIB string theory with ω(ˇ g) = Gr the fieldstrength of the ˇ RR 3-form and ω(h) = Hns the NSNS 3=form fieldstrength. The sum over the topologically trivial flat C-fields imposes (9.8), which in turn shows that (9.25)

[G0 ∧ Hns ]DR = [G0 ∧ Gr ]DR = [Gr ∧ Hns ]DR = 0

where G0 = ω(Cˇ0 ). The equations (9.25) are indeed standard consequences of the supergravity equations of motion. However, once these equations are satisfied there is a further constraint on the characteristic classes ag := a(ˇ g) ˇ and ah = a(h). Note that these classes need not be torsion classes, although, by (9.25) ag ah is a torsion class. Combining the cubic refinement law (9.3) with the bilinear identity (9.14) we find (9.26)

ˇ · tˇ2 + Cˇf ; Y ) Φ(Cˇ0 + gˇ · tˇ1 + h = haf , Sq 3 (ag + ah ) + ag ah ieiπf (af ) ˇ ˇ ˇ ˇ Φ(C0 + gˇ · t1 + h · t2 ; Y )

where Cˇf is a flat character on X9 , af = a(Cˇf ) and f (af ) is the mod 2 index on X9 × S+1 . By the bilinear identity eiπf (af ) = haf , P i where P depends only

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on the topology and spin structure of X9 . In particular, it is independent of ˇ Thus, we arrive at the SL(2, ZZ) invariant equation of motion: Cˇ0 , gˇ, h. (9.27)

Sq 3 (ag ) + Sq 3 (ah ) + ag ah = P

While this does not fully resolve the puzzle in section 11 of [7], it does constitute progress. 10. Application 1: The 5-brane partition function In addition to the electrically charged membranes, M-theory has magnetically charged 5-branes. In [25, 26] Witten has analyzed in detail topological considerations concerning the 5-brane partition function. In particular, he stated topological conditions necessary for the construction of a nonzero partition function. In this section we make contact with his work and the related work of Hopkins and Singer [9]. In particular we interpret a certain anomaly cancellation condition of Witten’s in terms of the class ΘX (a). As a preliminary, let us discuss some geometrical facts. The worldvolume of the 5-brane is denoted by W . We assume W is compact and oriented and embedded in an 11-dimensional spacetime ι : W ,→ Y . We may identify a tubular neighborhood of W in Y with the total space of the normal bundle N → W . The unit sphere bundle of radius r, X = Sr (N ) is then an associated S 4 bundle π : X → W , and we may construct an 11-manifold Yr with boundary X by removing the disk bundle of radius r, Yr = Y − Dr (N ). If X is oriented, compact, and spin, while W is orientable and compact, then one can show that the Euler class of the normal bundle vanishes and hence the Gysin sequence simplifies to give a set of short exact sequences (10.1)

π∗

π

0 → H k (W, Z) → H k (X, Z) →∗ H k−4 (W, Z) → 0.

¿From this we conclude that π ∗ : H 3 (W, Z) → H 3 (X, Z) is an isomorphism. Moreover, (10.2) H 4 (X, Z) ∼ = H 4 (W, Z) ⊕ Z The isomorphism is noncanonical, indeed, a choice of splitting is defined by 4 a choice of global angular form υ ∈ Hcpt (N ) with π∗ (υ) = 1. The general degree four class on X can then be written (10.3)

a = π ∗ (¯ a) + kυ.

with k ∈ Z. Finally, it follows from the above that (10.4) H 3 (X, U (1)) ∼ = H 3 (W, U (1)). 10.1. Statement of Anomaly inflow. The 5-brane is a magnetic source for the C-field. This means that if S 4 is a small sphere linking W in Y then Z (10.5) G=k S4

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for a “5-brane of charge k.” Here k is a nonzero integer. In the E8 model for the C-field this means that there is an E8 instanton on the 4-sphere of instanton number k. Consequently, the bundle P (a) → Yr cannot be smoothly prolonged to a bundle over Y . There are two ways to approach this difficulty. First, one may note that it is natural to associate the magnetic ˇ ) ∈ Zˇ 5 (Y ) to a single 5-brane wrapping current in differential cohomology δ(W W and view the C-field as a 4-cochain giving rise to this class. A second point of view, which we shall adopt here, proceeds by dividing up the physical system into two subsystems, the brane and the exterior. More precisely, we divide up Y into a small tubular neighborhood Dr (W ) of W and the complement Yr of this neighborhood. The two regions overlap on the 4sphere bundle X. The exterior of the 5-brane, Yr is referred to as the “bulk” and is described by 11-dimensional supergravity. In particular, the C-field path integral over Yr is the wavefunction (10.6)

Ψbulk ∈ Γ(L → Met(X) × EP (X))

discussed throughout this paper. In order to define the “partition function of M-theory with a 5-brane wrapping W ” we must define the 5-brane partition function ZM 5 together with a well-defined pairing (10.7)

hZM 5 , Ψbulk i

which can be integrated over, for example, the space of C-fields on X to produce the full partition function. The existence of a well-defined, gauge invariant function (10.7), is the so-called anomaly inflow cancellation statement. Note, in particular, that ZM 5 must be a section of a dual line bundle to L. Note that, since the 5-brane is a brane, ZM 5 should only depend on local data near W . In terms of the metric, it should be a function of the induced metric on W and its first few normal derivatives, such as the induced connection on the normal bundle N W . Moreover, ZM 5 should only depend on the “C-fields on W .” For this reason we should consider the limit that the radius r of the 4-sphere bundle goes to zero in discussing (10.7). The gauge equivalence classes of “C-fields on W ” will be shifted differential ¯ ∈H ˇ 41 (W ). In order to make the anomaly inflow statement characters [C] λ 2 W precise we must relate C-fields on W to C-fields on X. To that end we choose a C-field Cˇ0 so that we can map ˇ 41 (X) ˇ 41 (W ) → H (10.8) i:H 2

λW

2

λX

¯ = [Cˇ0 ] + π ∗ [C]. ¯ The choice of basepoint Cˇ0 is not canonical. Note via i[C] that it should be an unshifted character since π ∗ (λW ) = λX . The map (10.8) is compatible with the isomorphism (10.9)

J 3 (W ) → J 3 (X)

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where J 3 (X) is the connected component of the identity in H 3 (X, U (1)). Therefore, the construction of the line L over J 3 (X) can be used to construct a line L−1 over J 3 (W ). In section 3.2 of [25] (see especially p. 125) Witten uses the anomalyinflow statement to determine the line of which ZM 5 should be a section. He then notes that Hodge ∗ defines a complex structure on J 3 (W ) such that the curvature form of L−1 Z (10.10) ω(x1 , x2 ) := −2π x1 x2 X

is of type (1, 1). In this complex structure L−1 can be given the structure of a holomorphic line bundle which in turn gives J 3 (W ) the structure of a principally polarized abelian variety. The partition function ZM 5 is then identified, up to a metric-dependent constant, with the unique holomorphic section of this line bundle. The condition of holomorphy is interpreted as the condition of self-duality of the 3-form fieldstrength of the 5-brane. 10.2. Intrinsic and extrinsic definitions of the 5-brane partition function. In [25, 26], Witten has in fact given two definitions of the 5-brane partition function, which we will refer to as the “intrinsic” and “extrinsic” definitions. The extrinsic definition is the one based on anomaly inflow, as reviewed in the previous section. The intrinsic definition proceeds only from the data of a compact oriented Riemannian 6-manifold W . In the intrinsic definition we consider J 3 (W ) as a complex manifold with complex structure induced by ∗. We seek to define a holomorphic line bundle LM 5 → J 3 (W ) with Hermitian metric and curvature given by (10.10). In order to define the holomorphic line one must find a function (10.11)

ΩW : H 3 (W, ZZ) → ZZ2

which satisfies the cocycle relation: (10.12)

ΩW (x1 + x2 ) = ΩW (x1 )ΩW (x2 ) exp[iπ

Z

x1 x2 ] W

Different choices of ΩW correspond to lines LM 5 differing by tensoring with a flat holomorphic bundle of order 2. In [25, 26] Witten constructs such a function ΩW , when W is Spinc , making use of the Spinc structure. 20 It follows from (10.12) that ΩW is a homomorphism HT3 (W, ZZ) → ZZ2 ,→ Q/ZZ and hence by Poincar´e duality ΩW (x) = hθ, xi, for x ∈ HT3 (W ), is pairing with a distinguished element θ ∈ HT4 (W ). In [26], eqs. 5.5 et. seq., Witten then points out that in constructing a theta function as a sum over H 3 (W, ZZ), if θ 6= 0, the sum will 20This

construction is given in section 5.2 of [26]: One extends W × S 1 to a Spinc 8-fold R∂B = W × S 1 , and simultaneously extends x[dt] to a class z. Then ΩW (x) := exp[iπ B8 (z 2 + λz)], where 2λ = p1 − α2 and r2 (α) = w2 (B8 ).

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vanish, indicating the presence of an anomaly. Thus, Witten’s anomaly cancellation condition is θ = 0. In the next section we will interpret the class θ in terms of ΘX (a). 10.3. The decoupling conditions. It is widely believed that 5-brane theories exist independently of M-theory as distinguished (2, 0)-superconformal field theories in six dimensions. The “simple” theories are expected to be classified by ADE gauge groups and reduce, upon compactification on a circle and upon taking a long distance limit, to 5-dimensional nonabelian gauge theories with 16 supersymmetries. Nevertheless, the only known definition of the 5-brane theory is that given by taking a decoupling limit from the theory of the 5-brane in 11-dimensional supergravity. Therefore, we should only expect to be able to give an “intrinsic definition” to the 5-brane partition function when the brane can be consistently decoupled from the bulk. We will now interpret Witten’s torsion anomaly condition θ = 0 as a necessary condition for decoupling the 5-brane from the bulk. Let us now recall ˇ as C-field charge. If ΘX (C) ˇ 6= 0 then we cannot the interpretation of ΘX (C) decouple the 5-brane from the bulk. Effectively, open 2-branes must end on the 5-brane to satisfy charge conservation, and these spoil decoupling. Thereˇ = 0 is a necessary condition for fore, physical reasoning implies that ΘX (C) decoupling of the M5 brane from the bulk, i.e. a necessary condition for the existence of a nonzero “5-brane partition function.” In order to state the decoupling condition in terms intrinsic to W let us consider the integration over the fiber: π∗ (ΘX (a)) ∈ H 4 (W6 , ZZ). We use (10.3) to conclude that that a = π ∗ (¯ a) + υ for a single brane. Then we may split (noncanonically) π∗ (ΘX (a)) = 0 into its torsion and nontorsion components. In DeRham cohomology π∗ (ΘX (a)) = 0 implies that a¯ = 0. This is the well-known condition that the “C-field on W ” must have a fieldstrength which can be written as (10.13)

¯ = dh G

for some globally well-defined 3-form h. The form h is interpreted as the fieldstrength of the chiral 2-form field on W . When (10.13) is satisfied π∗ (ΘX (a)) is a torsion class. As we have seen, it is precisely this class which obstructs the definition of a well-defined line bundle L → J 3 (X) ∼ = J 3 (W ). When ΘX (a) = 0 we can identify (10.14)

ΩW (x) = ϕ(Cˇ0 , χ). ˇ

Here χ ˇ is a character such that ω(χ) ˇ = x. Of course, such lifts are ambiguous by flat characters, but, precisely because ΘX (a) = 0, this ambiguity drops out of the right hand side of (10.14). This is in accord with the argument in [26], section 5.3 . (The latter argument relied on several topological restrictions on the normal bundle of W in Y . )

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In general, the anomaly inflow argument implies that, for α ∈ H 2 (W, U (1)) we should identify hα, θi = ΩW (β∗ α), where β∗ is the Bockstein homomorˇ = hα, π∗ ΘX (C)i, ˇ phism, with Φ(X × S−1 , Cˇ + tˇπ ∗ α) = hπ ∗ (α), ΘX (C)i and ˇ thus we conclude that θ = π∗ (ΘX (C)). 11. Application 2: Relation of M-theory to K-theory In the previous section we have shown that if π∗ (ΘX ) 6= 0 then there must be 2-branes ending on a 5-brane. In type II string theory there is an analogous statement for D-branes. If a D-brane wraps a worldvolume W in isolation, i.e. with no other D-branes ending on it, then it is necessary that: (11.1)

(Sq 3 + [H])P D(W ) = 0.

This condition is closely related to the K-theoretic classification of D-branes, as explained in [7, 15, 27, 28, 29]. If Y11 is S 1 -fibered over a 10-manifold U10 , then we expect a relation between M theory on Y and type II string theory on U10 . Therefore, in this situation we expect a relation between ΘX (a) and the left hand side of (11.1). We will now show that this is indeed the case. For simplicity, suppose Y11 = U10 × S 1 . Then ∂Y11 = X10 , ∂U10 = V9 , so X10 = V9 × S 1 . A characteristic class of CˇX on X10 can be written as: (11.2)

a = π ∗ (¯ a) + π ∗ ([H])[dx11 ]

where a¯ ∈ H 4 (V9 ; ZZ), [H] ∈ H 3 (V9 ; ZZ), and π : X10 → V9 is the projection. The first result is that if X10 = V9 × S 1 has Ramond (nonbounding) spin structure on S 1 and [H] = 0 then ˜ X (a)) = Sq 3 (¯ (11.3) π ∗ (Θ a) Thus the Gauss law and the K-theoretic restriction on the RR fluxes are indeed closely related. To prove (11.3) we compute the pairing of the left hand side with α ∈ 2 H (V9 , U (1)). This is (11.4) hΘX (a), π ∗ (α)i = Φ(p∗ Cˇ + tˇp∗ (π ∗ α), ˇ X × S−1 ) ˇ 4 (X10 ), p : X10 × S 1 → X10 , and the coordinate on S 1 is t. where Cˇ ∈ H − − If Cˇ is pulled back from V9 then we may regard the 11-manifold Y = X10 × S−1 instead as Y = X 0 × S+1 , with X 0 = V9 × S−1 . Then we can apply to result of [7] to obtain ˜ X (a), αi = eiπ[f (¯a+dtβ(π∗ α))−f (dtβ(π∗ α)] (11.5) hπ∗ Θ where β : H 2 (V9 , U (1)) → HT3 (V9 ) is the Bockstein and f (a) is the E8 mod 2 index on X 0 = V9 × S−1 . Here we have used [7] eq. 8.24 and the fact that β(α) is flat, hence the local term cannot contribute because the local density does not contain dt. Now we use the bilinear identity of [7]. f (¯ a) = 0 since the S−1 is a bounding spin structure. Therefore (11.6)

˜ X (a), αi = e hπ∗ Θ

iπ

R

1 V9 ×S−

a ¯Sq 2 dtβ(α)

= hSq 3 (¯ a), αi

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where in the second equality we used the pairing H 7 (V9 , ZZ) × H 2 (V9 , U (1)) → U (1). This establishes (11.3). It is also of interest to know π∗ (ΘX (0)). The style of argument above shows that (11.7)

hα, π∗ (ΘX (0))i = eiπf (β(α)dt)

This formula shows that π∗ (ΘX (0)) is at most two-torsion. There seems to be no elementary formula for Rthis mod two index. If V9 = X8 × S+1 then one can show that 21 f (bdt) = r2 (b)Sq 2 r2 (b) for b ∈ H 3 (X8 , ZZ). The latter expression is, in general, nonzero (e.g. for X8 = SU (3) and b a generator of H 3 (SU (3); ZZ) ), and hence we conclude that π∗ (ΘX (0)) is in general nonzero. Now let us turn to the inclusion of an [H] flux. It is straightforward to show (11.8)

π∗ (ΘX (a)DR ) = [H]DR ∧ a ¯DR

There are two ways to lift this to a statement about the integral classes. First, using the bilinear identity (8.2) (with a2 → a2 − a1 , together with (11.3) we find that if ai = π ∗ a¯i + π ∗ [H][dx11 ] then (11.9)

π∗ (ΘX (a1 ) − ΘX (a2 )) = (Sq 3 + [H])(¯ a1 − a¯2 )

Moreover, using (8.3) we also have (11.10)

¯ π∗ (2ΘX (a)) = [H](2¯ a − λ)

Now, (11.10) and (11.9) together amount to the “moral” statement 1¯ (11.11) π∗ (ΘX (a))“ = 00 (Sq 3 + [H])(¯ a − λ) 2 (it is only a moral statement because the division by 2 is illegal in the presence of 2-torsion). Hence, the Gauss law is in harmony with the K-theoretic classification of RR fluxes. The above arguments extend the results of [7, 15] to manifolds with boundary. To complete the story one should demonstrate that the C-field wavefunction is related to the RR flux wavefunction in the expected manner. This is an interesting problem, but we leave it for future work. (Some preliminary remarks appear in [30].) 12. Application 3: Comments on spatial boundaries One of the primary motivations for developing the E8 formalism for the C-field is the desire to make precise the intuition that the C-field is related to some kind of topological gauge theory in the bulk of 11 dimensions, but 21Let

+ denote the nonbounding, − the bounding spin structure on S 1 . Note that f−− (bdt1 ) = f−− (bdt2 ) = 0 where the subscript denotes the spin structure. Note that under a diffeomorphism (t1 , t2 ) → (t1 , t1 + t2 ) we have f+− (bdt2 ) = f−− (bdt1 + bdt2 ). Now apply the bilinear identity.

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which becomes a dynamical theory upon the introduction of boundaries. In this section we make some simple comments on one realization of that idea. Suppose now we have a spatial boundary ι : X ,→ Y . The adjective “spatial” means that we will be regarding X as a Euclidean spacetime, rather than as a temporal slice. We need to choose boundary conditions on the Cfield. One natural choice of boundary condition on Cˇ = (A, c) is the condition (12.1)

ι∗ (c) = 0.

That is, we restrict attention to the C-fields: (12.2)

EP (Y, X) := {(A, c) ∈ EP (Y )|ι∗ (c) = 0}

Now, to define a physical theory, we need to describe the gauge symmetry. It is quite standard for boundary conditions to break a symmetry group G to a subgroup H. It is thus natural to consider the subgroup of G(Y ) of elements that preserve (12.2). In the present case, the group elements (α, χ) ˇ ∈ ∗ ∗ G(Y ) which preserve the entire set (12.2) must satisfy ι (α) = ι (ω(χ)) ˇ = 0. Such group transformations leave “too many” physical degrees of freedom. In particular, any two connections on ι∗ (P ) would be considered to be gauge inequivalent. It is here that the categorical viewpoint of section 3.4 becomes quite useful. Let Pˆ = ι∗ (P ). There is an obvious restriction functor r : EP (Y )//G(Y ) → EPˆ (X)//G(X). The simplest way to characterize the category defined by the boundary condition (12.1) is that it is the subcategory of EP (Y )//G(Y ) which maps under r to a category with the morphisms of the form (A, 0) → (Ag , 0), ˆ To say this a little more formally, we noted in section 3.4 for g ∈ AutP. above that ordinary bundle automorphisms of Pˆ define a subgroupoid of EPˆ (X)//G(X), in spite of the fact that Aut(Pˆ ) is not a subgroup of G(X). Let us fix a functor I from the standard gauge theory groupoid A(Pˆ )//Aut(Pˆ ) to EPˆ (X), by choosing I(A) = (A, 0) on objects. The groupoid of C-fields and gauge transformations defined by (12.1) is the fiber product EP (Y, X): r

(12.3)

EP (Y )//G(Y ) → EPˆ (X)//G(X) ↑ ↑I ˆ EP (Y, X) → A(P )//Aut(Pˆ )

The situation is best described as follows: This boundary condition breaks the topological gauge symmetry G. It is not possible to describe it as the breaking of a gauge group, but we can Rview it as a symmetry breaking of groupoids. Note in particular that while Y trF ∧ ∗F is not gauge R invariant, and hence there are no propagating gauge modes in the interior, X trF ∧ ∗F is gauge invariant and therefore, with the above notion of a groupoid of fields, we can define a gauge invariant theory with dynamical E8 gauge fields on the boundary.

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12.1. Relation to heterotic M-theory. Now that we have explained one way in which dynamical gauge fields on the boundary can be related to the topological gauge theory of the E8 gauge field in the interior we will indicate how it can be compatible with the outstanding example of M-theory on a manifold with spatial boundary, namely the Horava-Witten model [4, 5] of heterotic M-theory. Consider the E8 model of the C-field on an 11-manifold of the type X ×[0, 1] and impose boundary conditions ι∗L (c) = ι∗R (c) = 0. We learned in the previous section that we indeed find dynamical E8 gauge fields on each boundary. However, because the 11-dimensional spacetime provides a homotopy of the left and right connections the E8 bundles on the boundaries necessarily have characteristic classes aL = aR . Thus, this theory resembles the nonsupersymmetric model introduced by Fabinger and Horava [31]. The above simple observation raises a challenge to using the E8 formalism to describe the Horava-Witten model. We may overcome this difficulty as follows. Note that M-theory should be formulated without use of an orientation, but our formulation breaks parity automatically since a → λ − a under orientation reversal. We may give a parity invariant formulation of the M-theory C-field by passing from Y to Yd , the orientation double cover of Y , and defining a Cfield to be a parity invariant E8 cocycle on Yd . Let σ be the nontrivial deck transformation on Yd , a parity invariant E8 cocycle is one such that the shifted differential character satisfies: ˇ = [C] ˇP (12.4) σ ∗ ([C]) where we recall that the parity transform was defined in (8.4). If Y is orientable, this amounts to saying that a C-field is a pair {(A1 , c1 ), (A2 , c2 )} such that (12.5)

χ ˇ(A1 ,c1 ) = χˇ∗(A2 ,c2 )

The gauge group is G(Yd ) × G(Yd ). This gauge invariance preserves (12.5). So, again, no degrees of freedom are added when Y has no boundary. However, on a manifold with boundary this formulation leads to a natural boundary condition which again leads to dynamical gauge fields on the boundary. On X × [0, 1] we take (12.6)

i∗L (c1 ) = 0

i∗R (c2 ) = 0

Then, we get a dynamical gauge field AL with characteristic class aL on the left boundary and another one AR with aR on the right boundary. Note that ι∗L ([G]) = aL − 12 λ, while ι∗R ([G]) = −(aR − 12 λ) and hence aL + aR = λ, as expected. It is important to note that with our boundary conditions ι∗ (G) = ±(trF 2 − 1 trR2 ), on left- and right- boundaries, respectively, while dG = 0 throughout 2 the 11-manifold. This is to be contrasted with the equation one often finds

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in the literature on heterotic M-theory, namely, 1 1 (12.7) dG = δ(x) trF (AL )2 − trR2 ± δ(π − x) trF (AR )2 − trR2 . 2 2 In our view, such a Bianchi identity implies that the boundary carries nontrivial magnetic current, and a proper formulation of the C-field will involve a different geometrical construction from what we have used. It is also worth noting that since Φ is valued in the Pfaffian line bundle the E8 gauge anomalies cancel (locally) in a way which is manifest. The gravitational anomalies are more subtle, but we expect that the same will be true for them. We hope to discuss this elsewhere [23]. (See [32] for a different point of view.) 13. Conclusions and future directions In this paper we have given a precise mathematical formulation of the E8 model for the M-theory C-field. We have used it to write the Chern-Simons term of the 11-dimensional supergravity action and we have used it to describe the Gauss law for the C-field in precise terms, applicable to topologically nontrivial situations. In particular we have shown how to define the C-field electric charge induced by the nonlinear interactions of the C field, and by gravity, as an integral cohomology class. We have also given other applications to a clarification of the topological conditions for the existence of the 5-brane partition function and to the relation of M-theory flux quantization to IIA Ktheoretic flux quantization. Finally, we have sketched how one may formulate the topological field theory of the E8 gauge field to allow for dynamical gauge fields on the boundary of an 11-manifold. Many related problems remain open and issues remain to be resolved. We will survey some of these problems here. • Gravitino determinant. There are several nontrivial technical issues related to the gravitino path integral. We hope to address these elsewhere [23]. • Derivation of the K-theoretic classification of RR fields. The K-theoretic formulation of RR fields should be generalized to manifolds with boundary and the wavefunction of the C-field in IIA supergravity compared to the wavefunction of C-field in M theory. We made some progress in section 11 but the story remains to be completed. • Is the E8 formalism really necessary ? One of the virtues of the E8 formalism is that it allows us to define the action of 11-dimensional supergravity, and the Gauss law for wavefunctions of C-fields on manifolds with boundary. Nevertheless, the E8 gauge field plays a purely topological role and appears, in some sense, to be a “fake.” As we have mentioned, the category EP (X)//G is equivalent to the cateogry Zˇ 41 λ (X) of Hopkins-Singer cocycles. Indeed, 2 there is an alternative construction of the C-field and M-theory action which

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makes no use of E8 gauge fields [8]. What remains to be seen is whether any of these formalisms is really useful for physical investigations, and whether they lead to a useful reformulation of M-theory. • Electric-Magnetic duality. The E8 formalism is seemingly very asymmetric between 5-branes and 2-branes. Formally, one would expect a dual formuˇ 7 (X). lation of the theory in terms of Cheeger-Simons characters [CˇD ] ∈ H These objects would define the holonomies of the C-field on 5-brane worldvolumes. However, there is no obvious E8 model for a dual object CˇD . Since the theory is nonlinear, this duality transformation is not obvious. • Parity invariance. A basic axiom of M theory is that it is parity invariant. This is what allows us to gauge parity and produce chiral theories such as the heterotic string. In this paper, we have assumed that Y is an oriented 11-manifold and we have used the orientation heavily in writing integrals of differential forms, and in defining spinor bundles and η invariants. A very interesting open problem is the generalization of our formalism to unoriented and nonorientable 11-manifolds. • Anomaly cancellation. In spite of the shortcommings noted above, we believe that the present formalism should have many future applications to anomaly cancellation issues connected to 5-branes, G2 singularities, and frozen singularities in M-theory [33]. For example, the present formalism leads to a substantial simplification of the anomaly cancellation for normal bundle anomalies of the M5-brane described in [34]. Acknowledgements: We would like to thank J. Harvey, M. Hopkins, D. Morrison, G. Segal, and E. Witten for useful discussions on this material. G.M. and D.F. would like to thank H. Miller and D. Ravenel for the invitation to speak at the Newton Institute conference on elliptic cohomology. Portions of this work were carried out by G.M. at the Aspen Center for Physics. Portions of this work were carried out by D.F. and G.M. at the Isaac Newton Institute, Cambridge and at the KITP, Santa Barbara. The work of E.D. and G.M. is supported in part by DOE grant DE-FG02-96ER40949. The work of D.F. is supported in part by NSF grant DMS-0305505. References [1] An online version is available at http://online.kitp.ucsb.edu/online/mp03/moore1. [2] D. S. Freed, “K-theory in quantum field theory,” Current Developments in Mathematics 2001, International Press, Somerville, MA, 41-87 [arXiv:math-ph/0206031]. [3] D. S. Freed, “Dirac charge quantization and generalized differential cohomology,” Differ. Geom. VII (2000) 129–194 [arXiv:hep-th/0011220]. [4] P. Horava and E. Witten, “Eleven-Dimensional Supergravity on a Manifold with Boundary,” Nucl. Phys. B 475, 94 (1996) [arXiv:hep-th/9603142].

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[5] P. Horava and E. Witten, “Heterotic and type I string dynamics from eleven dimensions,” Nucl. Phys. B 460, 506 (1996) [arXiv:hep-th/9510209]. [6] E. Witten, “On flux quantization in M-theory and the effective action,” J. Geom. Phys. 22, 1 (1997) [arXiv:hep-th/9609122]. [7] D. E. Diaconescu, G. W. Moore and E. Witten, “E(8) gauge theory, and a derivation of K-theory from M-theory,” Adv. Theor. Math. Phys. 6, 1031 (2003) [arXiv:hepth/0005090]. [8] D. Freed and M. Hopkins, to appear. [9] M. J. Hopkins and I. M. Singer, “Quadratic functions in geometry, topology, and M-theory,” arXiv:math.at/0211216. [10] J. Evslin, “From E(8) to F via T,” arXiv:hep-th/0311235. [11] V. Mathai and H. Sati, “Some Relations between Twisted K-theory and E8 Gauge Theory,” arXiv:hep-th/0312033. [12] E. Witten, “Topological Tools In Ten-Dimensional Physics,” Int. J. Mod. Phys. A 1, 39 (1986). [13] S. S. Chern and J. Simons, “Characteristic forms and geometric invariants,” Ann. Math. 99 ( 1974) pp. 48–69. [14] D. S. Freed, Classical Chern-Simons Theory II, Houston J. Math. 28 (2002), no. 2, 293–310. [15] D. E. Diaconescu, G. W. Moore and E. Witten, “A derivation of K-theory from Mtheory,” arXiv:hep-th/0005091. [16] D. Morrison, Talk at Strings2002, http://www.damtp.cam.ac.uk/strings02/avt/morrison/. [17] P. Horava, “M-Theory as a Holographic Field Theory,” hep-th/9712130. [18] S. Melosch and H. Nicolai, “New canonical variables for d = 11 supergravity,” Phys. Lett. B 416, 91 (1998) [arXiv:hep-th/9709227]. [19] J. Lott, “R/Z index theory,” Comm. Anal. Geom. 2 (1994) 279. [20] G. W. Moore, “Some comments on branes, G-flux, and K-theory,” Int. J. Mod. Phys. A 16, 936 (2001) [arXiv:hep-th/0012007]. [21] J. Dupont, R. Hain and S. Zucker, Regulators and characteristic classes of flat bundles, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 47–92, Amer. Math. Soc., Providence, RI, 2000, alg-geom/9202023. [22] R. Stong, “Calculation of Ωspin 11 (K(Z, 4))” in Unified String Theories, 1985 Santa Barbara Proceedings, M. Green and D. Gross, eds. World Scientific 1986. [23] To appear. [24] X. Dai and D. S. Freed, “Eta-invariants and determinant lines,” J. Math. Phys. 35, 5155 (1994) [Erratum-ibid. 42, 2343 (2001)] [arXiv:hep-th/9405012]. [25] E. Witten, “Five-brane effective action in M-theory,” J. Geom. Phys. 22, 103 (1997) [arXiv:hep-th/9610234]. [26] E. Witten, “Duality relations among topological effects in string theory,” JHEP 0005, 031 (2000) [arXiv:hep-th/9912086]. [27] J. M. Maldacena, G. W. Moore and N. Seiberg, “D-brane instantons and K-theory charges,” JHEP 0111, 062 (2001) [arXiv:hep-th/0108100]. [28] J. Evslin and U. Varadarajan, “K-Theory and S-Duality: Starting Over from Square 3,” hep-th/0112084; Journal-ref: JHEP 0303 (2003) 026. [29] G. Moore, “K-theory from a physical perspective,” arXiv:hep-th/0304018. [30] G. W. Moore and E. Witten, “Self-duality, Ramond-Ramond fields, and K-theory,” JHEP 0005, 032 (2000) [arXiv:hep-th/9912279]. [31] M. Fabinger and P. Horava, “Casimir effect between world-branes in heterotic Mtheory,” Nucl. Phys. B 580, 243 (2000) [arXiv:hep-th/0002073]. [32] A. Bilal and S. Metzger, “Anomaly cancellation in M-theory: A critical review,” Nucl. Phys. B 675, 416 (2003) [arXiv:hep-th/0307152].

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[33] J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D. R. Morrison and S. Sethi, “Triples, fluxes, and strings,” Adv. Theor. Math. Phys. 4, 995 (2002) [arXiv:hep-th/0103170]. [34] D. Freed, J. A. Harvey, R. Minasian and G. W. Moore, “Gravitational anomaly cancellation for M-theory fivebranes,” Adv. Theor. Math. Phys. 2, 601 (1998) [arXiv:hepth/9803205]. Department of Physics, Rutgers University, Piscataway, New Jersey, 088550849 and Department of Mathematics, University of Texas at Austin

ALGEBRAIC GROUPS AND EQUIVARIANT COHOMOLOGY THEORIES J.P.C. GREENLEES Abstract. Many important cohomology theories are associated to algebraic groups, and this is clearest when equivariant theories are considered. The purpose of this article is to describe some familiar examples and to speculate on the existence of further examples.

Contents 1. Introduction. 2. K-theory and the multiplicative group. 3. The shape of a cohomology theory. 4. The non-split torus. 5. Elliptic cohomology and elliptic curves. 6. T-equivariant elliptic cohomology. 7. Shapes from projective varieties. 8. Rational equivariant cohomology theories. 9. The model for the circle group G = T. 10. Reflecting the group structure of the elliptic curve. References

89 91 93 96 97 99 100 103 105 107 109

1. Introduction. 1.A. The context. This article discusses a connection between the world of algebraic geometry and the world of algebraic topology. In the world of algebraic geometry, an algebraic group G is a functor associating a group G(k) to a commutative ring k. In the world of algebraic topology, a G-equivariant cohomology theory associates a graded abelian group EG∗ (X) to a space X with G-action, where G is the ambient compact Lie group of equivariance. Sometimes an algebraic group G corresponds to an equivariant cohomology theory EG∗ (·) in a way we will describe. Going from topology to geometry is the more elementary process, and we will describe several well-known examples: in fact when G is the circle group T, suitable cohomology theories ET∗ (·) give rise to one dimensional algebraic The author is grateful to M.Ando, M.J.Hopkins, H.R.Miller and N.P.Strickland for collaborations and discussions which have informed this work.

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groups G by evaluation on the appropriate spaces. More striking is the fact that we can sometimes reverse the process and move from geometry to topology. From the easy direction, we shall see that a one dimensional algebraic group G suggests the values of an equivariant cohomology theory ET∗ (·) on equivariant spheres. Rather remarkably, the algebraic model for rational Tequivariant cohomology theories [11] shows that, for rational theories, the values on spheres are very close to determining the cohomology theory, and in favourable cases they determine it completely. Going one step further, one can often construct a cohomology theory from its putative values on spheres, and this is often the case for values arising from algebraic groups. To put flesh on this philosophy we need some examples. 1.B. Some familiar examples. In the most familiar case, the multiplicative group Gm is associated to K-theory, with the association being clearest for KT∗ (·) where T is the circle group. This is discussed in detail in Section 2, and if the introduction seems too abstract, the reader should read Section 2 first. It is also familiar that the additive group Ga is associated to ordinary cohomology with the association being clearest for the T-equivariant Borel theory H ∗ (ET ×T (·)). Similarly, there is at least a formal group associated to any complex oriented theory E ∗ (·), with the association being clearest for the T-equivariant Borel theory E ∗ (ET ×T (·)). When the formal group is the formal completion of an ellipic curve around the identity we obtain elliptic cohomology theories, but again the association between the cohomology theory and the ellipitc curve is clearest for the T-equivariant version of the theory [19, 15]. It is the new behaviour of this T-equivariant elliptic theory, and the possibility of adapting its construction that led to the author writing this article. 1.C. Some putative examples. We will speculate on higher dimensional examples, at least when both the algebraic group and the compact Lie group of equivariance are abelian and connected. In this case it seems that if an algebraic group G of dimension d is associated to a cohomology theory EG∗ (·) then the compact Lie group G is of dimension d, and in general one might expect their ranks to agree. However we will see that it is very rare for EG∗ (·) to be complex orientable, so that these theories will be of an unfamiliar sort. 1.D. Some motivation. One reason for this investigation is to understand familiar objects better. Equivariant versions of elliptic cohomology behave in a way which is unfamiliar to topologists. To develop appropriate intuition, we need to put this into context. One might even hope that the new context would help us to find a truly geometric definition. History has shown that cohomology theories arising from algebraic groups are significant, having applications to index theory [32, 31], rigidity [25, 2, 3] and to geometric representation theory [10, 9]. This alone makes it worth seeking new ones.

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In any case, the strength of the connection with algebraic geometry was already the motivation for the introduction of equivariant elliptic cohomology: applying the functor to natural objects in representation theory gives interesting algebras. Similar constructions are straightforward for other cohomology theories based on algebraic groups. 2. K-theory and the multiplicative group. We will describe properties of the multiplicative group, K-theory and the relationship between them in some detail because the underlying phenomena are well known, and several of the issues arise in this case. 2.A. The multiplicative group and the multiplicative formal group. The multiplicative group Gm is the functor on commutative rings l defined by Gm (l) = Units(l) = Rings(Z[z, z −1 ], l). The first equality is the definition, and the second shows that Gm is represented by the ring Z[z, z −1 ]. We also say that Z[z, z −1 ] is the ring of functions on Gm and write OGm = Z[z, z −1 ]. The group operation is given by multiplication of units Units(l)×Units(l) = Rings(Z[z, z −1 ]⊗Z[z, z −1 ], l) −→ Rings(Z[z, z −1 ], l) = Units(l), and this induced by the ring homomorphism ∆ : Z[z, z −1 ] −→ Z[z, z −1 ] ⊗ Z[z, z −1 ] z 7−→ z ⊗ z. The identity element of Units(l) is 1 ∈ l and therefore the identity e of Gm is represented by the augmentation Z[z, z −1 ] −→ Z setting z = 1. Accordingly, the ideal of functions vanishing at e is the principal ideal (y) where y = 1 − z. We say that y is a coordinate at the identity. In terms of y, the coproduct then takes the more familiar form y 7−→ 1 ⊗ y + y ⊗ 1 − y ⊗ y. We may then complete around the identity to form the multiplicative formal group (Gm )∧e with ring of functions (Z[z, z −1 ])∧(y) = Z[[y]]. 2.B. T-equivariant K-theory and T-equivariant Borel K-theory. We can easily recover the multiplicative group from T-equivariant K-theory and the multiplicative formal group from T-equivariant Borel K-theory, where T is the circle group. Just as the multiplicative formal group is only an approximation to the multiplicative group itself, so Borel K-theory K ∗ (ET×T X) is only an approximation to the true equivariant theory KT∗ (X) for a Tspace X. Just as the multiplicative formal group is a completion of the multiplicative group, so the Atiyah-Segal completion theorem says Borel Ktheory is a completion of equivariant K-theory.

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The cohomology theory associated to the group Gm is T-equivariant Ktheory for the circle group T. Indeed, the ring of functions on Gm is realized by KT0 = R(T) = Z[z, z −1 ]. The coproduct is realized by the multiplication map µ : T × T −→ T in the sense that the coproduct is µ∗

0 KT0 −→ KT×T = KT0 ⊗ KT0 .

The cohomology theory associated to the formal group (Gm )∧e is Borel Ktheory for the circle group T. Indeed, the ring of functions on the formal group (Gm )∧(e) is realized by its value on a point K 0 (BT) = K 0 (ET ×T pt) = (KT0 )∧J = R(T)∧(e(z)) = (Z[z, z −1 ])∧(y) = Z[[y]], where the first equality is the Atiyah-Segal completion theorem [5] with J = ker(R(T) −→ Z) the augmentation ideal, and the other equalities hold since J = (e(z)) and the Euler class e(z) = 1 − z = y. The coproduct is realized by the multiplication map Bµ : BT × BT = B(T × T) −→ T since ˆ 0 (BT). K 0 (BT × BT) = K 0 (BT)⊗K 2.C. Reconstructing T-equivariant K-theory from the multiplicative group. To fully justify saying that K-theory is associated to Gm we would need to reverse the process of making the multiplicative group from K-theory. In general it is not known how to make such a construction, but it can be done rationally using the algebraic model for rational T-spectra [11]. We return to this in Section 9.C below. 2.D. A-equivariant formal groups and A × T-equivariant Borel Ktheory. There is an intermediate stage between the multiplicative group and its completion at the identity: we may complete Gm along a subgroup. This is known as an equivariant formal group [6, 14, 29]. More precisely, we choose a homomorphism ζ : A∗ −→ Gm of algebraic groups, where A is a finite abelian group and A∗ = Hom(A, S 1 ) is the dual group. The equivariant formal group is the formal neighbourhood (Gm )∧ζ of the image of ζ equipped with the map A∗ −→ (Gm )∧ζ . To obtain an interesting construction, we need some points of finite order. Thus we want ζ to be specified by a map A∗ −→ Units(k) for a ring k with more units than Z. Accordingly, we change base to the version of the multiplicative group Gm |k over k: this is the functor defined on k-algebras l and has ring of functions k[z, z −1 ]; by the Yoneda lemma, ζ is now given by a homomorphism ζ : A∗ −→ Units(k) = Gm |k (k) of ordinary groups. The homomorphism ζ is represented by the Hopf algebra map θ : k[z, z −1 ] −→ A∗ kQ with αth component θα (z) = 1 − ζ(α) for α ∈ A∗ . Thus θ has kernel ( α∈A∗ (1 − ζ(α))) and the ring of functions on (Gm |k )∧ζ is the completion O(Gm |k )∧ζ = k[z, z −1 ]∧(Qα∈A∗ (1−ζ(α))) .

The universal example of this is when k = R(A) = Z[A∗ ] and ζ(α) = α.

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This algebraic intermediate stage is also realized topologically, by using a suitable A × T-equivariant Borel version of A-equivariant K-theory. To 0 explain this, we start by observing that KA×T = R(A × T) = R(A)[z, z −1 ] is the ring of functions on the multiplicative group Gm |k over k = R(A), and the coproduct is still induced by µ : T × T −→ T. We can now construct the ring of functions on the A-equivariant formal group by using a Borel theory. This time we need the universal T-free A-space EA T, which is the total space of the universal bundle over the classifying space BA T for Aequivariant line bundles. We may then take the Borel theory KA0 (EA T ×T X) where X is an A × T-space. Again the Atiyah-Segal completion theorem [1] gives a connection with rings of functions on the equivariant formal group: the ring of functions on (Gm |k )∧ζ when k = R(A) is 0 0 0 KA0 (BA T) = KA×T (EA T) = (KA×T )∧(e(CA⊗z)) = (KA×T )∧(Qα (1−αz)) .

The components of the fixed point set (BA T)A correspond to the A-equivariant line bundles over a point (i.e., to points of A∗ ), and the resulting A-map A∗ −→ BA T induces the structure map A∗ −→ (Gm |k )∧ζ of the equivariant formal group. 3. The shape of a cohomology theory. Before we turn to other examples we must be more precise about what we mean by a cohomology theory being associated to an algebraic group. In this section we consider the example of K-theory in some detail. The description is phrased so that it can be easily transposed to give general principles in the following sections. 3.A. The definition of the shape. Non-equivariantly, the spheres S n play a dual role. They are the building blocks for spaces and are also invertible as spectra. In G-equivariant homotopy theory, nice spaces are built from the suspended homogeneous spaces S n ∧ G/K+ . These are not invertible unless K = G, and the group theory means that the cohomology groups E˜G∗ (S n ∧ ∗ G/K+ ) = E˜K (S n ) can be complicated as modules over the coefficient ring EG∗ . On the other hand, the one point compactification S V of a representation V is invertible as a spectrum, and E˜G∗ (S V ) is often rather simple. For this reason, we may view the values on representation spheres S V as a rather basic invariant of the homology or cohomology theory. Partly because of variance, it is simpler to concentrate on homology, but the analogue for cohomology is also useful. Definition 3.1. The shape of an equivariant homology theory E˜∗G (·) is the composite functor ˜ G (·) E

∗ AbGp RepC (G) −→ G-spectra −→

where RepC (G) is the category of unitary representations of G and inclusions.

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For example, if E˜∗G (·) is complex oriented then E˜∗G (S V ) is a shifted copy of the coefficient group E∗G . However, even in this case the shape includes useful information. 3.B. Equivariant K-theory. We concentrate on the circle group G = T. In this case, any complex representation is a sum of the one dimensional representations z n for integers n, where z is the natural representaton of T on C. Applying K-theory to the inclusion 0 −→ V gives a diagram ˜ T (S 0 ) = 0 − 7 → K 0 ↓ ↓ T ˜ V − 7 → K0 (S V ) ∼ =

R(T) = ↓ χ(V ) ∼ R(T) =

R(T ) ↓ iV 1 · R(T ). χ(V )

˜ 0T (S V ) ∼ The isomorphism K = R(T), and the fact that the inclusion S 0 −→ S V induces multiplication by the Euler class χ(V ) come from equivariant Bott periodicity. The last isomorphism has replaced multiplication by the Euler class χ(V ) by an inclusion, at the price of making 1/χ(V ) the generator of the free R(T )-module. For a 1-dimensional representation α we have χ(α) = 1−α, and χ(V ⊕ W ) = χ(V )χ(W ), so that we may easily calculate χ(V ) in general. In particular for V = z n then we find χ(z n ) = 1 − z n , which is familiar as the function on the multiplicative group Gm whose vanishing defines the n subgroup Gm [n] := ker(Gm −→ Gm ) of points of order dividing n. Writing OGm (Gm [n]) for the ideal of functions permitted a simple pole on Gm [n], we may summarize the discussion by the calculation ˜ 0T (S z n ) = 1 · R(T ) = OGm (Gm [n]). K 1 − zn Thus the shape of K-theory is specified by functions on the multiplicative group; more precisely, if V T = 0 we have ˜ 0T (S V ) = OGm (D(V )) K P P for a suitable divisor D(V ); indeed if V = n6=0 an z n then D(V ) = n6=0 an Gm [n]. Suspension by the trivial complex representation (i.e., double real suspension) corresponds to the tangent bundle (Ω1Gm )∨ . It is convenient to prove the dual statement that desuspension by corresponds to Ω1Gm . Since we have ˜ 0 (S W ) = K ˜ T (S −W ), we work in cohomology to prove K ˜ 0 (S 2 ) ∼ K = Ω1Gm . First 0 T T note that S 2z is obtained from S z by attaching a single T-free 3-cell, so that we have a cofibre sequence Σ2 T+ −→ S z −→ S 2z . ˜ 0 . Remembering that we are now working in cohomology, Now apply K T 0 z ˜ (S ) = J and K ˜ 0 (S z ) = J 2 , where J = OGm (−(e)) is the ideal of functions K T T vanishing at the identity, and we obtain the a short exact sequence ˜ 0 (Σ2 T+ ) ←− J ←− J 2 ←− 0. 0 ←− K T

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Hence the value of the non-equivariant theory on S 2 is the conormal bundle at the identity (i.e., the cotangent space at the identity since we are dealing with a point): ˜ 0 (S 2 ) = K ˜ 0 (Σ2 T+ ) ∼ K = J/J 2 = ωe1 . Accordingly, double suspension in cohomology is given by tensoring with ωe1 , and in particular the equivariant theory is the module of K¨ahler differentials ˜ 0 (S 2 ) = ω 1 ⊗K 0 K 0 ∼ K = Ω1 T

e

T

Gm

as claimed, where the final isomorphism is given by using the group structure on Gm to trivialize the module, or in more concrete terms, by using the function z. The fact that Ω1Gm is trivializable corresponds to the fact that KT∗ is 2-periodic. Since the K-theory of all complex spheres is in even dimensions, this describes all of K-theory. 3.C. Complex orientable theories. The above discussion also applies to complex orientable cohomology theories E ∗ (·) and their associated formal b Since different notation may be more familiar to some readers, we groups G. provide a brief comparison. The relevant equivariant cohomology is the Borel theory ET∗ (X) = E ∗ (ET ×T X), so we have ET∗ = E ∗ (BT) and

E˜T∗ (S V ) = E˜ ∗ (BTV ), where BTV denotes the Thom space of the bundle associated to the representation V . The ring E ∗ (BT) is complete for the skeletal topology, and is b and since we have a therefore the ring of functions on a formal scheme G, ∗ b completed K¨ unneth theorem for E (BT × BT), multiplication on T gives G the structure of a formal group. Now consider the ideal J = ker(ET∗ = ET∗ (pt) −→ ET∗ (T) = E ∗ ) of functions vanishing at the identity, which is principal by complex orientabilb and the coefficient ring is the ity. A generator y of J is a coordinate on G, ∗ ∗ power series ring ET = E [[y]]. The cofibre sequence T+ −→ S 0 −→ S z ˜ ∗ (BT z ). Since shows that y is the restriction of a Thom class in E˜T∗ (S z ) = E T n ∗ zn z = n z we see that a Thom class for BT is given by [n](y), the pull back of y along the nth power map Bn : BT −→ BT. In these terms, and remembering we are working in cohomology, we recover the usual formula b ˜ ∗ (S z n ) = E˜ ∗ (BT z n ) = ([n](y)) = O b (−G[n]). E T

G

Specializing, this gives ˜ ∗ (S z ) = E˜ ∗ (BT z ) = (y) = J = O b (−(e)), E T G

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and hence the behaviour under untwisted suspensions E˜ ∗ (S 2 ) ∼ = J/J 2 ∼ = ωe1 and E˜T∗ (S 2 ) ∼ = E˜ ∗ (BT+ ∧ S 2 ) ∼ = Ω1Gb . 3.D. What is special about K-theory and the multiplicative group? In algebra and topology there are several very special features of the examples described in this section. They are rather obvious in these cases, but the analogues become more interesting as we change the algebraic group and the cohomology theory: we will argue that the algebraic and topological special features correspond precisely. We concentrate on K-theory for simplicity, and we have in mind G = A × T for a finite abelian group A. • The ideal of functions on Gm vanishing at the identity is principal, and KG∗ (·) is complex orientable. • Gm is affine and KG∗ is in even degrees. • The group Gm is one dimensional and KG∗ is 2-periodic. We will see in Section 4 that the first property fails for the non-split torus, in Section 5 that the second property fails for the elliptic case, and in Section 7 that the third fails for abelian varieties of dimension ≥ 2. 4. The non-split torus. We describe a one dimensional algebraic group whose ideal of functions vanishing at the origin is not principal. There is a corresponding cohomology theory which is therefore not complex orientable. 4.A. The non-split torus and its formal group. Let C be a 1-dimensional torus over a Q-algebra k, with ring of functions O. The standard torus is the multiplicative group C = Gm with O = k[z, z −1 ], but there is also a non-split (non-deploy´e) form Gnd with O = k[a, b]/(a2 + b2 = 1), which we consider here. This is related to Gm since they become isomorphic when we adjoin a square root i of −1. We then have z = a + ib, and the usual formula (a1 + ib1 )(a1 + ib1 ) = (a1 a2 − b1 b2 ) + i(a1 b2 + a2 b1 ) gives the coproduct a 7−→ a ⊗ a − b ⊗ b b 7−→ a ⊗ b + b ⊗ a. Now the counit sets a = 1 and b = 0 so that the ideal of functions vanishing at the identity is (1 − a, b). This is not a principal ideal. Furthermore, even if we complete at the identity, the ideal (1 − a, b) is still not principal. Thus even the Borel theory of a theory associated to the non-split torus will not be complex orientable. One way to obtain Gnd is as a quotient of Gm |k(i) . Indeed, k[a, b]/(a2 + b2 = 1) is the fixed ring of k(i)[z, z −1 ] under the action of the group of order 2 acting by Galois automorphisms on k(i) and exchanging z with z −1 .

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4.B. Constructing the associated cohomology theory. Rationally, one may construct a T-equivariant cohomology theory associated to Gnd by the method of Section 9. It would be interesting to have a bundle theoretic construction. Complex orientable theories ET∗ (·) have the property that for any complex representation V the ET∗ -module E˜T∗ (S V ) is free on one generator. The cohomology theory associated to Gnd is therefore not complex orientable since its value on S z is the non-principal ideal of functions vanishing at the identity. 5. Elliptic cohomology and elliptic curves. We now replace the multiplicative group by an elliptic curve C over k. This makes both the algebra and the topology more complicated, but probably the biggest change is that C is no longer affine. For general background on elliptic curves and algebraic geometry see [28, 20]. 5.A. Elliptic curves and their formal completions. Because C is not affine we must replace the ring of functions by the sheaf of functions O = OC . Since all regular functions on C are constant, if we want to recover C from a knowledge of functions, we need to permit some poles. These are measured by considering formal sums of points of C, known as divisors. Given a divisor P D = P n(P )P on C we may form the line bundle O(D), of functions which have poles of order ≤ n(P ) at P . The Riemann-Roch theorem states that if D has degree ≥ 1 then P

dimk H 0 (C; O(D)) = deg(D),

where deg(D) = P n(P ). Thus in particular if we concentrate on the identity e and consider a multiple D = ne, the sheaf O(ne) allows functions with a pole of order n ≥ 1 at the identity e, and we obtain n independent functions since dimk H 0 (C; O(ne)) = n. Thus if n = 1 we only obtain the constants, for n = 2 we obtain a new function, call it x, and for n = 3 we obtain a new function, call it y. For n = 4 we may use x2 as the new function, and for n = 5 we may use xy. However for n = 6 we have two potential new functions, y 2 and x3 . There is therefore a linear relation amongst the 7 functions so far, and this may be put into the Weierstrass form y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 , for suitable a1 , a2 , a3 , a4 , a6 in k. It turns out that this implies all relations, and defines C. Indeed, we may consider the graded ring k(C, e) := {H 0 (C; O(ne))}n≥0 , and the above discussion my be restated briefly in geometric language as C = Proj(k(C, e)). In the Weierstrass form, we find there is one point at infinity, and we may choose this as the identity of the group C.

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5.B. Reflecting the group structure on functions. For an elliptic curve C the graded ring k(C, e) is not a Hopf algebra. This is because the pullback of a line bundle L, such as O(ne), along the multiplication map C × C −→ C does not decompose as a tensor product. However, for any line bundle L we may form M = M(L) = p∗1 L ⊗ p∗2 L, and then if we define ξ : C × C −→ C × C by ξ(x, y) = (x + y, x − y), the theorem of the square shows that ξ ∗ M ∼ = M⊗2 . This gives ξ ∗ : H 0 (C; L) ⊗ H 0 (C; L) −→ H 0 (C; L⊗2 ) ⊗ H 0 (C; L⊗2 ), and hence the multiplication on C is reflected in the functions. The properties satisfied by the function ξ are fairly complicated [24]. On the other hand, we may construct a formal group by completing C at the identity, and since C is a smooth curve the resulting representing ring is isomorphic k[[t]]. Since C × C is a smooth surface, if we complete it at its identity we obtain a ring of functions k[[1 ⊗ t, t ⊗ 1]]. Thus, the group operation on C gives a coproduct on k[[t]], and when we choose a suitable coordinate, this takes the familiar form of an elliptic formal group law. 5.C. Elliptic cohomology theories and elliptic formal groups. An elliptic spectrum in the sense of Ando-Hopkins-Strickland [4] is a triple (E, C, φ) where E is a 2-periodic even spectrum, necessarily complex orientable, an elliptic curve C over E 0 and an isomorphism φ between the formal group E 0 (BT) of E and Ce∧ . In the present terms, the T-equivariant Borel theory of E is associated to the formal group Ce∧ . 5.D. T-equivariant elliptic cohomology theories and elliptic curves. Since C is not determined by a ring of functions, it is now rather less clear what it means for a T-equivariant cohomology theory to be associated to an elliptic curve. However the way the graded ring k(C, e) was used in reconstructing C suggests that the sections of various line bundles should occur as values of the cohomology theory. The way the multiplication on C was reflected in ξ : C × C −→ C × C suggests that the T × T equivariant theory for C × C should play a role in describing this structure. All of this structure should reduce to elliptic formal groups in the Borel theory. In Section 10 we make precise the properties it is natural to hope for. Such a cohomology theory has been constructed [15] by the methods described in Section 8 below. 5.E. A-equivariant elliptic cohomology theories and elliptic formal groups. Just as we constructed A-equivariant formal groups from Gm for a finite abelian group A, we may construct them by completing around the image of ζ : A∗ −→ C. Strickland has recently shown [30] that provided 2 is inverted, these correspond to complex oriented A-equivariant cohomology

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theories. Devoto earlier constructed a version for the Jacobi quartic when the group order is inverted [7] and investigated its role in index theory. 6. T-equivariant elliptic cohomology. If we are looking for other interesting shapes in the sense of Definition 3.1, it is natural to consider other ways of getting a graded abelian group from a representation, and we might consider replacing the triangulated category of spectra by a different one, such as the category of sheaves on an algebraic variety. 6.A. Sheaves on a curve. As with K-theory, we expect to associate to an invertible (i.e., one dimensional) representation α, an invertible sheaf (i.e., a line bundle) Lα = OC (D(α)) on the curve C. Sums of representations should correspond to sums of divisors or tensor products of bundles: α 7 → − Lα = OC (D(α)) V ⊕W − 7 → LV ⊗ LW = OC (D(V ) + D(W )). One also hopes to make the tensor product of representations correspond to some structure in the category of sheaves, but this seems less straightforward. One can then obtain a potential shape of a homology theory E˜∗T (·) by taking sheaf cohomology of the associated line bundle ˜ T (S V ) = H i (C; OC (D(V ))) E −i

for suitable values of i. Since C is a curve, the sheaf cohomology on the right will only be non-zero for i = 0, 1, so we expect to deal with other values of i by twisting the sheaf with the line bundle L = OC (D()) associated to the trivial representation of T on C; just as in the case of the multiplicative group, we expect L to have a rather different character to the line bundles associated to representations with no fixed points. When C is an elliptic curve, the identity element e provides a preferred divisor, and the example of the multiplicative group suggests we should choose D(z) = e. We then observe that z n = n∗ z and hence choose n

D(z n ) = n∗ e = C[n] = ker(C −→ C), and this too is analogous to the case of the multiplicative group. Again the example of the multiplicative group suggests the trivial representation should be associated to the tangent sheaf L = (Ω1C )∨ . However, since C is a group, this tangent sheaf is trivializable. Thus we have the association 0 z zn V

7−→ 7−→ 7−→ 7−→ 7−→

L 0 = OC L = (Ω1C )∨ Lz = OC (e) Lz n = OC (C[n]) LV = OC (D(V )); P P as with K-theory and Gm , if V = n6=0 an z n , we take D(V ) = n an C[n].

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6.B. An associated cohomology theory. In fact one may associate a Tequivariant cohomology theory to an elliptic curve. Theorem 6.1. [15] If C is an elliptic curve over a field of characteristic 0 then there is a T-equivariant cohomology of the shape described in Subsection 6.A. In particular, it is 2-periodic and for i = 0, 1 we have E˜ T (S V ) = H i (C; O(D(V ))) −i

and

E˜Ti (S V ) = H i (C; O(−D(V ))).

˜ T (S nz ) = k n for n ≥ 1, so that the theory is not complex For example E 0 orientable, and E˜iT (S 0 ) = k for all i, so that the theory is not concentrated in even degrees. The construction depends on an auxiliary piece of structure: a coordinate divisor, which is the divisor of a function te vanishing to the first order at e with all zeros and poles at points of finite order. For instance if the points of order 2 are P, P 0 , P 00 then e + P − P 0 − P 00 is a coordinate divisor. From the function te one can construct functions ts vanishing to the first order at points of exact order s for each s ≥ 2. Indeed, the space of functions vanishing at the ns points of exact order s and with a pole of order ≤ ns at e is one dimensional, and we may normalize by requiring that ts /tne s has the value 1 at e. In fact, the connection between elliptic curves and T-equivariant cohomology goes a bit further. The representing spectrum EC is a commutative ring spectrum, and we may consider modules over it. Inverting equivariant equivalences (which are equivalences in T-fixed points and in T[n]-fixed points for all n ≥ 1), we obtain its homotopy category Ho(T-EC-mod). On the other hand we may consider sheaves of modules over OC ; here we only consider sheaves as taking values on open sets which are the complements of complete sets of points of specified finite order. Inverting strong equivalences (which are equivalences in all H ∗ (C; O ⊗ (·)) and in H ∗ (C; O(C[n]) ⊗ (·)) for all n ≥ 1), we obtain its derived category D(OC -mod). These algebraic and topological derived categories are equivalent. Theorem 6.2. [15] There is an equivalence of triangulated categories D(OC -mod) ' D(T-EC-mod). 7. Shapes from projective varieties. Encouraged by the success described in the previous section, we seek suitable shapes associated to curves of higher genus, and discuss the properties of a cohomology theory with such a shape. At present the author does not know if cohomology theories with this shape exist, but for suitable curves, one can construct what would be T-equivariant restrictions of them. The geometry is too intricate to be described here in detail, but we give an outline.

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7.A. Sheaves on a Jacobian. If C is a curve of genus g, we may consider shapes associated to the Jacobian J = Jac(C) (for background see [23, 22, 21]). If the genus is 1 then C = J, and we are reduced to the case of an elliptic curve, as discussed in Subsection 6.A above. Suppose C is a curve over the complex numbers. The genus g of C may be calculated either topologically via the singular homology H1 (C; Z) ∼ = Z2g 0 1 ∼ g or geometrically, via the space H (C; ΩC ) = C of holomorphic 1-forms. We may construct the Jacobian Jac(C) as follows. A topological 1-chain Z on R C defines a linear form on the holomorphic 1-forms by integration ω 7−→ Z ω; this defines an embedding H1 (C; Z) −→ H 0 (C; Ω1C )∨ , and Jac(C) is the quotient. The most important features of J are (1) that it is a group, (2) that it has a preferred divisor Θ (the theta divisor), uniquely specified up to translation and (3) that it has the structure of an algebraic variety. Any other principally polarized abelian variety of dimension g would do as well for the general discussion, although special properties of the abelian variety will probably required to actually construct such a theory. Since J is of dimension g, one might expect the associated group of equivariance to be a torus Tg of dimension g. The group of torsion points of J is Jtors ∼ = (Q/Z)2g as an abelian group, whereas (Tg )∗ ∼ = Zg , so we may choose a complete level structure (Tg )∗ ⊗ (Q/Z × Q/Z) ∼ = Jtors . For the Jacobian we have a natural isomorphism Jtors = H1 (C; Q/Z), so it is natural to choose the factors of Tg to correspond to a decomposition of H1 (C; Z) into a sum of 2-dimensional hyperbolic forms; in more concrete terms we choose copies of Q/Z × Q/Z to correspond to pairs of curves on C with a single point of intersection. For some purposes, it may be more appropropriate to use the torus T = Hom(N S(J), T) dual to the NeronSeveri group as the group of equivariance. Here N S(J) := im(H 1 (J; O× ) −→ H 2 (J; Z)) is a free abelian group. Its rank is called the Picard number, and this is ≤ g 2 . With this structure available, we may attempt to define a suitable shape. It turns out that the divisors n∗ Θ and n2 Θ define isomorphic line bundles. However, the divisor n∗ Θ is usually not decomposable. It is therefore better to replace it by a sum of translates of Θ. Indeed, if z is a generating one dimensional representation, our level structure provides P an embedding z˜ : Q/Z × Q/Z −→ Jtors , and we may consider Θz,n := na=0 Θz˜(a) , where Θx denotes the translation of Θ by x. The divisor Θz,n is of degree n2 , and linearly equivalent to n2 Θ since {˜ z (a) | na = 0} is a subgroup of Jtors with the property that the sum of its elements is the identity. This leads to the

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following association. 0 z zn V

7 → − 7−→ 7−→ 7−→ 7−→

L0 L Lz Lz n LV

= = = = =

OJ ? OJ (Θ) OJ (Θz,n ) O(D(V ))

The constraints on the choice of L are suggested by the discussion in Subsection 7.B. Note in particular that (Ω1J )∨ is not a line bundle for g ≥ 2, so we need another candidate. Since J is of dimension g, the sheaf cohomology of these line bundles will be in degrees between 0 and g, so we now only expect a spectral sequence g E i,2j = H i (J; OJ (D(V )) ⊗ Lj ) ⇒ E˜ T (S V ), 2

∗

where odd rows are zero. The cohomology of all these sheaves is well-known, and for g = 2, a spectral sequence of this form necessarily collapses for some natural choices of L . This leads to predictions for the values of the cohomology theory on spheres. 7.B. Further constraints. We argue here that if there is a cohomology theory associated to an abelian surface, it cannot be 2-periodic. Suppose then that C is a curve of genus 2, and that ET∗ 2 (·) is a cohomology theory associated to it. Since g = 2, we may identify the theta divisor Θ with the curve C embedded in J by the Abel-Jacobi map (once we have picked a basepoint on C). We choose a circle subgroup T and restrict attention to the T-equivariant cohomology theory. Now we consider the two cofibre sequences S 0 −→ S z −→ ΣT+ and S z −→ S 2z −→ Σ3 T+ , where the second is obtained by smashing the first with S z . The point to note is the untwisting T+ ∧ S z ' T+ ∧ S 2 that has been used to identify the last term in the second cofibre sequence. These cofibre sequences should correspond to the exact sequences O −→ O(C) −→ Q1 and O(C) −→ O(2C) −→ Q2 of sheaves over J, where the second is obtained by tensoring the first with O(C). Here we note that Q1 and Q2 are supported on C. Indeed, we have Q1 ∼ = i∗ q1 and Q2 ∼ = i∗ q2 for suitable sheaves q1 and q2 on C. We even expect q1 and q2 to be line bundles on C. Since the second exact sequence is obtained from the first by tensoring with O(C), we find Q2 ∼ = Q1 ⊗ O(C) ∼ = i∗ (q1 ⊗ O(C)|C ). On the other hand, by the untwisting we have Q2 ∼ = Q1 ⊗ L ∼ = i∗ (q1 ⊗ (L )|C ).

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This gives

L |C ∼ = OJ (C)|C ∼ = Ω1C , where the final equality is the adjunction formula. An alternative argument adapts the discussion in Subsection 3.B to show L |C is the normal bundle of C in J; we see that this is Ω1C , since the canonical bundle on J is trivial. Since C is of genus 2, the line bundle Ω1C is of degree 2, and H 0 (C; (Ω1C )⊗n ) depends on n (it is of dimension 2 for n = 1 and of dimension 2n − 1 for n ≥ 2). This shows that even the non-equivariant form E ∗ (·) = ET∗ (T × (·)) of the theory will not be 2-periodic. 8. Rational equivariant cohomology theories. So far we have only presented evidence for a one-way connection between algebraic groups and equivariant cohomology theories. In the one dimensional case, suitable cohomology theories ET∗ (·) give rise to one dimensional algebraic groups G by evaluation on suitable spaces. On the other hand, we have argued that a one dimensional algebraic group G suggests the values of an equivariant cohomology theory ET∗ (·) on spheres (i.e., the shape of the theory). In this section we show that the shape of a cohomology theory is very close to determining a cohomology theory over the rationals, and that it does determine a cohomology theory in favourable cases. 8.A. Algebraic models for categories of rational cohomology theories. The idea is that for any compact Lie group G there is an abelian category A(G) modelling rational G-equivariant cohomology theories. As usual, it is worth working with the category of objects representing the cohomology theories, namely rational G-spectra. This category admits the structure of a Quillen model category, and its homotopy category gives the cohomology theories. Thus homotopy equivalence classes of G-spectra correspond to Gequivariant rational cohomology theories, and all cohomology theories are of the form EG∗ (X) = [X, E]∗G . On a practical level, we want to be able to calculate in this homotopy category, but if we understand the category completely we can also construct interesting new cohomology theories. The idea is that objects of A(G) should be a rather small, and based on information easily accessible from the cohomology theories they represent. One natural way to construct simpler cohomology theories from EG∗ (·) is to take Borel cohomology of fixed points. Indeed, if X is a G-space we may form the H-fixed point space X H ; the group WG (H) = NG (H)/H acts on this, as does the maximal torus T WG (H) of its identity component. We may therefore form the Borel theory E˜T∗ WG (H) (ET WG (H)+ ∧ X H ). At least when G is abelian, this is closely related to the shape of the cohomology theory. Indeed, each of the two steps can be performed (up to extension) using the

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shape. For the first step, we use the formal localization theorem, which states that in suitable circumstances X H may be obtained by inverting formal Euler classes. To explain this, we suppose H is normal, and consider the infinite sphere S ∞V (H) = lim S V . We say that G has enough representations to → V H =0

detect H if (S ∞V (H) )K is contractible when K does not contain H. When H is normal in G and G has enough representations to detect H, obstruction theory gives [T H , X H ]G/H = [T, X ∧ S ∞V (H) ]G as required. For the second step, we note that for any torus T we may form the universal free T -space ET from spheres. More precisely, if we choose a direct product decomposition T = T×· · ·×T and z1 , z2 , . . . , zr are the natural representations of the factors, we have ET ' S(∞z1 ) × S(∞z2 ) × · · · × S(∞zr ). The factors are related to spheres relevant to the shape by the cofibre sequences S(∞z)+ −→ S 0 −→ S ∞z , so that we obtain a model for ET+ like the stable Koszul complex. Conjecture 8.1. For a compact Lie group G there is an abelian category A(G) and a Quillen equivalence G-spectra/Q ' dgA(G) such that (1) A(G) is abelian (2) InjDim(A(G)) = rank(G) (3) The category consists of sheaves of modules over a space of closed subgroups of G; the object corresponding to a cohomology theory EG∗ (·) has fibre over H built from the Borel theory ET∗ WG (H) (ET WG (H)+ ∧ X H ). The additional structure is built from these Borel theories using their relation under localization and inflation. (4) The model of EG∗ (·) is built from its shape and a little extra structure. 8.B. Consequences of the conjecture. Note immediately that if the conjecture holds we have an equivalence of homotopy categories Ho(G-spectra/Q) ' D(A(G)) as triangulated categories. This reduces to algebra the problem of classifying rational equivariant cohomology theories and the process of calculation with them. Furthermore, it provides a universal homology theory π∗A(G) : G-spectra −→ A(G) and an Adams spectral sequence A(G) Ext∗,∗ (X), π∗A(G) (Y )) ⇒ [X, Y ]G ∗ A(G) (π∗

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for calculation. Finally, because of the injective dimension of A(G), the Adams spectral sequence is only non-zero on s-line for 0 ≤ s ≤ r, so the calculation is very accessible. 8.C. Status of the conjecture. • G finite. The conjecture is true. Indeed, all rational cohomology theories are ordinary. This follows easily from three facts: (1) rational cohomology is rational homotopy, (2) the rational cohomology of a finite group is trivial, and (3) rational Mackey functors are all injective. The idempotents in the rationalized Burnside ring allow us to express this in terms of representation theory, so that Y QWG (H)-mod. A(G) = (H)

• The circle group G = T. Again the theorem is true. Indeed, [11] constructs A(G) and shows that there is a triangulated equivalence of homotopy categories. Shipley [27] upgraded this to a Quillen equivalence. The model is briefly described in Section 9 below. • The groups G = O(2), SO(3) and their double covers. In this case the equivalence of homotopy categories is proved in [12, 13]. • The tori G = Tg . The Adams spectral sequence exists [16], and in [17, 18] we work towards showing that the Quillen equivalence holds. 9. The model for the circle group G = T.

9.A. The category A(T). The objects of A(T) are sheaves over the discrete set F ∪ {T}, where F is the set of finite subgroups. It is thus natural that the structure sheaf should be the constant sheaf Q[c], with global sections Y R = map(F, Q[c]) = Q[c]. n

Next we need to discuss suspension. If w : F −→ Z≥0 is a function zero almost everywhere, we may form Y Σw R = Σ2w(n) Q[c]. n

For example, if W is a complex representation with W T = 0 we might take w(n) = dimC (W T[n] ), and Σw would correpond to suspension by W ; this explains the factor of 2. The inclusion R −→ Σw R corresponds to the inclusion S 0 −→ S W , and hence we think of it as division by a universal Euler class. Now form [ t= Σw R = E−1 R, w

where the last equality refers to the idea that the map R −→ Σw R is multiplication by an Euler class, so that in forming the union we have inverted the multiplicative set of Euler classes.

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The objects M of A(T) are of the form β

M = (N −→ t ⊗ V ), where N is an R-module, and β is a map of R-modules which becomes an isomorphism when E is inverted. The morphisms are diagrams M = (N −→ t ⊗ V ) ↓ θ↓ ↓1⊗φ 0 0 M = (N −→ t ⊗ V 0 ) It is not hard to show that A(T) is abelian and of injective dimension 1. Theorem 9.1. [11, 27] There is a Quillen equivalence T-spectra/Q ' dgA(T). 9.B. Rigid even objects. Consider an object M = (N −→ t ⊗ V ) of A(T). If N is concentrated in even degrees and E-torsion free we have N = ker(t ⊗ V −→ T ) where T isLthe E-torsion module (t ⊗ V )/N . Furthermore, it then turns out that T = n Tn where Tn is a torsion module over Q[c]. We may thus define the object M by using the map M q : t ⊗ V −→ Tn , n

and this always defines an object M provided q is surjective. Now if M represents the homology theory E∗T (·) it turns out that we can very nearly recover M from the shape of E∗T (·): • V = lim E˜ T (S W ) • T =

∗ → W T =0 ˜ T (S 0 ) cok(E ∗

i

−→ lim

→ W T =0

E˜∗T (S W )).

The reason that the shape does not quite determine the object is that the map q also encodes information about division by Euler classes. Summary 9.2. If i is injective, we may almost write down the object ME of A(T ) representing the homology theory E∗T (·) purely in terms of the shape. 9.C. Construction of a cohomology theory from an algebraic group. When C is a 1-dimensional algebraic group we may now describe the construction of a 2-periodic rational T-equivariant cohomology theory ET∗ (·) associated to it. Indeed, since the theory is 2-periodic we need only describe the model in degrees 0 and 1, and we take the odd part to be zero. In degree 0 we take V0 = lim

→ W T =0

˜ T (S W ) = lim E ∗

→ W T =0

H 0 (C; O(D(W )) = KC ,

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where KC is the ring of meromorphic functions with poles only at points of finite order. For the torsion we take M 1 T0 = H 0 (C; KC /OC ) = HChsi (C), s

1 where Chsi consists of the points of exact order s and HChsi (C) is the local cohomology, and therefore consists of the principal parts of functions at points of exact order s. Finally, to define the map q : t ⊗ V0 −→ T0 we must choose some additional structure. For each s ≥ 1 we must choose a function ts vanishing to the first order at points of Chsi and with poles only at points of finite order. The geometry of C may suggest natural ways to make this choice but for now we assume the choice has been made. For instance if G = Gm we may take ts to be the sth cyclotomic polynomial φs (z), and for an elliptic curve we may use the functions constructed from the coordinate divisor in Section 6. Having made this choice, we may define q(cw ⊗ f ) for a meromorphic function f to have as sth component w(s)

qs (cw ⊗ f ) = ts

f.

w(s)

In fact this is legitimate since ts f is regular on Chsi for all but finitely many values of s, and in fact q is determined by the values on these elements. 9.D. The shape of a rigid even spectrum. The sphere S W is modelled by (Σw R −→ t ⊗ Q. ¿From this it is easy to calculate HomA(T) (S W , ME) and ExtA(T) (S W , ME). This gives E˜0T (S W ) = E˜T0 (S −W ) = ker(q : cw ⊗ KC −→ T ), and

T E˜−1 (S W ) = E˜T1 (S −W ) = cok(q : cw ⊗ KC −→ T ). Since functions are regular if they are regular at all points, it is clear that H 0 (C; O(D(W ))) = ker(q : c−w ⊗ KC −→ T ) = E˜ 0 (S W ), T

T as required. The argument for E˜−1 (S W ) involves a little more geometry [15, Section 5].

10. Reflecting the group structure of the elliptic curve. We explain here how the map ξ ∗ of Subsection 5.B should be realized in topology. Since we do not yet have a complete algebraic model of rational T × T-equivariant spectra, we can only show that the structures arising in algebra would be precisely mirrored in topology under the assumption that the model works as expected. Suppose there exists a T × T-equivariant cohomology theory with shape suggested by C × C. Constructing such a theory is significantly easier than constructing a T × T-equivariant theory for an arbitrary abelian surface. To

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the representation w i ⊗ z j of T × T we associate the divisor C[i] × C[j] and extend this to arbitrary representations in the usual way. The theory E(C × C)∗T×T (·) should then come with a spectral sequence T×T

^ H ∗ (C × C; OC×C (D(W ))) ⇒ E(C × C)∗

(S W ).

Since some line bundles have cohomology in degree 2, this does not determine T×T ^ E(C × C)∗ (S W ) in general. However when OC×C (D(W )) has no cohomology in dimension 2 we would find T×T

^ E(C × C)0

(S W ) = H 0 (C × C; OC×C (D(W ))).

Next, the map ξ : C × C −→ C × C is an isogeny with kernel ∆A[2] = {(a, a) | a + a = e}. We also consider the corresponding group homomorphism ξˆ : T × T −→ T × T, ˆ z) = (wz, w/z), which is surjective with kernel defined by ξ(w, ∆T[2] = {(z, z) | z 2 = 1}. To minimize confusion, we identify the second T × T with T × T = (T × T)/∆T[2]. The map ξ should correspond to a map E(C × C) −→ E(C × C) ξi∗ : inf T×T T×T (i for inflation) of T × T-spectra or adjointly, to a map ξf∗ : E(C × C) −→ E(C × C)∆T[2] (f for fixed point) of T × T-spectra. Lemma 10.1. For any representation W of T × T, the map ξf induces ξf∗ : E(C × C)T×T (S W ) −→ E(C × C)0T×T (S W ). 0 Proof: The map ξf∗ induces ξf∗ : [S 0 , S W ∧ E(C × C)]T×T −→ [S 0 , S W ∧ E(C × C)∆T[2] ]0T×T , 0 T×T

^ so it suffices to identify the domain and codomain. By definition E(C × C)0 T×T 0 W [S , S ∧ E(C × C)]0 so we turn to the codomain and calculate

(S W ) =

[S 0 , S W ∧ E(C × C)∆T[2] ]0T×T = [S −W , E(C × C)∆T[2] ]0T×T = [S −W , E(C × C)]0T×T = [S 0 , S W ∧ E(C × C)]0T×T T×T ^ = E(C × C) (S W ). 0

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To model M = p∗1 L ⊗ p∗2 L with L = O(D(W )) for a representation W of T we take W = (W ⊗ 1) ⊕ (1 ⊗ W ). Direct sum of representations corresponds to tensor product of line bundles and to sums of divisors, so if W corresponds to the line bundle L and the divisor D(W ), then W corresponds to the line bundle p∗1 L⊗p∗2 L and the divisor [D(W )×C]+[C×D(W )]. Viewed as a representation of T × T by pullback along ξˆ we find ξˆ∗ (W ) = ξˆ∗ W ⊕ ξˆ∗ W. 1

2

In particular if W = z we find ξˆ∗ (W ) = (w n ⊗ z n ) ⊕ (w n ⊗ z −n ). n

Finally, we need to observe that for any n, the bundles associated to (w n ⊗ z n ) ⊕ (w n ⊗ z −n ) and (w 2n ⊗ 1) ⊕ (1 ⊗ z 2n ) are isomorphic: this is precisely the same argument as showed ξ ∗ M ∼ = M2 above. With L = O(D(W )), we thus expect a commutative diagram ξ∗

H 0 (C; L)⊗2 = H 0 (C × C; p∗1 L ⊗ p∗2 L) −→ H 0 (C × C; p∗1 L2 ⊗ p∗2 L2 ) = H 0 (C; L2 )⊗2 ↓ ↓ T×T

^ E(C × C)0

(S ) W

ξf∗

−→

T×T

^ E(C × C)0

(S W ).

References [1] J.F.Adams, J.-P.Haeberly, S.Jackowski and J.P.May “A generalization of the AtiyahSegal completion theorem.” Topology 27 (1988) 1-6. [2] M. Ando “Power operations in elliptic cohomology and representations of loop groups” Trans. American Math. Soc. 352 (2000) 5619-5666 [3] M. Ando and M. Basterra “The Witten genus and equivariant elliptic cohomology.” Math. Z. 240 (2002) 787-822. [4] M. Ando, M.J.Hopkins and N.P.Strickland “Elliptic spectra, the Witten genus and the theorem of the cube.” Inventiones Math. 146 (2001) 595-687 [5] M.F.Atiyah and G.B.Segal “Equivariant K-theory and completion” J.Diff. Geom. 3 (1969) 1-18. [6] M.Cole, J.P.C.Greenlees and I.Kriz ‘Equivariant formal group laws.’ Proc. LMS 81 (2000) 355-386 [7] J.Devoto “Equivariant elliptic homology and finite groups.” Michigan Math. J. 43 (1996) 3-32 [8] L.G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) “Equivariant stable homotopy theory.” Lecture notes in maths. 1213, Springer-Verlag (1986) [9] V.Ginzburg (notes by V. Baranovsky) “Geometric methods in the representation theory of Hecke algebras and quantum groups.” NATO Sci. Ser. C Math.Phy. Sci. 514, Kluwer (1998) 127-183 (AG/9802004) [10] V.Ginzburg, M. Kapranov, E. Vasserot “Elliptic algebras and equivariant elliptic cohomology” Preprint (1995) (q-alg/9505012)

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[11] J.P.C.Greenlees “Rational S 1 -equivariant stable homotopy theory.” Mem. American Math. Soc. 661 (1999) xii + 289pp [12] J.P.C.Greenlees “Rational O(2)-equivariant stable homotopy theory.” Fieldds Inst. Comm. 19 AMS (1998) 103-110 [13] J.P.C.Greenlees “Rational SO(3)-equivariant cohomology theories.” Contemporary Maths. 271 AMS (2001) 99-125 [14] J.P.C.Greenlees “Equivariant formal group laws and complex oriented cohomology theories.” HHA 3 (2001) 225-263 [15] J.P.C.Greenlees “Rational S 1 -equivariant elliptic cohomology.” Topology (to appear) 67pp. [16] J.P.C.Greenlees “Rational torus equivariant cohomology theories I: calculating groups of stable maps.” (Submitted for publication) 25pp [17] J.P.C.Greenlees “Rational torus equivariant cohomology theories II: the algebra of localization and inflation.” (Submitted for publication) 22pp [18] J.P.C.Greenlees and B.E. Shipley “An algebraic model for rational torus equivariant stable homotopy.” (Submitted for publication) 41pp [19] I. Grojnowski “Delocalized elliptic cohomology.” Yale University Preprint (1994). [20] R. Hartshorne “Algebraic geometry” Springer-Verlag, 1977 [21] H.Lange and Ch.Birkenhake “Complex abelian varieties.” Springer-Verlag (1992) [22] D.Mumford “Abelian varieties.” Oxford University Press (1974) [23] D.Mumford “Curves and their Jacobians.” University of Michigan Press (1975), reprinted as Appendix to Lecture Notes in Mathematics 1358 Springer-Verlag (1999) [24] D.Mumford “On the equations defining abelian varieties, I” Inventiones, (1966) 287354 [25] I.Rosu “Equivariant elliptic cohomology and rigidity.” American J. Math. 123 (2001) 647-677. [26] J.-P.Serre “Algebraic groups and class fields” Springer-Verlag (1988) [27] B.E.Shipley “An algebraic model for rational S 1 -equivariant stable homotopy theory.” Q.J.Math. 54 (2002) 803-828 [28] J.H.Silverman “The arithmetic of elliptic curves.” Springer-Verlag (1986) [29] N.P.Strickland “Multicurves and equivariant cobordism.” Preprint (2002) 55pp. [30] N.P.Strickland “Arithmetic abelian-equivariant elliptic cohomology.” (In preparation) [31] H.Tamanoi “Elliptic genera and vertex operator super-algebras.” Lecture Notes in Mathematics, 1704 Springer-Verlag, Berlin, 1999. vi+390 pp [32] E.Witten “Elliptic genera and quantum field theory.” Comm. Math. Phys. 109 (1987), no. 4, 525–536. Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK. E-mail address: [email protected]

IAN GROJNOWSKI’S “DELOCALIZED EQUIVARIANT ELLIPTIC COHOMOLOGY”

In 1994 Ian Grojnowski distributed a preprint sketching the construction of a model for a complex circle-equivariant elliptic cohomology theory. This preprint has had a great impact on the subject, and we are grateful to Grojnowski for allowing us to reproduce it here. We provide a brief setting for this paper, and mention some of the work which has flowed from it. Since rational cohomology theories are ordinary, it is natural to try to express equivariant rational (or complex) cohomology theories in terms of the ordinary rational cohomology of fixed point data. This idea was put into effect for abelian groups by Paul Baum, Jean-Luc Brylinksi, and Bob MacPherson [BBM85], who expressed complex equivariant K-theory in terms of a sheaf over the group. Grojnowski took a different approach and a more difficult example, giving a construction of a complex circle-equivariant cohomology theory starting from an elliptic curve E over C and taking values in sheaves of algebras over E. (At around the same time, Victor Ginzberg, Mikhail Kapranov, and Eric Vasserot [GKV95] released a sketch of the general properties one should expect of an equivariant elliptic cohomology theory, using a general compact Lie group but omitting an explicit construction.) Ioanid Rosu later [Ros03] closed the circle by providing a model for complexified torus-equivariant K-theory along these lines, taking values in sheaves over C× . Short as it is, Grojnowski’s paper sketches more than just the construction of his circle-equivariant elliptic cohomology. He also constructs the “pushforward” map in his theory, and he indicates how elliptic cohomology is related to representations of loop groups. We focus on subsequent developments which also use these features. In his thesis [Ros01], Rosu interpreted the work of Raoul Bott and Cliff Taubes [BT89] as the construction of a Thom class (or “orientation”) in Grojnowski’s elliptic cohomology whose push-forward is the equivariant Ochanine genus. This gives a proof of the rigidity of the Ochanine genus, fulfilling a proposal for such a proof by Haynes Miller [Mil89]. More recently, Matthew Ando and Maria Basterra [AB02, And03] carried out the analogous program for the Witten genus, and constructed a circle-equivariant analogue of the “string orientation” of Hopkins et al [Hop95, AHS01, Hop02]. Thus the rigidity of the Witten genus is an aspect of the equivariant string orientation in Grojnowski’s theory, just as the modularity of the Witten genus is an aspect of the string orientation of nonequivariant elliptic cohomology. 111

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The push-forward in Grojnowski’s theory can also be used to give a coherent account of a number of formulae in the literature. In their paper in this volume [AF], Ando and Chris French use it to produce the orbifold elliptic genus of [BL02]. More recently, Ando, French, and Ganter [AFG] have used it to give an account of the two-variable elliptic genus [EOTY89, Kri90, H¨oh91, BL02] and the level-N genus [Hir88]. Another very influential feature of Grojnowski’s paper is his introduction of Eduard Looijenga’s work [Loo76]. Grojnowski mentions it only briefly at the end of his paper, but he recognized that it was the key to understanding the relationship between equivariant elliptic cohomology and representations of loop groups, as explained in [And00]. Looijenga’s work expresses in a direct way the relationship between the second Chern class (and so “string structures”) and elliptic curves, and it has played an important role in the development of the subject. Matthew Ando Haynes Miller June, 2006 References [AB02]

Matthew Ando and Maria Basterra. The Witten genus and equivariant elliptic cohomology. Mathematische Zeitschrift, 240(4):787–822, 2002, arXiv:math.AT/0008192. [AF] Matthew Ando and Christopher P. French. Discrete torsion for the supersingular orbifold sigma genus, arXiv:math.AT/0308068. [AFG] Matthew Ando, Christopher P. French, and Nora Ganter. The Jacobi orientation and the two-variable elliptic genus, arXiv:math.AT/0605554. [AHS01] Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. Elliptic spectra, the Witten genus, and the theorem of the cube. Inventiones Mathematicae, 146:595–687, 2001, DOI 10.1007/s002220100175. [And00] Matthew Ando. Power operations in elliptic cohomology and representations of loop groups. Trans. Amer. Math. Soc., 352(12):5619–5666, 2000. [And03] Matthew Ando. The sigma orientation for analytic circle-equivariant elliptic cohomology. Geometry and Topology, 7:91–153, 2003, arXiv:math.AT/0201092. [BBM85] Paul Baum, Jean-Luc Brylinski, and Robert MacPherson. Cohomologie ´equivariante d´elocalis´ee. C. R. Acad. Sci. Paris S´er. I Math., 300(17):605–608, 1985. [BL02] Lev A. Borisov and Anatoly Libgober. Elliptic genera of singular varieties, orbifold elliptic genus and chiral de Rham complex. In Mirror symmetry, IV (Montreal, QC, 2000), volume 33 of AMS/IP Stud. Adv. Math., pages 325–342. Amer. Math. Soc., Providence, RI, 2002, arXiv:math.AG/0007126. [BT89] Raoul Bott and Clifford Taubes. On the rigidity theorems of Witten. J. of the Amer. Math. Soc., 2, 1989. [EOTY89] H. Eguchi, H. Ooguri, A. Taormina, and S.-K. Yang. Superconformal algebras and string compactification on manifolds with SU (N ) holonomy. Nucl. Phys. B, 315, 1989.

Ian Grojnowski’s “Delocalized equivariant elliptic cohomology” [GKV95] [H¨ oh91]

[Hir88]

[Hop95]

[Hop02]

[Kri90] [Loo76] [Mil89]

[Ros01] [Ros03] [Wit87]

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V. Ginzburg, M. Kapranov, and E. Vasserot. Elliptic algebras and equivariant elliptic cohomology. 1995. Preprint. Gerald H¨ ohn. Komplexe elliptische Geschlechter und S 1 -¨ aquivariante Kobordimustheorie (complex elliptic genera and S 1 -equivariant cobordism theory), 1991, arXiv:math.AT/0405232. Bonn Diplomarbeit. Friedrich Hirzebruch. Elliptic genera of level N for complex manifolds. In Differential geometrical methods in theoretical physics (Como, 1987), volume 250 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 37–63. Kluwer Acad. Publ., Dordrecht, 1988. Michael J. Hopkins. Topological modular forms, the Witten genus, and the theorem of the cube. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), pages 554–565, Basel, 1995. Birkh¨ auser. M. J. Hopkins. Algebraic topology and modular forms. In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291– 317, Beijing, 2002. Higher Ed. Press, arXiv:math.AT/0212397. Igor M. Krichever. Generalized elliptic genera and Baker-Akhiezer functions. Mat. Zametki, 47(2):34–45, 158, 1990. Eduard Looijenga. Root systems and elliptic curves. Inventiones Math., 38, 1976. Haynes Miller. The elliptic character and the Witten genus. In Mark Mayowald and Stewart Priddy, editors, Algebraic topology (Northwester University, 1988), volume 96 of Contemporary Math. Amer. Math. Soc., 1989. Ioanid Rosu. Equivariant elliptic cohomology and rigidity. Amer. J. Math., 123(4):647–677, 2001, arXiv:math.AT/9912089. Ioanid Rosu. Equivariant K-theory and equivariant cohomology. Math. Z., 243(3):423–448, 2003. Edward Witten. Elliptic genera and quantum field theory. Comm. Math. Phys., 109, 1987.

DELOCALISED EQUIVARIANT ELLIPTIC COHOMOLOGY I. GROJNOWSKI

Abstract. In this paper we construct an equivariant elliptic cohomology theory over C. As defined, the level k equivariant cohomology of a point with respect to a compact group G is just the span of the characters of c at level k. the loop group LG

In this paper we construct a ‘delocalised’ equivariant elliptic cohomology over C. The construction here suffers from several obvious disadvantages—it is unwieldy to work with, and clearly misses completely the point of elliptic cohomology [8]. However, it suffices for the construction of the elliptic affine algebras (see below), and does produce the ‘correct’ results. The result was inspired by [2] (which in turn is a child of [3]). We essentially define equivariant elliptic cohomology by using the Chern character E(X ×T ET ) → H(X ×T ET ), using the topology of the abelian variety ET to avoid completions. This construction produces non-trivial bundles on ET (for example ES 1 (P1 )), and, compatibly, the Gysin homomorphism we define involves twisting the bundles still further. In section 2.4 we define ES 1 (X), a sheaf over a fixed totally marked elliptic curve E. The definition works by identifying a neighbourhood of the elliptic curve around e ∈ E with its tangent space, and taking the equivariant cohomology of X e , the fixpoints of e on X. In 2.6 this is generalised to an arbitrary compact connected group. In section 2.5 we define the pushforward, or Gysin, homomorphism. This involves a choice of local coordinate on the elliptic curve, i.e. an analytic map s : U ⊆ E → C, where U is a neighbourhood of 1. This data is precisely that of a complex orientation in topology, and defines in a standard manner a homology theory with pushforward maps [1]. When this standard pushforward is applied locally on the elliptic curve, the effect of this is to twist the sheaf ES 1 (X), so that π : X → Y induces a map from a twist of ES 1 (X) to ES 1 (Y ). This behaviour is forced upon us, as elliptic cohomology satisfies a ‘locality’ property: the Mayer-Vietoris exact sequence. As ES 1 (X) is usually a nontrivial sheaf, if we can partition X into contractible pieces, a long exact sequence must necessarily twist some of the cohomology of the pieces in order Date: Feb 1994.

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to produce ES 1 (X). This is indeed what happens (consider, for example ES 1 (P1 )). It is immediate from definitions that all standard properties of a cohomology theory hold; in section 3 we mention a few that are specific to elliptic cohomology. Finally, we consider what happens when we vary the elliptic curve. It turns out that the fixpoints X e(z,τ ) make sense for a point on the universal elliptic curve (no extra marking is necessary). It follows that if the given fixed complex orientation s is defined on a curve with some marking, than so is our cohomology theory. For example, if s is the orientation class of [8], the S 1 -equivariant cohomology is a sheaf over the universal elliptic curve with a marked point of order 2. Taking the formal neighbourhood of 1 on this marked universal curve, we recover the usual elliptic cohomology E(X ×T ET ) of [8]. (See also [11] for an explanation of why this is the ‘wrong’ object.) However, instead of this one may take global sections on the universal curve of tensor products of EG (X) with certain line bundles (see section 3). These global objects are the ‘correct’ definition of a level k elliptic cohomology theory. In sharp contrast to HG (X) and KG (X) they are finite dimensional.

This note is part of a project of the author to construct certain generalisations of quantum groups and Hecke algebras which I call elliptic affine algebras. These algebras depend on a marked elliptic curve as well as a point on the curve (the analog of q in the quantum group), and theta constants occur in the structure coefficents of the algebra. One proceeds by applying the equivariant elliptic cohomology constructed here to certain well known varieties to produce elliptic Hecke algebras, elliptic quantum groups and their finite dimensional representations. The usual case is obtained by applying equivariant K-theory to these same varieties (see [6]). The construction detailed here produces certain ‘twisted’ algebras: coherent sheaves A on an abelian variety, equipped with a multiplication A ⊗ A ⊗ L → A, where L is a certain line bundle. The representations of A are easily described as in [5], even at points of finite order. The unfinished task is then to produce an honest algebra out of A, with a similar representation theory. I believe I now understand how to do this, though at the glacial pace at which I work it will take some time to get the details correct. In any case, the polite interest expressed by the people who have seen this note (which has been circulating since February, 1994) suggest that it may be worth publishing as is.

Finally, it is worth mentioning what we have really done. We have produced a theory such that the level k elliptic cohomology of a point, with respect to a

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group G, is precisely the span of the characters of the level k representations f of LG. We have done this purely finite dimensionally, by a cheap trick. However, it is morally clear how to do this in general. One must define a certain f equivariant vector bundles on the loopspace LX with the semicategory of LG infinite topology. Pushforward maps then become Euler characteristics in semi-infinite cohomology [4]. I believe this is not too difficult to do rigorously. 2. Elliptic Cohomology 2.1. Fix τ ∈ C, Im(τ ) > 0. Let E = Eτ denote the marked elliptic curve C/(Z + τ Z). There is a continuous isomorphism of groups E → S 1 × S 1 , induced from the map C → S 1 × S 1 , x + τ y 7→ (e2πix , e2πiy ). Let T be a compact torus (product of S 1 ’s); Y (T ) = Hom(S 1 , T ) its lattice of cocharacters. Then T ∼ = Y (T ) ⊗Z S 1 ; define ET = Y (T ) ⊗Z E and t = Y (T ) ⊗Z C. We identify t with both the complexified Lie algebra of T , and the tangent space to ET at 1. Write Ot for the sheaf of complex valued analytic functions on t. The map E → S 1 × S 1 gives rise to a continuous isomorphism of groups ET → T × T ; for e ∈ ET denote its image under this map as (e1 , e2 ). If V is a small neighbourhood of 0 ∈ t, there is a neighbourhood U of 1 ∈ ET and an isomorphism exp : V → U with inverse log : U → V . We will write log∗ for the corresponding map from sheaves on V to sheaves on U . Now suppose T acts on a topological space X. Define the fixpoint set X e , for e ∈ ET to be X e1 ,e2 = {a ∈ X | e1 a = e2 a = a}. For H a connected subgroup of T , put X(H) = {a ∈ X | stab(a)0 = H}. Here, stab(a)0 denotes the identity component of the subgroup of T that fixes a. Then for x ∈ t define the fixpoint set X x as X x = ∪H:x∈(LieH)C X(H). If X is compact and T acts smoothly, then for each e ∈ ET there exists an open neighbourhood U of e such that X f ⊆ X e for all f ∈ U . This is essentially a result of Mostow (see [2, 1.3]). Note that for e in a small neighbourhood of 1 ∈ ET we have X e = X log e , and more generally, for f in a small neighbourhood of e we have X f = (X e )log(f −e) . We will systematically use this fixpoint notation (though neither the Abelian variety or the Lie algebra act, we have made perfect sense of their fixpoints). For e ∈ ET let te : ET → ET , e0 7→ e0 + e be the map ‘translation by e’. 2.2. Recall there is a functor, equivariant cohomology, from the category of pairs (G, X), where G is a topological group and X a topological space on which G acts continuously to the category of Z-graded super-commutative complex algebras, (G, X) 7→ HG (X), with the following properties: i) HG (X) is a graded super-commutative algebra over HG = HG (point). If T is a compact torus, then HT = S(t∗ ) canonically, where S(t∗ ) is the algebra of polynomial functions on t, graded so the generators

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are in degree 2. If G is compact connected, T ⊆ G a maximal torus, W = NG (T )/T the Weyl group, then HG = HTW , the W -invariant polynomial functions on t. We often regard HG (X) as a sheaf over Spec(HG ). In the case G is compact connected, we can also regard HG (X) as a W -equivariant sheaf on t. ii) If G is compact and connected, HG (X) is determined by X and the Lie algebra of G. We denote it Hg (X). Further, if G is a general 0 compact group, HG (X) = Hg (X)G/G , and if G → G0 is a homotopy equivalence, where G’ is an arbitrary topological group, the induced map HG (X) → HG0 (X) is an isomorphism of graded algebras. iii) Let T be a compact torus. Then if x ∈ t and U is a sufficiently small neighbourhood of x, the inclusion i : X x ,→ X induces an isomorphism i∗ : Ht (X) ⊗Ht Γ(U, Ot ) ∼ = Ht (X x ) ⊗Ht Γ(U, Ot ). More generally, if x ∈ g is semisimple, where G is a possibly disconnected compact Lie group, then the inclusion i : (Gx , X x ) ,→ (G, X) induces a map HG (X) ⊗HG HGx → HGx (X x ) which becomes an isomorphism when both sides are localised at x ∈ SpecHGx . This is the “non-Abelian localisation” of Atiyah-Segal. iv) If tx : t → t, y 7→ y + x is the translation by x map, then it induces a functor from sheaves on t to sheaves on t, denoted (as always) by t∗x . Then t∗x Ht (X x ) ∼ = Ht (X x ). We indicate the proof. Write t = h ⊕ h0 , where h is the line of multiples of x. Then Ht (X x ) = Hh ⊗C Hh0 (X x ), and tx acts only on Hh . More generally, if x ∈ g is semisimple, then t∗x : HGx (X x ) → HGx (X x ) is an isomorphism, as x is in the center of gx . 2.3. Let O = OET denote the sheaf of complex valued analytic functions on ET ; Γ(U, O) = Γ(U ) its sections over U . Recall that to specify a sheaf A of O-modules on ET it is enough to give a Γ(U, O) module AU for each U in some open cover of ET by sufficiently small sets, and for each U, V with U ∩ V 6= ∅ a Γ(U ∩ V ) isomorphism φU V : Γ(U ∩ V ) ⊗ΓU AU → Γ(U ∩ V ) ⊗ΓV AV such that if U, V, W are such that U ∩ V ∩ W 6= ∅, then φV W φU V = φU W . Clearly it also suffices to only give this glueing data φU V when V ⊆ U if the open cover is closed under finite intersection. Similarly, if A is a sheaf of Z/2-graded super-commutative O-algebras on × ET , then elements λU V ∈ Γ(U ∩ V ) ⊗ΓU A× U (where AU denotes the commutative group of invertible elements in the ring AU ) such that λV W λU V = λU W defines an element [λ] of H 1 (ET , A× ) and a sheaf A[λ] , the “twist” of A by λ. Here, Γ(U, A[λ]) = AU and the glueing isomorphisms are φ0U V = λU V φU V . The isomorphism class of A[λ] depends only on the class of [λ] in H 1 (ET , A× ) (A notational warning: t∗e , log∗ refer to the pullback of sheaves. On the other hand, if π : X → Y is a map, we also write π ∗ to denote pullback

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in cohomology. These two uses are married in the definition below, most particularly in 2.6). 2.4. Now define ET (X), a Z/2 graded sheaf of supercommutative algebras over O. If e ∈ ET , and U is a sufficiently small neighbourhood of e, define Γ(U, ET (X)) = t∗e log∗ {Ht (X e ) ⊗Ht Γ(Ot , log(t−e U )}. Write Ht (X e )U −e for Ht (X e ) ⊗Ht Γ(Ot , log(t−e U ). If f ∈ U , and V ⊆ U is a small enough neighbourhood of F , define φU V : Γ(U, ET (X)) ⊗ΓU ΓV → Γ(V, ET (X) as the composition of the following isomorhisms Γ(U, ET (X)) ⊗ΓU ΓV ∼ = t∗ log∗ {Ht (X e )U −e ⊗Γ(log(t U,O )) Γ(log(t−e V ), Ot )} e

−e

t

i∗ ∼

→= te log∗ {Ht ((X e )log(f −e) )V −e )} ∼ Ht (X f )V −e } = t∗ log∗ {t∗ f

log(e−f )

∼ = t∗f log∗ {Ht (X f )V −f } = Γ(V, ET (X)) where i denotes the inclusion (X e )log(f −e) = X f ,→ X e . We have i∗ is an isomorphism by localisation in equivariant cohomology (2.2,iii), and the last line is an isomorphism by (2.2,iv). It is clear that φU V satisfy the cocycle condition, and so by the discussion above we have defined a sheaf over ET . Similarly, if π : X → Y is a T -map, then π ∗ : Ht (Y e ) → Ht (X e ), e ∈ ET , induces a map of O-algebras, also denoted π ∗ , π ∗ : ET (Y ) → ET (X). (This is a map of sheaves by naturality of π ∗ and by (2.2,iv) above; the diagram chase is omitted). We remark that if L is a T -equivariant local system on X, or even a complex in DT (X), the derived category of T -equivariant sheaves on X, then the same procedure serves to define elliptic cohomology with coefficents in L, ET (X, L) and π ∗ : ET (Y, π ∗ L) → ET (X, L) (see [10]), as the localisation theorem (2.2,ii) is still true in this case. 2.5. Consider the local ring at 1 of ET (X); ET (X)1 @ > exp >> Ht (X)0 = Ht (X) ⊗Ht (Ot )0 ,→ H(X ×T ET ), where BT = ET /T is the classifying space of T . In the case X is a point, we may define s(x) = s exp = exp∗ (s) ∈ H(BS 1 ) as an orientation class, and regard it as an element of Γ(U, ES 1 ) for sufficiently small U . Here s : U ⊆ E → C is a local coordinate. The usual machinary of algebraic topology means that from this data we get Gysin morphisms π!E : Ht (X)0 → Ht (Y )0 for π : X → Y a proper weakly complex oriented map [1], as well as Todd classes s(x)/x and a Riemann-Roch isomorphism relating π!E and π!H , where π!H is the usual Gysin morphism in cohomology. Now let π : X → Y be a proper weakly complex oriented map. If e ∈ ET , denote π ˜ : X e → Y e the induced map. This is still a proper weakly complex oriented map.

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The map π defines a cohomlogy class λ(π) = [π] ∈ H 1 (ET , ET (X)× ) as follows. Let e ∈ U , f ∈ V ⊆ U , e 6= f be points on ET and small neighbourhoods containing them. Let i : X f ,→ X e be the inclusion. Then f f ∗ E i∗ iE ! : Ht (X )U −e → Ht (X )U −e is well defined, and i i! 1, which we write e f f e as e(X /X ), the Euler class of X ,→ X gives an invertible element of Ht (X f )V −f . (It is invertible as for each connected component X 0 of X f , the normal bundle in X e to X 0 does not contain the trivial T -bundle, and the orientation s(x) is a local coordinate on E; i.e. an analytic function with an isolated simple zero at 0 ∈ t). Then if π ˜ : X f → Y f is the induced map, π ˜ ∗ e(Y e /Y f ) · e(X e /X f )−1 is f an invertible element of Ht (X )V −e , so applying t∗e log∗ (i∗ )−1 to it gives an element λU V ∈ Γ(U ∩ V ) ⊗ΓU Γ(U, ET (X)× ). It is clear that if W ⊆ V , W a neigbourhood of f 0 that λV W λU V = λU W and so (λU V ) defines a cohomology class, λ(π). Now we define π! : ET (X)λ(π) → ET (Y ) by defining, for e ∈ U , U sufficiently small, Γ(U, π! ) := t∗e log∗ π ˆ!E , where π ˆ : X e ,→ Y e . It follows from the definitions and the “excess intersection formula” in a generalised cohomology theory that this is well defined (again we leave the diagram chase to the reader), and a map of ET (Y )-modules. 2.6. More generally, suppose G is a compact connected Lie group, T ,→ G the maximal torus, W = NG (T )/T the Weyl group. We define EG (X), a Z/2graded sheaf on ET /W as follows. Write p : ET → ET /W for the canonical projection. Then if U ⊆ ET is a small open neighbourhood, e ∈ U is such that W e U = U , wU ∩ U = ∅ if w 6∈ W e we define we

Γ(pU, EG (X)) = (⊕w∈W/W e t∗we log∗ {HGwe (X we ⊗HtW we Γ(log(t−we U ), Ot )W })W e ∼ = t∗ log∗ {HGe (X e ) ⊗ W e Γ(log(t−e U ), Ot )W }. e

Ht

e

Observe that HGe = HtW , and that neighbourhoods of this form cover ET , as W is finite. If V ⊆ U is a neighbourhood of f (so W f ⊆ W e ) such that W f V = V and wV ∩ V = ∅ if w 6∈ W f , define φU V : Γ(pU, EG (X)) ⊗ΓU W e f (ΓV )W → Γ(pV, EG (X)) as the composition of the obvious maps and the maps induced by i : (Gf , X f = (X e )log(f −e) ) ,→ (Ge , X e ) and translation t∗log(e−f ) as in (2.4). Then φU V is an isomorphism of rings over Γ(pV, ET /W ) = f (ΓV )W by non-Abelian localisation (2.2,iii), and the cocycle condition is satisfied. Similarly, if f : (G, X) → (H, Y ) is a map of (compact connected groups, spaces) inducing a map TG → TH of the maximal tori of G to that of H, and hence a map f˜ : ETG /WG → ETH /WH , we get an induced map of sheaves f ∗ : f˜∗ EH (Y ) → EG (X). Further, if h : (H, Y ) → (K, Z) is another such map of (groups,spaces) we have h∗ f ∗ = (f h)∗ . The obvious diagram chases required to verify this are omitted.

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Finally, by repeating word for word the discussion in (2.5) we see that for a proper weakly complex oriented map of G-spaces π : X → Y we have a cohomology class λ(π) ∈ H 1 (ET /W, EG (X)× ) and maps of Z/2-graded OET /W modules π! : EG (X)λ(π) → EG (Y ). (This is not a ring homomorphism!). One may check that this is functorial in the appropriate sense; i.e. that if π 0 : Y → Z is also a proper weakly complex oriented map of G-spaces, then 0 π ∗ λ(π 0 ) · λ(π) = λ(π 0 π) and (π 0 π)! : EG (X)λ(π π) → EG (Z) is the composite of π!0 and the map induced from π! . 3. Remarks 3.1. We leave to the reader the task of making a systematic list of the properties of EG . Most follow immediately from the definitions and the corresponding properties of HG , and are obvious analogues of the usual properties of a cohomology theory. The following remarks are some indications of properties more particular to EG . 3.2. Let X, X 0 be smooth projective G-spaces, with H odd (X) = H odd (X 0 ) = 0. Then the natural morphism EG (X) ⊗EG EG (X 0 ) → EG (X × X 0 ) is an isomorphism if and only if the centralizer of every pair of commuting semisimple elements of G is connected (see [7]). (This is immediate from the definition of EG , as with these hypotheses on X, X 0 such a Kunneth theorem holds in equivariant cohomology for arbitrary connected G). Essentially the only groups G with this property are products of GLn ’s and tori. Note that a Kunneth theorem holds in equivariant K-theory if and only if the centraliser of every semisimple element is connected; i.e. if and only if G is simply connected, by a theorem of Steinberg. The usual proof of this fact (Kazhdan-Lusztig, Hodgkins) relies on another theorem of Steinberg; clearly the ‘delocalised’ technique we use gives a different proof. 3.3. Let G be a compact group, with a fixed invariant non-degenerate symmetric bilinear form on g. This data defines a line bundle L on ET with this form as its Chern class [9]. For example, if G is simple and simply connected, and L has degree the order of the center of G, then the Weyl denominator for the affine Kac-Moody algebra b g is a section of ΓLg , where g is the dual Coxeter number for G [9].. One may then consider a “level k” elliptic cohomlogy of X as Γ(EG (X) ⊗ Lk ). 3.4. Modularity. Let H = {τ ∈ C | Imτ > 0}. Recall the group SL2 Z acts on C × H by ( ac db ) · (z, τ ) = (z/(cτ + d), (aτ + b)/(cτ + d)). Hence SL2 Z acts on t × H also. Denote by e(z, τ ) the image of (z, τ ) ∈ t × H in ET τ = t ⊗C Eτ . Then if X is a T -space, the fixpoints X e(z,τ ) depend only on the orbit of (z, τ ) under SL2 Z. (This is clear for ( −1 1 ) and ( 1 11 ), which generate SL2 Z). Hence the modular properties of EG (X) depend only on the modular properties of s(z, τ ), the local coordinate around 1. We can thus regard equivariant elliptic cohomology as defined on the moduli of marked elliptic curves, with marking

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determined by the chosen s. For example, for s the orientation class of [8], EG (X) is defined over the curve S = H/Γ0 (2). (In such a case, all sections Γ should be interpreted as sections over the universal marked curve.) References [1] J. F. AdamsStable homotopy and generalised homology University of Chicago press Chicago lectures in mathematics 1974 [2] J. Block and E. Getzler Equivariant cyclic homology and equivariant differential forms preprint 1993 [3] P.Baum, J. L. Brylinski and R. MacPherson Cohomologie ´equivariante d´elocalise C.R. Acad. Sci. Paris, Serie I 3001985605–608 [4] P. Etingof and I. FrenkelCentral extensions of current groups in two dimensions Communications in Math Physics [5] I. GrojnowskiRepresentations of affine Hecke algebras (and affine quantum GL n ) at roots of unity International Math. Research Notes 41994215–217 [6] I. GrojnowskiAffinizing quantum algebras: from D-modules to K-theory Preprint, 1994 [7] M. J. Hopkins, N. J. Kuhn and D. C. RavenelGeneralised group characters and complex oriented cohomology theories preprint [8] P. S. Landweber (Ed.) Elliptic curves and modular forms in algebraic topology Springer LNM 13261988 [9] E. LooijengaRoot systems and elliptic curves Invent. Math38197617–32 ´ [10] G. LusztigCuspidal local systems and graded Hecke algebras I Inst. Hautes Etudes Sci. Publ. Math 1988 67 145–202 [11] H. MillerThe elliptic character and the Witten genusContemp. Math961989281–289 Yale University, New Haven, CT 06520

ON FINITE RESOLUTIONS OF K(n)-LOCAL SPHERES HANS-WERNER HENN Dedicated to the memory of Dieter Puppe

Abstract. Let p be an odd prime and Gn be the n-th (extended) Morava stabilizer group. We construct finite resolutions of the trivial Gn -module Zp by (direct summands of) permutation modules with respect to finite p-subgroups of Gn . Furthermore we discuss the problem of realizing these resolutions by finite resolutions of the K(n)-local sphere by spectra which are (direct summands of) wedges of homotopy fixed point spectra for the action of these finite p-subgroups on the Lubin-Tate spectrum En .

0. Introduction Let p be a prime and let K(n) be the n - th Morava K-theory at p. The category of K(n)-local spectra is a basic building block of the stable homotopy category of p-local spectra and, of course, the localization of the sphere, LK(n) S 0 , plays a central role in this category. The homotopy of LK(n) S 0 can be studied by the Adams-Novikov spectral sequence: if En denotes the periodic Landweber exact spectrum En whose coefficients in degree 0 classify deformations (in the sense of Lubin and Tate) of the Honda formal group law over Fpn then, up to a Galois extension, the E2 -page of the spectral sequence can be identified, by the Morava change of rings isomorphism, with the continuous cohomology of the Morava stabilizer group Sn with coefficients in (En )∗ . In this paper we discuss homological properties of the groups Sn , in particular resolutions of the trivial module Zp , and show how they can be used to construct finite resolutions of LK(n) S 0 in terms of spectra which are easier to understand. For example, if p − 1 does not divide n, then the mod-p cohomological dimension cdp (Sn ) of Sn is finite, equal to n2 , the trivial module Zp admits a projective resolution of length n2 and the E2 -term of the Adams-Novikov spectral sequence has a horizontal vanishing line. In homotopy theory this This paper is a sequel to the joint paper [GHMR1]. It is inspired by that paper and the author is happy to acknowledge the influence of numerous discussions with Paul Goerss, Mark Mahowald and Charles Rezk on this subject. Thanks are also due to the referee for his suggestions. 122

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allows us to construct a finite En -resolution of LK(n) S 0 in the sense of Miller [Mi] (at least up to a Galois extension in case n is still divisible by p). If p−1 divides n, then the mod-p cohomological dimension of Sn is infinite, so Zp does not admit a finite projective resolution, the E2 -term has no horizontal vanishing line and a finite En -resolution for LK(n) S 0 cannot exist. However, in analogy with discrete groups of finite virtual cohomological dimension one can hope to construct resolutions of the trivial module Zp by permutation modules on finite subgroups of Sn and then hope to realize those by finite resolutions (which will not be En -resolutions) of LK(n) S 0 via homotopy fixed point spectra of the form EnhF for suitable finite subgroups of Sn . This is in fact the main subject of this paper. There are various advantages of such resolutions. For instance, such a finite resolution gives rise to a spectral sequence with a horizontal vanishing line starting from the homotopy of the homotopy fixed point spectra and converging to π∗ (LK(n) S 0 ). This spectral sequence should be more manageable than the Adams-Novikov spectral sequence. For example, some of the delicate differentials in the latter might already be accounted for by more transparent phenomena in the homotopy of the homotopy fixed point spectra E2hF . In fact, this is essentially what happened in the calculation of the homotopy of the Toda-Smith complex V (1) at the prime p = 3 carried out in [GHM] (completing earlier work of Shimomura [Sh]). The resolutions can also be used to analyze the exotic part of Hopkins’ Picard groups (cf. [HMS]), i.e. the group of homotopy equivalence classes of invertible spectra in the category of K(n)-local spectra whose Morava module is isomorphic to that of the sphere S 0 . This will be pursued in a separate paper. On a more philosophical level, one can say that these resolutions capture to what extent the presence of finite p-subgroups in the stabilizer group influences the homotopy of π∗ (LK(n) S 0 ), very much in the same way as finite subgroups in a discrete group of finite virtual cohomological dimension influence the cohomology of the group. Here is an outline of the paper. In section 1 we recall background material on K(n)-localization, the stabilizer groups and homotopy fixed point spectra. Section 2 discusses algebraic and homotopy theoretic resolutions in the case n 6≡ 0 mod p − 1; there is a general existence result (Theorem 4) and beyond that we discuss the few cases n = 1 and p > 2 as well as n = 2 and p > 3 in which finite resolutions are known in an explicit form. The results in this section are mostly reformulations or reinterpretations of results which have been known for more than 25 years. They are included for completeness and in order to help develop our ideas on the interplay between homological properties of the groups Sn and homotopical properties of LK(n) S 0 . In section 3 we discuss the much more difficult case n ≡ 0 mod p − 1. We start by explaining how the

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well-known fibration Ψ3 −id

LK(1) S 0 → KOZ2 −→ KOZ2 (where KOZ2 is 2-adic real K-theory) can be regarded as the realization of a particular permutation resolution of the trivial S1 -module Z2 . Then we use this example as a role model which suggests possible generalizations for larger n. In section 3.3 we survey recent joint work with Goerss, Mahowald and Rezk in which the case n = 2, p = 3 was studied. In section 3.4 we comment on joint work in progress with the same coauthors which concerns the case n = 2, p = 2. In the final two sections we present new general results on the existence of finite resolutions of the trivial module Zp by permutation modules, at least for p > 2 (Proposition 17). If n = p − 1, we show how these algebraic resolutions can be realized by finite resolutions of LK(n) S 0 (Theorem 25 and Theorem 26). 1. Background 1.1 Localization with respect to Morava K-theory. 1.1.1. Let E be a spectrum and E∗ be the generalized homology theory determined by E. We recall that Bousfield localization with respect to E∗ is a functor LE from the homotopy category of spectra to itself together with natural maps λ : X → LE X which are terminal among all E∗ -equivalences. By [B] LE exists for each E. Classical examples are given by localization with respect to the Moore spectra MZ(p) , for the p-local integers Z(p) , resp. MQ, for the rationals, in which case LE is the homotopy theoretic version of arithmetic localization with respect to Z(p) resp. with respect to Q. 1.1.2. In this paper we will be concerned with the localization functors LK(n) with respect to Morava K-theory K(n). We refer to [HS] for a good general reference. Here we recall that K(0) = HQ is the rational Eilenberg-Mac Lane spectrum and is independent of p. Otherwise, for any fixed prime p we have K(n)∗ = Fp [vn±1 ] with generator vn in degree 2(pn − 1). Furthermore, K(n)∗ is a multiplicative periodic cohomology theory which admits a theory of characteristic classes, and the associated formal group law Γn is the Honda formal group law of height n. Localization with respect to K(n) plays a prominent role in stable homotopy theory because the functors LK(n) are elementary “building blocks” of the stable homotopy category of finite p-local complexes in the following sense. • The localization functor LK(n) is “simple” in the sense that the category of K(n)-local spectra contains no nontrivial localizing subcategory [HS, Theorem 7.5].

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• There is a tower of localization functors . . . → Ln → Ln−1 → . . . → L0 (with Ln = LK(0)∨...∨K(n) ) together with compatible natural maps X → Ln X such that X ' holimn Ln X for every finite p-local spectrum X [Ra2, Theorem 7.5.7]. Furthermore, for every X there is a homotopy pullback diagram (a “chromatic square”) LnX −→ LK(n) X y y Ln−1 X −→ Ln−1 LK(n) X

i.e. Ln is built from LK(n) and Ln−1 . (The diagram is easily established by using that LK(n) Ln−1 Z ' ∗ for any Z. Its existence is implicit in [Ho].) 1.2 The stabilizer groups.

1.2.1. The functors LK(n) are controlled by cohomological properties of the Morava stabilizer group Sn . We recall that Sn is the group of automorphisms of the p-typical formal group law Γn over the field Fq (with q = pn ) whose n [p]-series is given by [p](x) = xp . The group Sn can be extended to a slightly larger group Gn . In fact, because Γn is already defined over Fp the Galois group Gal(Fq /Fp ) of the finite field extension Fp ⊂ Fq acts on Aut(Γn ) = Sn and Gn can be identified with the semidirect product Sn o Gal(Fq /Fp ). In the sequel we will recall some of the basic properties of the group Sn resp. Gn . The reader is referred to [Ha] or [Ra1] for more details (see also [He] for a summary of what will be important in this paper). 1.2.2. The group Sn is equal to the group of units in the endomorphism ring of Γn , and this endomorphism ring can be identified with the maximal order On of the division algebra Dn over Qp of dimension n2 and Hasse invariant 1 . In more concrete terms, On can be described as follows: let WFq denote n the Witt vectors over Fq . Then On is the non-commutative ring extension of WFq generated by an element S which satisfies S n = p and Sw = w σ S, where w ∈ WFq and w σ is the image of w with respect to the lift of the Frobenius automorphism of Fq . The element S generates a two sided maximal ideal m in On with quotient On /m = Fq . Inverting p in On yields the division algebra Dn , and On is its maximal order. The action of the Galois group Gal(Fq /Fp ) on Sn is realized by conjugation by S inside D× n , the group of units of Dn , and the semidirect product Gn can

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therefore be described as the quotient of D× n by the central subgroup generated n × n ∼ by S , i.e. Gn = Dn /hS i. 1.2.3. Reduction mod m induces an epimorphism On× −→ F× q . Its kernel will be denoted by Sn and is also called the strict Morava stabilizer group. The group Sn is equipped with a canonical filtration by subgroups Fi Sn , i = nk , k = 1, 2, . . ., defined by Fi Sn := {g ∈ Sn |g ≡ 1 mod S in } . The intersection of all these subgroups contains only the element 1 and Sn is complete with respect to this filtration, i.e. we have Sn = limi Sn /Fi Sn . Furthermore, we have canonical isomorphisms Fi Sn /F 1 Sn ∼ = Fq i+ n

induced by x = 1 + aS in 7→ a ¯. Here a is an element in On , i.e. x ∈ Fi Sn and a ¯ is the residue class of a in On /m = Fq The associated graded object grSn with gri Sn = Fi Sn /Fi+ 1 Sn , i = n1 , n2 , . . . n becomes a graded Lie algebra with Lie bracket [¯ a, ¯b] induced by the commutator xyx−1 y −1 in Sn . Furthermore, if we define a function ϕ from the positive real numbers to itself by ϕ(i) := min{i + 1, pi} then the p - th power map on Sn induces maps P : gri Sn −→ grϕ(i) Sn which define on grSn the structure of a mixed Lie algebra in the sense of Lazard [La, Chap. II.1]. If we identify the filtration quotients with Fq as above then the Lie bracket and the map P are explicitly given as follows (cf. Lemma 3.1.4 in [He]). Lemma 1. Let a ¯ ∈ gri Sn , ¯b ∈ grj Sn . Then a) ni nj [¯ a, ¯b] = a ¯¯bp − ¯b¯ ap ∈ gri+j Sn .

b)

ni (p−1)ni ¯1+p +...+p a ni (p−1)ni Pa ¯= a ¯ + a¯1+p +...+p a ¯

if i < (p − 1)−1 if i = (p − 1)−1 if i > (p − 1)−1 .

1.2.4. Next we record some basic facts about finite p-subgroups of Sn . First of all, all finite abelian p-subgroups of Sn are cyclic. Sn is known to contain a cyclic subgroup of order pk if and only if pk−1 (p − 1) divides n, and then such a cyclic subgroup Cpk ⊂ Sn is unique up to conjugacy. Furthermore, if p > 2, or p = 2 and n is odd, then all finite p-subgroups are cyclic.

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The structure of centralizers of cyclic subgroups will be of importance for us. To get at it we note that the centralizer CDn (Cpk ) of Cpk in Dn is again a division algebra. It is central over the cyclotomic extension of Qp generated by Cpk and its dimension over its center is m2 , if n = mpk−1 (p − 1). Then the centralizer CSn (Cpk ) of Cpk in Sn can be identified with the group of units in the maximal order of CDn (Cpk ). 1.2.5. Recall that the cohomological p-dimension cdp (G) of a profinite group G is defined as cdp (G) = sup{n ∈ N|H n (G, M) 6= 0 for some finite continuous G−module M} where, here and elsewhere in this paper, H ∗ (G, M) is always continuous cohomology. Later on M may be a continuous profinite module over the completed group algebra Zp [[G]] := limU Zp [G/U] (with U running through all open normal subgroups of G). We refer to [SyW] for a discussion of the relevant homologial algebra. The cohomological p-dimension of the group Sn resp. Gn is n2 unless Sn resp. Gn contain non-trivial finite p-subgroups in which case it is infinite. By 1.2.4 this happens for Sn iff p − 1 divides n, and in the case of Gn this happens iff p or p − 1 divides n. However, even in these cases Sn and Gn are still virtually of finite cohomological p-dimension (i.e. they contain a finite index subgroup of finite cohomological p-dimension) and its virtual cohomological p-dimension vcdp (Sn ) remains n2 . The reader is referred to [La] or [SyW] for more details on these notions. 1.3 Homotopy fixed point spectra. 1.3.1. By Hopkins-Miller (cf. [Re]) the group Gn acts on the Lubin-Tate spectrum En ; we recall that En is the Landweber exact spectrum given by the 2-periodic theory with coefficients π∗ (En ) = π0 (En )[u±1 ] (with u ∈ π−2 (E)) whose associated formal group law over π0 (En ) is a universal deformation of Γn in the sense of Lubin and Tate [LT]. In particular there is a (non-canonical) isomorphism between π0 (En ) and WFq [[u1 , . . . , un−1 ]], the ring of formal power series over WFq in the variables u1 , . . . , un−1 . We can and will choose the fn to be p-typical with p-series universal deformation Γ [p]Γe n (x) = px +Γe n u1 xp +Γe n . . . +Γe n un−1 xp

n−1

n

+Γe n xp ,

in other words the classifying map BP∗ → π∗ (En ) sends the Araki generator i n vi to ui u1−p , if i < n, vn to u1−p , and vi to 0 if i > n. Let OGn be the orbit category of Gn , i.e. the objects of OGn are orbits Gn /K where K is a closed subgroup of Gn and morphisms are continuous Gn equivariant maps. By Devinatz-Hopkins [DH2] there is a contravariant functor

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from OGn to K(n)-local spectra which assigns to Gn /K the homotopy fixed point spectrum EnhK and this spectrum comes with an associated homotopy fixed point spectral sequence E2s,t = H s (K; πt (En )) =⇒ πt−s (EnhK ) . Furthermore, EnhGn can be identified with LK(n) S 0 and the Adams-Novikov spectral sequence for LK(n) S 0 can be identified with the associated homotopy fixed point spectral sequence E s,t ∼ = H s (Gn , (En )t ) =⇒ πt−s LK(n) S 0 . 2

Finally, EnhGn can be identified with the iterated homotopy fixed trum (EnhSn )hGal(Fq /Fp ) , the Galois group acts on the homotopy

point specfixed point

spectral sequence E2s,t ∼ = H s (Sn , (En )t ) =⇒ πt−s (EnhSn ) , and the action on the whole spectral sequence is coinduced. Thus we get isomorphisms π∗ LK(n) S 0 ∼ = π∗ (EnhSn )Gal(Fq /Fp ) , π∗ EnhSn ∼ = π∗ LK(n) S 0 ⊗Zp WFpn , and we may therefore say that EnhSn is equal to LK(n) S 0 , up to a Galois extension. 1.3.2. Hopkins and Devinatz also showed that for any closed subgroup K ⊂ Gn there is an isomorphism π∗ (LK(n) (En ∧ E hK )) ∼ = maps (Gn /K, (En )∗ ) n

cts

(where mapcts denotes continuous maps). The isomorphism is functorial on OGn . It is compatible with the obvious (En )∗ -module structures on both sides as well as with the actions of Gn on both sides, which is via the action on En on the left hand side and via the diagonal action on the space of continuous maps on the right hand side. In other words, the isomorphism is one of Morava modules where a Morava module M is a complete (En )∗ -module with a continuous action of Gn such that g(ax) = g(a)g(x) for g ∈ Gn , a ∈ (En )∗ , x ∈ M . By abuse of notation we will also say that M is a (twisted) (En )∗ [[Gn ]]-module. Typical examples of such modules are given by π∗ (LK(n) (En ∧ X)), at least under suitable conditions on X, e.g. if K(n)∗ X is evenly graded (see [HS], or [GHMR1] for a summary of what is important for us). In order to keep our notation compact we will write in the sequel (En )∗ X instead of π∗ (LK(n) (En ∧ X)). 1.3.3. We will need information about maps between various homotopy fixed point spectra. In the following F stands for function spectrum. We recall the following results from [GHMR1].

On finite resolutions of K(n)-local spheres

129

1.3.3.1. Let U be an open subgroup of Gn . Functoriality of the homotopy fixed point spectra construction of [DH2] gives us a map EnhU ∧ Gn /U+ → En where as usual Gn /U+ denotes Gn /U with a disjoint base point added. Together with the product on En we obtain a map En ∧ EnhU ∧ Gn /U+ → En ∧ En → En whose adjoint induces an equivalence of En -module spectra Y LK(n) (En ∧ EnhU ) → En Gn /U

realizing the isomorphism of 1.3.2 above. Now let FEn be the function spectrum in the category of En -module spectra (see [EKMM] for details). If we apply FEn (−, En ) to this last equivalence we obtain another equivalence of En -module spectra Y FE n ( En , En ) → FEn (En ∧ EnhU , En ) Gn /U

which can be rewritten as an equivalence (still of En -module spectra) En ∧ Gn /U+ ' F (EnhU , En ) . The same reasoning shows that we can replace En in the target of the function spectrum by LK(n) (En ∧ I) where I is any spectrum and we obtain an equivalence LK(n) (En ∧ I) ∧ Gn /U+ ' F (EnhU , LK(n) (En ∧ I)) . 1.3.3.2. More generally, let K be any closed subgroup of GnT. Then there exists a decreasing sequence Ui of open subgroups Ui with K = i Ui and by [DH2] we have EnhK ' LK(n) hocolimi EnhUi . By passing to the limit we obtain an equivalence LK(n) (En ∧ I)[[Gn /K]] ' F (EnhK , LK(n) (En ∧ I)) where we have used the convention that if E is a spectrum and X = lim i Xi is an inverse limit of a sequence of finite sets with each Xi finite then Q E[[X]] is given as holimi E ∧(Xi )+ , i.e. as the fibre of the usual self map of i E ∧(Xi )+ . Note that if X is such a profinite set with continuous K-action and if E is a K-spectrum then E[[X]] is a K-spectrum via the diagonal action. If we concentrate (for simplicity) on the case I = S 0 and take homotopy fixed points in this equivalence with respect to another finite subgroup of Gn then we get the following result.

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Proposition 2 ([GHMR1, Prop. 2.6]). a) Let K1 be a closed subgroup and K2 a finite subgroup of Gn . Then there is a natural equivalence (where the homotopy fixed points on the left hand side are formed with respect to the diagonal action of K2 ) En [[Gn /K1 ]]hK2 ' F (EnhK1 , EnhK2 ) . b) If K1 is also an open subgroup then there is a natural decomposition Y En [[Gn /K1 ]]hK2 ' EnhKx K2 \Gn /K1

where Kx = K2 ∩ xK1 x−1 is the isotropy subgroup of the coset xK1 and K2 \Gn /K1 is the finite (!) set of double cosets. T c) If K1 is a closed subgroup and K1 = i Ui for a decreasing sequence of open subgroups Ui then Y F (EnhK1 , EnhK2 ) ' holimi En [[Gn /Ui ]]hK2 ' holimi EnhKx,i K2 \Gn /Ui

where Kx,i = K2 ∩ xUi x−1 is, as before, the isotropy subgroup of the coset xUi . Remark If Ui ⊂ Uj then the map Y EnhKx,i → K2 \Gn /Ui

Y

EnhKx,j

K2 \Gn /Uj

in the inverse system of part (c) of the proposition can be described as follows: if x ∈ Gn /Ui has isotropy group Kx,i and its image x0 ∈ Gn /Uj has isotropy group Kx0 ,j then the restriction of the map to the factor determined by x sends hK 0 hK En x,i via the transfer to the factor En x ,j determined by x0 . In particular, this implies that on homotopy groups the inverse system is Mittag-Leffler. In the next result Hom(En )∗ [[Gn ]] (−, −) denotes homomorphisms of Morava modules. Proposition 3 ([GHMR1, Prop. 2.7]). Let K1 and K2 be closed subgroups of Gn and suppose that K2 is finite. Then there is an isomorphism K ∼ = (En )∗ [[Gn /K1 ]] 2 −→ Hom(En )∗ [[Gn ]] ((En )∗ EnhK1 , (En )∗ EnhK2 )

such that the following diagram commutes

K2 hK2 π∗ En [[G −→ (En )∗ [[G n /K1 ]] n /K1 ]] ∼ ∼ y = y = hK1 hK2 π∗ F (En , En ) −→ Hom(En )∗ [[Gn ]] ((En )∗ EnhK1 , (En )∗ EnhK2 )

On finite resolutions of K(n)-local spheres

131

where the top horizontal map is the edge homomorphism in the homotopy fixed point spectral sequence, the left-hand vertical map is the isomorphism given by Proposition 2 and the bottom horizontal map is the En -Hurewicz homomorphism. 2. The case n 6≡ 0 mod p − 1 In this section we begin our discussion of LK(n) S 0 . The case n = 0 is both exceptional and trivial: K(0) = MQ and LK(0) S 0 is the rationalized sphere spectrum. From now on we will therefore assume n > 0. 2.1 Explicit examples I: the case n = 1 and p > 2. 2.1.1. We briefly review the case n = 1 which is well understood. In this case we have E1 = KZp (p-adic complex K-theory). The group G1 = S1 can be identified with Z× p , the group of units in the p-adic integers. If p is odd then ∼ C × Z , Z× = p−1 p where Cp−1 denotes the cyclic group of order p − 1 given by p the roots of unity in Zp . The homotopy fixed points E1hG1 can be formed in two steps, first with respect to Cp−1 and then with respect to Zp . Thus we obtain the following fibration (cf. [HMS]) in which ψ p+1 is the appropriate hC Adams operation and KZp p−1 can be identified with the Adams summand of p-adic complex K-theory ψ p+1 −id

p−1 p−1 . −→ KZhC LK(1) S 0 → KZhC p p

2.1.2. This fibration can also be considered as a suitable realization of a projective resolution of the trivial Zp [[G1 ]]-module Zp , and it is this point of view which turns out to be useful for finding generalizations of the above fibration for larger n. To get at this projective resolution we start with the following obvious short exact sequence of modules over the power series ring Zp [[t]] ×t

0 → Zp [[t]] −→ Zp [[t]] → Zp → 0 . Now recall that there is an isomorphism of complete algebras Zp [[t]] ∼ = Zp [[Zp ]] induced by sending t to g − e ∈ Zp [[Zp ]] if g is a topological generator of Zp , ∼ e.g. if g is the image of p + 1 ∈ Z× p = G1 in the quotient group G1 /Cp−1 = Zp . Therefore we can consider this sequence as an exact sequence of Zp [[Zp ]]modules, or even as an exact sequence of Zp [[G1 ]]-modules, and as such we can write it as g−e G1 1 0 → Zp ↑G Cp−1 −→ Zp ↑Cp−1 → Zp → 0

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1 where M ↑G Cp−1 denotes the induced module of a Zp [Cp−1 ]-module M, i.e. the b Zp [Cp−1 ] M. Because the trivial Zp [Cp−1 ] completed tensor product Zp [[G1 ]]⊗ 1 module Zp is projective we find that Zp ↑G Cp−1 is a projective Zp [[G1 ]]-module and the exact sequence is a projective resolution of the trivial Zp [[G1 ]]-module Zp which is even split as a sequence of continuous Zp -modules. So if we apply the functor Homcts (−, (KZp )∗ ) of continuous homomorphisms into (KZp )∗ to this sequence then we obtain another short exact sequence which, by 1.3.2, can be identified with the (`a priori long) exact sequence which is associated to our fibration: 0 → (KZp )∗ (LK(1) S 0 ) ∼ = (KZp )∗ → (KZp )∗ KZhCp−1 → (KZp )∗ KZhCp−1 → 0 .

p

p

2.2 The general case. 2.2.1. Following Miller [Mi] we say that a K(n)-local spectrum I is En -injective if the map I = S 0 ∧ I → LK(n) (En ∧ I) induced by the unit in En is split. Furthermore, a sequence X 0 → X → X 00 of K(n)-local spectra is En -exact if the composition X 0 → X 00 is trivial and if [X 0 , I] ←− [X, I] ←− [X 00 , I] is an exact sequence of abelian groups for each En -injective spectrum I. Finally an En -resolution of a K(n)-local spectrum X is an En -exact sequence of K(n)local spectra ∗ → X → I0 → I1 → . . . (i.e. every three term subsequence is En -exact) such that each I s , s ≥ 0, is En -injective. 2.2.2. The following result is a folk theorem whose roots can be traced back to the work of Morava [Mo]. Theorem 4. If n is neither divisible by p − 1 nor by p then LK(n) S 0 admits an En -resolution of length n2 . In fact, each of the En -injectives in the resolution can be chosen to be a direct summand of a finite wedge of En ’s. Proof. The idea of the proof is to start with information in homological algebra and use this to construct an En -resolution. If neither p − 1 nor p divides n then cdp (Gn ) = n2 . Therefore the trivial Zp [[Gn ]]-module Zp admits a projective resolution P• : 0 → P n 2 → . . . → P 0 → Z p → 0 of length n2 , and by [La] we may assume that each projective is finitely generated as Zp [[Gn ]]-module.

On finite resolutions of K(n)-local spheres

133

We want to construct an En -resolution X• of LK(n) S 0 X• : ∗ → X−1 → X0 → . . . → Xn2 → ∗ with X−1 = LK(n) S 0 such that the complex Homcts (P• , (En )∗ ) is isomorphic, as a complex of Morava modules, to the complex (En )∗ X• . For this we note that if F r is a free Zp [[Gn ]]-module of rank r then we have an isomorphism of Morava modules r _ r ∼ (1) Homcts (F , (En )∗ ) = (En )∗ ( En ) j=1

and the En -Hurewicz homomorphism (2)

[En , En ] → Hom(En )∗ [[Gn ]] ((En )∗ En , (En )∗ En )

is an isomorphism. In fact, (1) resp. (2) follow immediately from 1.3.2 resp. from Proposition 3. Property (2) allows us now to construct both the spectra Xs , s = 0, 1, . . . , n2 , (by lifting idempotents on finitely generated free Zp [[Gn ]]-modules to homotopy idempotents on corresponding wedges of En ’s) as well as the required maps between these spectra. Because En is K(n)-local and En is a ring spectrum, the spectra Xs will all be En -injective. It remains to show that the sequence is En -exact. For this it is enough to show that for any spectrum Z of the form Z := LK(n) (En ∧ I) with some spectrum I the complex [X• , Z] is exact. By our discussion in 1.3.3 we know that [En , Z] ∼ = limi (π0 (Z) ⊗ Zp [[Gn /Ui ]]) . = π0 (Z[[Gn ]]) ∼ Now assume first that π0 (Z) is p-complete, i.e. π0 (Z) ∼ = limj π0 (Z)/pj . Then we even have [En , Z] ∼ = π0 (Z[[Gn ]]) ∼ = limi,j (π0 (Z) ⊗ Z/pj [[Gn /Ui ]]) . e denote the This can be restated as follows. For a fixed abelian group A let A⊗− functor from profinite Zp -modules and continuous homomorphisms to abelian groups which sends the profinite Zp -module M ∼ = limα Mα , where each Mα is finite, to limα (A ⊗ Mα ). Therefore, as long as π0 (Z) is p-complete, we can write e p [[Gn ]] . [En , Z] ∼ = π0 (Z)⊗Z Lr More generally, if P is a direct summand in j=1 Zp [[Gn ]] and if X is the Wr corresponding direct summand in j=1 En then e [X, Z] ∼ = π0 (Z)⊗P

and we even obtain an isomorphism of complexes e •. [X• , Z] ∼ = π0 Z ⊗P

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Because P• is split as a complex of continuous Zp -modules, [X• , Z] is exact provided π0 (Z) is p-complete. In addition, under this hypothesis on Z this complex is even naturally split in Z. In the general case we use that Z = LK(n) (En ∧I) can be written as homotopy inverse limit of a sequence Zn of spectra with p-complete homotopy groups, even bounded p-torsion homotopy groups (cf. [HS, Proposition 7.10]). Because [X• , Zn ] is naturally split in n, we see that limin [X• , Zn ] is split for i = 0, 1, in particular exact, and therefore [X• , Z] is exact, i.e. the sequence X• is En -exact. Remark 1 This result is a pure existence result. It says nothing about an explicit form of such a resolution. Remark 2 If n is divisible by p (but not by p−1) we can offer the two following substitutes of Theorem 4. Either we can use the existence of a finite projective resolution of the trivial n Zp [[Sn ]]-module Zp to get one for the induced Zp [[Gn ]]-module Zp ↑G Sn . This resolution can then be realized as in the proof of Theorem 4 to give an En resolution for EnhSn . which by 1.3.1 is, up to a Galois extension, equal to LK(n) S 0 . Or, if we insist on a resolution of LK(n)SS 0 , we can consider the formal group Γn over the algebraic extension K := r≥0 Fqpr of Fq with Galois group Gal(K, Fq ) ∼ = Zp . This will have the effect of replacing Gn = Sn o Gal(Fq /Fp ) by the group Gn (K) := Sn o Gal(K/Fp ). The advantage of doing this is that while Gn has elements of order p and therefore infinite mod-p cohomological dimension, the group Gn (K) has no elements of order p and its mod-p cohomological dimension is finite, equal to n2 + 1. As a consequence one gets a projective resolution of the trivial Gn (K)-module Zp of length n2 +1. If we also replace En by the corresponding Lubin-Tate spectrum En (K) whose homotopy groups in degree 0 classify deformations of Γn over K then our proof carries over verbatim: we only need to remark that 1.3.2 and the two properties (1) and (2) in the proof of Theorem 4 hold with En and Gn replaced by En (K) and Gn (K). 1 2.3 Explicit examples II: the case n = 2 and p > 3. As before we let q = pn . We start with an observation valid for all n > 1 (cf. section 1.3 of [GHMR1]). The reduced norm Sn → Z× p admits a canonical × extension Gn → Zp × Gal(Fq /Fp ) and by composing with the evident projection we obtain a homomorphism Gn → Z× p . Furthermore, if we identify the 1I

would like to thank Ethan Devinatz for a reassuring discussion of this point.

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On finite resolutions of K(n)-local spheres

quotient of Z× p by its subgroup of elements of finite order by Zp we obtain a homomorphism Gn → Zp . The kernel of this homomorphism will be denoted G1n and the kernel of its restriction to Sn will be denoted by Sn1 . G1n contains a cyclic group Cq−1 of order q − 1 (the roots of unity in WFq ⊂ On ) and this subgroup is invariant with respect of the action of Gal(Fq /Fp ). Therefore G1n contains the semidirect product Cq−1 o Gal(Fq /Fp ). We will denote this finite subgroup by Fn(q−1) in the sequel. hFn(q−1)

For more information on π∗ (En

) we refer to the appendix.

Theorem 5. Assume n = 2 and p > 3. a) There exists a fibration hG12

LK(2) S 0 → E2

hG12

−→ E2

and an En -resolution hG12

∗ → E2

hF2(q−1)

where X ' Σ2(p−1) E2

hF2(q−1)

→ E2

∨ Σ2(p

hF2(q−1)

→ X → X → E2

2 −p)

hF2(q−1)

E2

→∗

.

b) There exists an En -resolution of the form ∗ → LK(2) S 0 ' E2hG2 →E2

hF2(q−1)

hF2(q−1)

→ E2

∨X →

hF2(q−1)

→ X ∨ X → X ∨ E2

hF2(q−1)

→ E2

→∗.

Given Theorem 4 this is a fairly straightforward consequence of the following purely algebraic result in which λp−1 denotes the Zp [F2(q−1) ]-module whose underlying Zp -module is WFq , on which Cq−1 ⊂ W× Fq acts via Cq−1 × WFq → WFq , (g, w) 7→ g p−1 w and on which the group Gal(Fq /Fp ) acts via the lift of Frobenius (cf. appendix). The algebraic result is in turn a consequence of the calculation of the cohomology of the relevant Morava stabilizer algebra in [Ra1, Theorem 6.3.22]. Proposition 7 below is the group theoretic version of Ravenel’s result. Theorem 6. Assume n = 2 and p > 3. a) There exists a short exact sequence of Zp [[G2 ]]-modules 2 2 0 → Z p ↑G → Z p ↑G → Zp → 0 G1 G1 2

2

and a projective resolution of the trivial Zp [[G12 ]]-module Zp G1

G1

G1

G1

2 2 2 2 0 → Zp ↑F2(q−1) → λp−1 ↑F2(q−1) → λp−1 ↑F2(q−1) → Zp ↑F2(q−1) → Zp → 0 .

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b) There exists a projective resolution of the trivial Zp [[G2 ]]-module of the form G2 G2 2 0 → Z p ↑G F2(q−1) →(λp−1 ⊕ Zp ) ↑F2(q−1) → (λp−1 ⊕ λp−1 ) ↑F2(q−1) → G2 2 →(Zp ⊕ λp−1 ) ↑G F2(q−1) → Zp ↑F2(q−1) → Zp → 0 .

Proposition 7. Assume n = 2 and p > 3. a) H ∗ (S21 ; Fp ) is a 3-dimensional Poincar´e duality algebra and H 3 (S 1 ; Fp ) ∼ = Fp 2

is trivial as module over F2(q−1) . b) There is a canonical isomorphism of Zp [F2(q−1) ]-modules H 1 (S 1 ; Fp ) ∼ = λp−1 ⊗Z Fp . 2

p

c) The Bockstein homomorphism induces an isomorphism of Zp [F2(q−1) ]-modules H 1 (S21 ; Fp ) ∼ = H 2 (S21 ; Fp ) . Proof of the Proposition. We start by proving (b). S21 is a torsionfree p-adic Lie group of dimension 3; so by [La, V.2.5.8] H ∗ (S21 ; Fp ) is a 3-dimensional Poincar´e duality algebra which is therefore additively determined by H 1 (S21 ; Fp ) resp. its dual H1 (S21 ; Fp ). The filtration of S2 described in section 1.2.3 induces one of S21 . From the splitting S2 ∼ = S21 × Zp we deduce that gri S21 = gri S2 if i = k2 with k odd and that gri S21 is given by the kernel of the trace map Fq → Fp if i = k2 with k even. In particular, if i = k2 with k odd, there is an element b ∈ gr1 S21 with (b

p2i

− b) 6= 0, and then Lemma 1a shows that the commutator map gri S21 × p2i

gr1 S21 → gri+1 S21 given by [a, b] = a(b −b) is onto. Furthermore, if i = k2 with k odd and j = 2l with l odd then the commutator map gri S21 ×grj S21 → gri+j S21 p given by [a, b] = ab − bap is again onto. This implies that there is a canonical isomorphism H1 (S 1 ; Zp ) ∼ = F1/2 S 1 /F1 S 1 ∼ = Fq ∼ = H1 (S 1 ; Fp ) . 2

2

2

2

Furthermore, the conjugation action of ω ∈ Cq−1 is induced by (ω, 1 + aS) 7→ ω(1 + aS)ω −1 = 1 + ω 1−p aS while the action of Frobenius is induced by (S, 1 + aS) 7→ S(1 + aS)S −1 = 1 + aσ S and this implies that our isomorphism is an isomorphism of Zp [F2(q−1) ]-modules: H1 (S 1 ; Zp ) ∼ = H1 (S 1 ; Fp ) ∼ = λ1−p ⊗Z Fp . 2

2

p

By dualizing we get an isomorphism of Zp [F2(q−1) ]-modules H 1 (S 1 ; Fp ) ∼ = (λ1−p )∗ ⊗Z Fp . 2

p

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On finite resolutions of K(n)-local spheres

Now λp−1 (and λ1−p ) are self-dual (the isomorphism is induced by the pairing (x, y) 7→ xy −1 + (xy −1 )σ ) and they are both isomorphic (the isomorphism is given by x 7→ xσ ) and therefore we obtain (b). Because of H1 (S21 ; Zp ) ∼ = H1 (S21 ; Fp ) ∼ = Fq we see that the Bockstein homo1 1 morphism β : H2 (S2 ; Fp ) → H1 (S2 ; Fp ) is onto, and hence β : H 1 (S21 ; Fp ) → H 2 (S21 ; Fp ) is mono. On the other hand Poincar´e duality gives an additive isomorphism H 1 (S21 ; Fp ) ∼ = H 2 (S21 ; Fp ) and hence the Bockstein gives an isomorphism H 1 (S21 , Fp ) ∼ = H 2 (S21 ; Fp ) which is clearly Zp [F2(q−1) ]-linear and thus (c) is proved. To prove (a) we consider the subgroup F1 S21 of S21 and note that this subgroup is invariant by the conjugation action of F2(q−1) . Using Lemma 1a as above shows that the closure of the commutator subgroup of F1 S21 is F 5 S21 2 and then Lemma 1b gives that the closure of the subgroup generated by pth powers and commutators is F2 S21 . It follows that H1 (F1 S21 ; Zp ) is torsion (∼ = Z/p2 ⊕ Z/p⊕ Z/p) and that we have an isomorphism of Zp [F2(q−1) ]-modules H1 (F1 S 1 ; Fp ) ∼ = gr1 S 1 ⊕ gr 3 S 1 . 2

2

2

2

Identifying the Zp [F2(q−1) ]-module structure on gr1 S21 ⊕ gr 3 S21 as before shows 2 that we have an isomorphism of Zp [F2(q−1) ]-modules Tr H1 (F1 S21 ; Fp ) ∼ = (Ker : Fq −→ Fp ) ⊕ λp−1

where Cq−1 acts trivially on the first summand and Frobenius acts by multiplication by −1. Again this module is self-dual so that we have an isomorphism of Zp [F2(q−1) ]-modules Tr H 1 (F1 S21 ; Fp ) ∼ = (Ker : Fq −→ Fp ) ⊕ λp−1 .

Now we use that F1 S21 is ´equi-p-valu´e in the sense of Lazard, hence its cohomology is the exterior algebra on H 1 (F1 S21 ; Fp ) [La, Proposition V.2.5.7.1)]. So if α is any endomorphism of H 1 (F1 S21 ; Fp ) then the induced homomorphism on H 3 (F1 S21 ; Fp ) ∼ = Fp is given by multiplication with the determinant of α. In particular, for the action of ω ∈ Cq−1 the determinant is 1; it is obviously Tr 1 on Ker : Fq −→ Fp and it is 1 on λp−1 because ω acts via multiplication by ω p−1 and hence its determinant is a (p − 1)-st power in F× p . For the action of Tr

Frobenius the determinant is again 1 because it is −1 on both Ker : Fq −→ Fp and on λp−1 . Therefore H 3 (F1 S21 ; Fp ) is trivial as Zp [F2(q−1) ]-module. Because H1 (F1 S21 ; Zp ) is a torsion group we deduce from the mod-p calculation and the universal coefficient theorem an isomorphism H 3 (F1 S21 ; Zp ) ∼ = Zp . 3 1 ∼ Likewise we find H (S2 ; Zp ) = Zp . Now a restriction-transfer argument shows that the Zp [F2(q−1) ]-module structure on H 3 (S21 ; Zp ) ∼ = Zp is trivial if and only 3 1 if it is trivial on H (F1 S2 ; Zp ). We have already seen that the latter is trivial after mod-p reduction and this implies that it was trivial before.

138 Proof of Theorem 6. modules (3)

Hans-Werner Henn a) The existence of the exact sequence of Zp [[G2 ]]2 2 0 → Z p ↑G → Z p ↑G → Zp → 0 G1 G1 2

2

∼ is an immediate consequence of the isomorphism G2 /G12 ∼ = Z× p /Cp−1 = Zp . (Note that this sequence is essentially the same exact sequence as that in section 2.1.2.) The projective resolution of Zp as Zp [[G12 ]]-module is now constructed by G12 using Proposition 7 as follows. The map Zp ↑F2(q−1) → Zp is just the G12 -linear extension of the identity of Zp (considered as an F2(q−1) -linear map). If N0 is its kernel then we can compute H0 (S21 ; N0 /(p)) from the long exact homology sequence associated to the short exact sequence G1

2 0 → N0 → Zp ↑F2(q−1) → Zp → 0

of Zp [F2(q−1) ]-modules and identify it with H1 (S21 ; Fp ) ∼ = λp−1 ⊗Zp Fp . Because λp−1 is projective as Zp [F2(q−1) ]-module (the order of F2(q−1) is prime to p!) we can lift the resulting map from λp−1 → H0 (S21 ; N0 /(p)) to an Zp [F2(q−1) ]-linear map λp−1 → N0 . G1

2 → N0 . By Let N1 be the kernel of the Zp [[G12 ]]-linear extension λp−1 ↑F2(q−1) a Nakayama Lemma type argument with H0 we see that this extension is onto (cf. Lemma 4.3 of [GHMR1]) and then we find an isomorphism of Zp [F2(q−1) ]modules H0 (S21 ; N1 /(p)) ∼ = H2 (S21 ; Fp ).

By iterating the procedure we construct a Zp [[G12 ]]-linear surjection G1

2 → N1 λp−1 ↑F2(q−1)

whose kernel N2 satisfies H0 (S21 ; N2 /(p)) ∼ = Fp as Zp [F2(q−1) ]= H3 (S21 ; Fp ) ∼ 1 module and Hi (S2 ; N2 /(p)) = 0 if i > 0. Finally by using the Nakayama Lemma once more we see that the G12 -linear G12 extension Zp ↑F2(q−1) → N2 of the F2(q−1) -linear projection Zp → H0 (S21 ; N2 /(p)) is an isomorphism. By splicing together the short exact sequences we obtain the projective resolution of Theorem 6a. b) We take the projective resolution obtained in (a) and induce it up to get 2 one of the Zp [[G12 ]]-module Zp ↑G . Then we use the exact sequence (3) and G12 construct the obvious double complex whose columns are these induced projective resolutions. The resulting double complex gives the projective resolution of (b). Proof of Theorem 5. a) For the fibration we can refer to Proposition 7.1 in [DH2]. In fact, the fibration “realizes” (in the same sense as before) the short

On finite resolutions of K(n)-local spheres

139 hG1

exact sequence of Zp [[G2 ]]-modules in Theorem 6a. The En -resolution of En 2 is now obtained as in the proof of Theorem 4 as the realization of the projective 2 resolution of Zp ↑G which is induced from the one given in Theorem 6a. To G12 finish the proof of (a) it remains to identify the spectrum which corresponds G12 to the module λp−1 ↑F2(q−1) . For this we refer to the appendix. b) The En -resolution of part (b) is nothing but the realization of the resolution of Theorem 6b obtained via (the proof of) Theorem 4. Remarks a) Ravenel [Ra1, Theorem 6.3.31] resp. Yamaguchi [Y] have also studied H ∗ (S3 , Fp ) for p ≥ 5 resp. p ≥ 3. In principle this can be used to obtain an explicit resolution for LK(3) S 0 if p ≥ 3. b) No other explicit resolutions seem to be known if n 6≡ 0 mod p − 1. 3. The case n ≡ 0 mod p − 1 3.1 Explicit examples III: the case n = 1 and p = 2. This case is again ∼ well understood. The isomorphism G1 = S1 ∼ = Z× 2 = C2 × Z2 allows, as before, to form the homotopy fixed points in two stages and we obtain the following fibration (cf. [HMS]) in which ψ 3 is again given by the appropriate Adams operation: (4)

ψ 3 −id

2 2 LK(1) S 0 → KZhC −→ KZhC . 2 2

2 The homotopy fixed points KZhC can be identified with 2-adic real K-theory 2 KOZ2 . Note, however, that this is not an example of Theorem 4. In fact, an En -resolution of finite length cannot exist in this case because the cohomological dimension cd2 (S1 ) is infinite. Nevertheless this is a very good substitute of such a resolution.

3.2 The general problem. The natural question arises whether there are generalizations of the fibre sequence (4) for higher n. What should they look like? In other words, can we explain the appearance of KOZ2 in (4) so that it fits into a more general framework? A good point of view is again provided by homological algebra as follows: the fibre sequence (4) is a homotopy theoretic analogue of the exact sequence of Z2 [[G1 ]]-modules G1 1 0 → Z 2 ↑G C2 → Z 2 ↑C2 → Z 2 → 0 .

This is not a free (neither a projective) resolution of the trivial module Z2 but rather a resolution by permutation modules.

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This suggests that we should look for a resolution of the trivial Gn -module Zp in terms of something like permutation modules on finite subgroups and try to realize those by appropriate homotopy fixed point spectra where realization is again in the sense of the isomorphism of Morava modules of 1.3.2 which gives us for each finite subgroup F of Gn a canonical isomorphism En∗ En hF ∼ = Homcts (Zp ↑FGn , En∗ ). This leads to the following questions. Questions. (Q1) Are there resolutions of finite length of the trivial Gn -module Zp by (finite) direct sums of permutation modules on finite (p)-subgroups of Gn ? (Q2) Can these algebraic resolutions be realized by “resolutions” of spectra? What do we mean by a “resolution” of a spectrum? (Q3) If the answers to (Q1) and (Q2) are yes, how unique are these resolutions? Remark The group Sn resp. Gn is of finite virtual cohomological p-dimension. If G is a discrete group which is virtually of finite cohomological dimension, then a permutation resolution of finite length can be obtained from the cellular chains of a contractible finite dimensional G-CW -complex on which G acts with finite stabilizers. Such spaces always exist and hence such resolutions always exist [Se]. In case G is profinite and vcdp (G) < ∞ then some sort of positive answer to (a) may be given by algebraically mimicking Serre’s construction: one considers a finite index open subgroup H with cdp (H) < ∞, then one takes a projective resolution of finite length of the trivial Zp [[H]]module Zp and finally one obtains the desired resolution by tensor induction from H to G. This ensures existence, but the drawback of this construction is that it tends to be not very efficient. In particular, the length of the resolution would be much larger than necessary (the vcd?) and the modules in the resolution would not be finitely generated. In the case of the stabilizer groups we will describe a construction which gives better qualitative (and in favorable cases quantitative) information on the form of such resolutions. However, before we turn to the general theory we will survey recent joint work with Goerss and Mahowald [GHM] resp. with Goerss, Mahowald and Rezk [GHMR1] in which we construct explicit and efficient resolutions in the case n = 2 and p = 3. 3.3 Explicit examples IV: the case n = 2 and p = 3. Throughout this section we assume n = 2 and p = 3. In this case there are two different explicit resolutions which we will call duality resolution resp. centralizer resolution. We will see in sections 3.5 and 3.6 below that the centralizer

On finite resolutions of K(n)-local spheres

141

resolution can be generalized both algebraically and homotopy theoretically. The justification for its name will become clear in section 3.5. The duality resolution has the advantage of being more efficient and having an intriguing symmetry. In the sequel we describe both resolutions. For more details in the case of the duality resolution we refer to [GHMR1]. The centralizer resolution is implicit in [GHM] and is a special case of Theorem 26 below. 3.3.1. In the following results we use the phrase “resolution of spectra” in the following weak sense: we call a sequence of spectra ∗ → X−1 → X0 → X1 → . . . a resolution of X−1 if the composite of any two consecutive maps is nullhomotopic and if each of the maps Xi → Xi+1 , i ≥ 0 can be factored as Xi → Ci → Xi+1 such that Ci−1 → Xi → Ci is a cofibration for every i ≥ 0 (with C−1 := X−1 ). We say that the resolution is of length n if Cn ' Xn and Xi ' ∗ if i > n. 3.3.2. Before we can describe our resolutions we need to introduce certain finite subgroups of G12 and some of their representations (cf. [GHMR1] for more details). The group G12 contains a group G24 of order 24 which is isomorphic to the semidirect product C3 oQ8 such that the quaternion group Q8 acts non-trivially on C3 . If ω is a primitive 8-th root of unity in WFq then C3 is generated by the element s = − 12 (1 + ωS) while Q8 is generated by ω 2 and ωS. G12 contains also the semidirect product C8 o Gal(F9 /F3 ) generated by ω and S. This group which was denoted F2(32 −1) in section 2.3 can be identified with the semidihedral group SD16 of order 16. It has a unique non-trivial one dimensional representation χ over Z3 which is trivial on the subgroup < ω 2 , ωS >. (χ agrees with the representation λ4,− that is discussed in the appendix.) 3.3.3. The duality resolution. The following results are proved in [GHMR1]. Theorem 8 (Algebraic duality resolution). There exists a short exact sequence of Z3 [[G2 ]]-modules 2 2 0 → Z 3 ↑G → Z 3 ↑G → Z3 → 0 , G1 G1 2

an exact complex of

Z3 [[G12 ]]

G1

2

- modules G1

G1

G1

2 2 0 → Z3 ↑G224 → χ ↑SD → χ ↑SD → Z3 ↑G224 → Z3 → 0 , 16 16

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and an exact complex of Z3 [[G2 ]] - modules G1

G2 G2 2 2 0 → Z 3 ↑G G24 →χ ↑SD16 ⊕ Z3 ↑G24 → (χ ⊕ χ) ↑SD16 → G1

G2 2 2 →Z3 ↑G G24 ⊕ χ ↑SD16 → Z3 ↑G24 → Z3 → 0 .

Theorem 9 (Homotopy theoretic duality resolution). There exists a fibration hG12

LK(2) S 0 → E2

hG12

−→ E2

and resolutions of spectra of length 3 resp. 4 hG12

∗ → E2

→ E2hG24 → Σ8 E2hSD16 → Σ40 E2hSD16 → Σ48 E2hG24 → ∗

resp. ∗ → LK(2) S 0 → E2hG24 → E2hG24 ∨ Σ8 E2hSD16 → Σ8 E2hSD16 ∨ Σ40 E2hSD16 → → Σ40 E2hSD16 ∨ Σ48 E2hG24 → Σ48 E2hG24 → ∗ . Remarks a) The spectrum E2hG24 is a version of the Hopkins-Miller higher real K-theory spectrum EO2 at p = 3. Its coefficients are described in detail in [GHMR1]. The coefficients of E2hSD16 are given by the completion of Z3 [v1 ][v2±1 ] with respect to the ideal generated by v14 v2−1 (cf. the discussion in the appendix). b) The appearance of the 8-fold suspension is forced by the character χ, while the 40-fold suspension is there for purely aesthetic reasons (note that Σ40 E2hSD16 ' Σ8 E2hSD16 by periodicity), so that the homotopy theoretic resolution displays a similar kind of duality as the algebraic resolution. The appearance of the 48-fold suspension, however, is a genuinely homotopy theoretic phenomenon and cannot be avoided. c) The resolution appears to be self dual but there is no satisfactory explanation of this duality yet. And it is not at all clear whether there are generalizations, say to the case n = p − 1, p > 3, and what they could look like. 3.3.4. The centralizer resolution. For describing the centralizer resolution we will make use of the Z3 [SD16 ]-module λ2 (cf. 2.3 and appendix) and the unique non-trivial one dimensional representation χ e of G24 over Z3 which is trivial on s and on ωS. The following results are implicit in [GHM]. We will give a proof in section 3.6.6.

Theorem 10 (Algebraic centralizer resolution). There exists a short exact sequence of Z3 [[G2 ]]-modules 2 2 → Z3 → 0 , 0 → Z 3 ↑G → Z 3 ↑G G1 G1 2

2

143

On finite resolutions of K(n)-local spheres an exact complex of Z3 [[G12 ]] - modules G1

G1

G1

G1

G1

2 2 2 0 → Z3 ↑SD → λ2 ↑SD → χ ↑SD ⊕χ e ↑G224 → Z3 ↑G224 → Z3 → 0 16 16 16

and an exact complex of Z3 [[G2 ]] - modules G1

G1

G1

G1

2 2 2 0 → Z3 ↑SD → (Z3 ⊕ λ2 ) ↑SD → (λ2 ⊕ χ) ↑SD ⊕χ e ↑G224 → 16 16 16

G1

G1

G1

2 → χ ↑SD ⊕ (e χ ⊕ Z3 ) ↑G224 → Z3 ↑G224 → Z3 → 0 . 16

Theorem 11 (Homotopy theoretic centralizer resolution). There exists a fibration hG1 hG1 LK(2) S 0 → E2 2 −→ E2 2 and resolutions of spectra of length 3 resp. 4 hG12

∗ → E2

→ E2hG24 → Σ8 E2hSD16 ∨ Σ36 E2hG24 → → Σ4 E2hSD16 ∨ Σ12 E2hSD16 → E2hSD16 → ∗

resp. ∗ → LK(2) S 0 → E2hG24 → Σ8 E2hSD16 ∨ Σ36 E2hG24 ∨ E2hG24 → → Σ4 E2hSD16 ∨ Σ12 E2hSD16 ∨ Σ8 E2hSD16 ∨ Σ36 E2hG24 → → E2hSD16 ∨ Σ4 E2hSD16 ∨ Σ12 E2hSD16 → E2hSD16 → ∗ . 3.4 Work in progress (the case n = p = 2). In this case we do have an algebraic duality resolution but we have not yet completely succeeded in realizing it. However, there is an algebraic centralizer resolution which can be realized. To describe the homotopy theoretic resolutions we note that, for p = 2, S2 contains a subgroup of order 24, also denoted G24 (but not isomorphic to the group with the same label that we used in the last section). For p = 2 the group G24 is isomorphic to the semidirect product Q8 o C3 of the quaternion group Q8 with the cyclic group C3 or order 3 which cyclically permutes i, j and k. G24 contains cyclic subgroups of order 2, 4 and of order 6, We fix such subgroups and denote them by C2 resp. C4 resp. C6 . Then we have the following results which, up to Galois extension, give resolutions of LK(2) S 0 at p = 2. Details will appear in [GHMR2]. Theorem 12 (Centralizer resolution). There exists a fibration hS12

E2hS2 −→ E2

hS12

−→ E2

and a resolution of spectra of length 3 hS12

∗ → E2

→ E2hG24 ∨ E2hG24 → E2hC6 ∨ E2hC4 → E2hC2 → E2hC6 → ∗ .

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Theorem 13 (Duality resolution). There exists a fibration hS12

E2hS2 −→ E2

hS12

−→ E2

and a resolution of spectra of length 3 hS12

∗ → E2

→ E2hG24 → E2hC6 → E2hC6 → X3 → ∗ together with an isomorphism of Morava modules E∗ (X3 ) ∼ = E∗ (E2hG24 ).

Remarks a) The spectrum E2hG24 is a version of the higher real K-theory spectrum EO2 at p = 2. We have not been able yet to further identify the spectrum X3 . b) We expect that the resolutions described in this and the previous section will help to better understand the Shimomura-Wang calculation [ShW] of π∗ LK(2) S 0 at p = 3 and that they will be crucial for calculating π∗ LK(2) S 0 at p = 2. 3.5 Permutation resolutions in the case n = k(p − 1) for p odd. In this section we will give a positive answer to question (Q1) and the algebraic part of question (Q3) of section 3.2 above, at least if p is odd. 3.5.1. We start by introducing some relative homological algebra (cf. [EM]) in a form which parallels Miller’s discussion of En -injective spectra and En injective resolutions of spectra (cf. section 2.2.1). Let p be a fixed prime. If G is a profinite group we denote the collection of finite p-subgroups of G by Fp (G), or simply by F (G) or even F if G and p are clear from the context. Throughout this section we will make the following Assumption: G contains only finitely many conjugacy classes of finite psubgroups. We recall that all our modules will be profinite continuous modules for the completed group algebras and that induced modules are formed by using the completed tensor product. A Zp [[G]]-module P will be called F -projective if the canonical Zp [[G]]-linear map M P ↑G F→ P (F )∈F

is a split epimorphism (where the sum is taken over conjugacy classes of finite p-subgroups). It is clear that each Zp [[G]]-module which is induced from a Zp [F ]-module for some F ∈ F is F -projective, and that a Zp [[G]]-module

On finite resolutions of K(n)-local spheres

145

P if and only if P is a retract of some module of the form L is F -projective G (F )∈F MF ↑F where each MF is a Zp [F ]-module.

The class of F –projectives determines in the usual way a class of F -exact sequences: a sequence of Zp [[G]]-modules M 0 → M → M 00 is called F -exact if the composition M 0 → M 00 is trivial and HomZp [[G]] (P, M 0 ) → HomZp [[G]](P, M) → HomZp [[G]](P, M 00 )

is an exact sequence of abelian groups for each F -projective Zp [[G]]-module P . It is obvious that the category of Zp [[G]]-modules has enough F -projectives. Finally an F -resolution of a Zp [[G]]-module M is a sequence of Zp [[G]]modules . . . → P1 → P0 → M → 0 where each Ps is F -projective and each 3-term subsequence is F -exact. Then it is clear that each module M admits an F -resolution and that any homomorphism of Zp [[G]]-modules M → N is covered by a map of F -resolutions which is unique up to chain homotopy. We will be interested in constructing F -resolutions of finite length of the trivial module Zp such that all modules in the resolution are finitely generated. 3.5.2. The construction of our F -resolutions will rely on the following three results. Lemma 14. Suppose G is a profinite group and H is a normal finite psubgroup. a) If P is F (G/H)-projective, then considered as a Zp [[G]]-module via the canonical projection π : G → G/H, P is F (G)-projective. b) If M 0 → M → M 00 is a sequence of Zp [[G/H]]-modules which is F (G/H)exact, then considered as a sequence of Zp [[G]]-modules via the canonical projection π : G → G/H, M 0 → M → M 00 is F (G)-exact. Proof. a) This follows immediately from the following observation. If F is a G/H finite p-subgroup of G/H and M is a Zp [F ]-module then M ↑F , considered as Zp [[G]]-module via π, is isomorphic to M ↑G π −1 F . b) If F is a finite p-subgroup of G and N is a Zp [F ]-module then there is a G/H ∼ natural isomorphism (N ↑G F )⊗Zp [H] Zp = (N ⊗Zp [F ∩H] Zp ) ↑F/F ∩H ) of Zp [[G/H]]modules. This implies that − ⊗Zp [H] Zp sends F (G)-projectives to F (G/H)projectives which in turn implies (b). Lemma 15. Suppose G is a profinite group and K is a closed subgroup of G which contains only a finite number of conjugacy classes of finite subgroups.

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a) If P is F (K)-projective, then P ↑G K is F (G)-projective. b) Assume that sequence of open normal subgroups U n of T there is a decreasing 0 G such that n Un = {1}. If M → M → M 00 is a sequence of Zp [[K]]-modules G 00 G which is F (K)-exact, then M 0 ↑G K → M ↑K → M ↑K is F (G)-exact. Proof. a) This is trivial. b) It is enough to show that for each F ∈ F (G) and each Zp [F ]-module L G 00 G the functor HomZp [F ](L, −) sends the sequence M 0 ↑G K → M ↑K → M ↑K to an exact sequence of abelian groups. This is in fact a consequence of the Mackey decomposition formula which describes the restriction of an induced module. In the case of profinite groups this requires some care so that it seems appropriate to give some details. To simplify notation we let Kn = K/K∩Un , Gn = G/Un and Fn = F/F ∩Un . Note that if F is finite then F = Fn if n is sufficiently large. Now let N be any profinite Zp [[K]]-module. Then we can write N = limi Ni (where i runs through some directed set I, not necessarily a countable sequence) with Ni finite and acted on trivially by K ∩ Uλ(i) for some increasing function λ : I → N. Then we have the classical Mackey decomposition formula for the Zp [Kλ(i) ]-modules Ni (where as usual (−) ↓G F denotes the restriction of a Zp [[G]]-module to a Zp [F ]-module) M G Gλ(i) ∼ gKλ(i) g −1 Fλ(i) g Ni ↑Kλ(i) ↓ ( N ) ↓ = i gKλ(i) g −1 ∩Fλ(i) ↑gKλ(i) g −1 ∩Fλ(i) λ(i) Fλ(i) g∈Fλ(i) \Gλ(i) /Kλ(i)

and by passing to the limit we obtain Gλ(i) Gλ(i) G ∼ G ∼ b N ↑G K ↓F = (Zp [[G]]⊗Zp [[K]] N ) ↓F = limi Ni ↑Kλ(i) ↓Fλ(i) M gKλ(i) g −1 Fλ(i) ∼ (g Ni ) ↓gKλ(i) = limi g −1 ∩Fλ(i) ↑gKλ(i) g −1 ∩Fλ(i) . g∈Fλ(i) \Gλ(i) /Kλ(i)

Now we consider our F (K)-split sequence M 0 → M → M 00 and factor the first homomorphism via the kernel M of M → M 00 as M 0 → M → M. It is enough G G G to show that M 0 ↑G K ↓F → M ↑K ↓F induces a surjection on HomZp [F ] (L, −). First we note that p : M 0 → M → 0 is F (K)-exact. This implies that for any finite p-subgroup H of K there exists an H-linear splitting s : M → M 0 , i.e. there exist increasing functions α : I → I, β : I → I and compatible families of Zp [H]-linear maps pi : (M 0 )α(i) → M i representing p and sα(i) : (M)βα(i) → (M 0 )α(i) representing s such that the composition (M )βα(i) → (M )i is the map in the given system for M . By explicitly choosing conjugations we may assume that we have such a splitting (with the same α and β) for all finite p-subgroups in the conjugacy class of H. And because we assume that there are only finitely

On finite resolutions of K(n)-local spheres

147

many conjugacy classes of finite p-subgroups in K we may assume that the functions α and β work for all H ∈ F (K). 0 The Mackey decomposition formula for (M)βα(i) and Mα(i) gives us therefore, for any choice of double coset representatives in Gλβα(i) resp. Gλα(i) , well defined Zp [F ]-linear maps G

G

G

G

0 ↓ λβα(i) → Mα(i) ↑Kλα(i) ↓ λα(i) . seα(i) : (M )βα(i) ↑Kλβα(i) λβα(i) Fλβα(i) λα(i) Fλα(i)

Now the elementary theory of profinite sets (cf. Lemma 5.6.7 in [RZ]) tells us that the quotient map G → F \G/K admits a continuous section and any such section gives us a compatible choice of double coset representatives in Gn for all n. (We note that it is here that we use the assumption on the existence of the decreasing sequence of subgroups Un .) For any such choice the associated Zp [F ]-linear maps seα(i) are compatible with respect to i so that they patch together and define a Zp [F ]-linear map on the level of inverse limits. This G G G shows that M 0 ↑G K ↓F → M ↑K ↓F is split as a map of Zp [F ]-modules and hence we are done. Lemma 16. Suppose G is a profinite group and M is a Zp [[G]]-module which admits a finite projective resolution and which is projective as a Zp -module. Then M is projective as a Zp [F ]-module for every F ∈ F . Proof. By induction on the length of a finite projective resolution it is enough to show that for a finite p-group F a short exact sequence 0 → M1 → M2 → M3 → 0 of Zp [F ]-modules splits if M1 is projective as a Zp [F ]-module and if the sequence splits as a sequence of Zp -modules. The existence of a Zp -splitting of the inclusion M1 → M2 implies that any Zp [F ]-linear map ϕ from M1 to the coinduced module Hom(Zp [F ], M1 ) can be extended to an Zp [F ]-linear map ϕ e : M2 → Hom(Zp [F ], M1 ). Next, if M1 is projective as a Zp [F ]-module then it is a direct summand in the induced module M1 ↑F{1} , and because F is finite the induced module is isomorphic to the coinduced module. Now we take for ϕ any Zp [F ]-split inclusion of M1 into Hom(Zp [F ], M1 ). Then the composition of ϕ e with a Zp [F ]-linear splitting of ϕ provides the desired splitting. 3.5.3. Here is the promised answer to question (Q1) and the algebraic part of (Q3). Proposition 17. Suppose G is a virtually profinite p-group and S is a closed normal subgroup which is a profinite p-group. Furthermore assume that (1) H ∗ (S; Fp ) is a finitely generated Fp -algebra, (2) all finite p-subgroups of S are cyclic and there is a bound on their order,

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(3) the trivial module Zp admits a projective resolution of finite type and finite length over the p-completed group algebra of G/S and all its closed subgroups, T (4) there is a sequence of open subgroups Un of G such that n Un = {1}. Then the trivial Zp [[G]]-module Zp admits an F -resolution of finite length.

Furthermore, all F -projectives in this resolution can be chosen to be summands in finite direct sums of modules of the form Zp ↑G F with F ∈ F .

Remarks a) Assumption (1) implies by a recent result of Minh and Symonds [MS] that S has only a finite number of conjugacy classes of finite p-subgroups. In particular the bound in assumption (2) is already a consequence of assumption (1). Furthermore recent work of Symonds suggests that finite length F -resolutions should exist even if we skip condition (2). b) Assumption (3) implies that G/S has no elements of order p, in other words every finite p-subgroup of G is already contained in S. c) It is well-known that H ∗ (Sn ; Fp ) is a finitely generated algebra. As already mentioned in 1.2.4 and 1.2.5, Sn has finite p-subgroups if and only if n is divisible by p − 1. Furthermore, if p is odd, or p = 2 and n is odd, then all finite p-subgroups are cyclic and their conjugacy class is unique. The p-Sylow subgroup Sn of Sn is normal in Gn of index (pn − 1)n which is of order prime to p if n = k(p − 1) with k 6≡ 0 mod p. Therefore, for such n the assumptions of the proposition hold with G = Gn and S = Sn . If n = k(p − 1) with k divisible by p then we replace S Gn as in remark 2 of section 2.2 by Gn (K) := Sn o Gal(K/Fp ) where K := r Fqpr . This has the effect that all p-torsion elements of Gn (K) are already contained in Sn resp. its normal Sylow subgroup Sn . In this case it is clear that the assumptions hold with G = Gn (K) and S = Sn . In fact, G/S has a finite normal subgroup of order prime to p with quotient Zp . The somewhat artificially looking assumptions on the pair (G, S) in Proposition 17 have been introduced in order to cover this case. d) If n is even and p = 2 then Sn has finite 2-subgroups which are not cyclic. Thus for n even and p = 2 the proposition does not give any information. Nevertheless, if n = p = 2 the algebraic centralizer resolution mentioned in section 3.4 gives an explicit finite length F -resolution of the trivial Z2 [[S2 ]]module Z2 . Proof. The proof will be by induction over the order of the largest finite p-subgroup of G. We distinguish the following cases. Case 1: G has no non-trivial finite p-subgroups.

On finite resolutions of K(n)-local spheres

149

In this case Quillen’s F -isomorphism theorem [Q] implies that H ∗ (S; Fp ) is a finite Fp -algebra. Because S is a profinite p-group this implies cdp (S) < ∞ and then assumption (3) implies cdp (G) < ∞. Finally, (3) and the spectral sequence of the group extension 1 → S → G → G/S → 1 show that H i (G, M) is a finite group for every finite discrete continuous Zp [[G]]-module M and this ensures the existence of a projective resolution of Zp of finite length in which all projectives are finitely generated (cf. Proposition 4.2.3 in [SyW]). Case 2: G has a normal finite p-subgroup F . By [MS] H ∗ (S/F, Fp ) is still a finitely generated Fp -algebra and hence G/F and S/F still satisfy our assumptions. Therefore, by induction hypothesis, the trivial G/F -module Zp admits an F (G/F )-resolution of finite length such that all F (G/F )-projectives are of the required form. Then Lemma 14 implies that the very same resolution is also an F (G)-resolution of finite length for the trivial G-module Zp , and all F (G)-projectives are as required. Case 3: The general case. We may suppose that G does not contain any finite normal p-subgroups. In this case we consider the short exact sequence of Zp [[G]]-modules M f ε (5) 0 → K −→ Zp ↑ G NG (E) −→ Zp → 0 (E)

where the sum is over conjugacy classes of non-trivial elementary abelian psubgroups of G (which by our assumption are all of order p), ε is the canonical augmentation and K is its kernel. First we note that the exact sequence (5) is F -exact. In fact, if F is a finite p-subgroup of G then it has a non-trivial central element of order p and F is contained in the normalizer of the elementary abelian p-subgroup E 0 generated by this element of order p. This means, that the action of F on G/NG (E 0 ) has a fixed point. Such a fixed point determines a Zp [F ]-linear splitting of the surjection in (5). In other words, the exact sequence splits upon restriction to every F ∈ F and this is equivalent to saying that the sequence is F -exact. L G It will therefore be enough to construct F -resolutions for (E) Zp ↑NG (E) and for K where the F -projectives have the required form. In fact, if we have two such resolutions we can lift f to a map of resolutions and then the resulting double complex will be an F -resolution for Zp with all F -projectives as required. The pair (NG (E), S ∩ NG (E)) satisfies the same assumptions as (G, S): assumptions (2), (3) and (4) are obvious. For (1) we note that S ∩ NG (E)) = NS (E) agrees with the centralizer CS (E) because E is of rank 1. In fact, because the p-rank of E is always 1, the p-group NS (E)/CS (E) injects into

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Aut(E) ∼ = Z/(p − 1) and hence it is trivial. Now CS (E) satisfies (1) because it is given by a component of Lannes’ T -functor (by Theorem 2.6 of [He]) and Lannes’ T -functor takes unstable finitely generated Fp -algebras to unstable finitely generated Fp -algebras ([DW, Theorem 1.4]). Furthermore, by [MS] the groups NS (E) and hence also NG (E) have only a finite number of conjugacy classes of finite p-subgroups. Therefore we can apply case 2 and deduce that for each E ∈ F (G) the trivial Zp [[NG (E)]]-module Zp admits an F (NG (E))resolution of finite length. Inducing this resolution gives, by Lemma 15, an F (G)-resolution of Zp ↑G NG (E) with all F -projectives as required. Finally consider K. The group G/S acts on the finite set of conjugacy classes of elementary abelian p-subgroups of S and the stabilizer of such a subgroup E is the image of NG (E) in G/S. In particular this image is of finite index in G/S and this implies that for each E ⊂ G we get an isomorphism of Zp [[S]]-modules M ∼ Zp ↑SNS (E 0 ) Zp ↑ G NG (E) = (E 0 )<S

where the direct sum is taken over conjugacy classes of E 0 ’s in S which become conjugate to E in G. Furthermore, as before the normalizer NS (E 0 ) can be identified with the centralizer CS (E 0 ). Therefore we end up with an isomorphism of Zp [[S]]-modules M

(E) 1. 3.6.1. Let us start by looking at the F -resolution of the trivial Zp [[Gn ]]-module constructed in the previous section. By 1.2.4 there is a unique conjugacy class of subgroups of Sn of order p. We pick a representative and denote it by E and its normalizer in Gn simply by N . So the exact sequence of the proof of Proposition 17 takes the following form (6)

f

n 0 → K −→ Zp ↑G N −→ Zp → 0 .

ε

By 1.3.2 this sequence can be realized by the cofibration LK(n) S 0 ' EnhGn −→ EnhN → C ι

where ι is induced by the inclusion of N into Gn and C is the cofibre of ι. Of course, as before realizing means that the map Homcts (ε, (En )∗ ) agrees with the map ι∗ Homcts (Zp , (En )∗ ) ∼ = Homcts (Zp ↑Gn , (En )∗ ) = (En )∗ LK(n) S 0 −→ (En )∗ E hN ∼ 2

N

induced by ι in (En )∗ . This map is injective and therefore E∗ C can be identified with Homcts (K, (En )∗ ). This suggests that we should start by constructing resolutions for EnhN and C. 3.6.2. We begin with C. Lemma 18. C admits an En -resolution of finite length in which each spectrum is a summand of a finite wedge of En ’s. Proof. We know from section 3.5 that K has a projective Zp [[Gn ]]-resolution P• : 0 → P d → . . . P 0 → K of finite length d in which each Ps is finitely generated. Let Q• be the following exact complex of Zp [[Gn ]]-modules n Q• : 0 → P d → . . . → P 0 → Z p ↑G N → Zp → 0

obtained by splicing the resolution of K and the exact sequence (6). By Proposition 3 there is a sequence of maps between spectra of the following form Y• : ∗ → LK(n) S 0 ' EnhGn −→ EnhN → X0 → . . . → Xd → ∗ ι

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where each Xi is a summand in a finite wedge of En ’s such that the complex Homcts (Q• , (En )∗ ) is isomorphic, as a complex of Morava modules to the complex (En )∗ (Y• ). Again by Proposition 3 the composite LK(n) S 0 → EnhN → X0 is null and we obtain a sequence of maps between spectra X• : ∗ → C → X0 → . . . → Xd → ∗ such that Homcts (P• , (En )∗ ) is isomorphic, as a complex of Morava modules to the complex (En )∗ X• . We claim that X• is, in fact, an En -resolution of C in the sense of 2.2.1. In fact, it is clear that all Xs for s ≥ 0 are En -injective and En -exactness of Y• is seen as in the proof of Theorem 4. En -exactness of X• follows immediately from that of Y• . 3.6.3. The case of EnhN is substantially more difficult and unlike in the case of C the resolution will not be an En -resolution although it will still have some good properties. In this subsection we will make the algebraic resolution of n Zp ↑ G N explicit. 3.6.3.1. First we investigate the structure of the group N = NGn (E). We choose a primitive (pn − 1)-st root of unity ω ∈ W× Fq ⊂ Sn and note that the element X := ω

p−1 2

S ∈ Dn satisfies X n = −p.

Lemma 19. The subfield Qp (X) of Dn is isomorphic to the cyclotomic extension Qp (ζp ) generated by a primitive p-th root of unity ζp and this isomorphism restricts to an isomorphism Zp [X]/(X n + p) ∼ = Zp [ζp ]. Proof (outline). In fact, this is a straightforward consequence of local class field theory but for the convenience of the reader we outline an elementary and direct proof. R := Zp [X]/(X n +p) is a local ring with maximal ideal generated by X. The powers of the maximal ideal induce, as in section 1.2.3, a decreasing complete filtration Fi with i = nk , k = 1, 2, . . ., of the group of units of R via Fi := {g ∈ R× |g ≡ 1 mod X in } and with filtration quotients gri := Fi /Fi+ 1 which are all canonically isomorn phic to Fp . Furthermore the p-th power map induces, as in the case of the groups Sn , maps gri → grϕ(i) which via these isomorphisms are given by the identity if i 6= n1 and by a 7→ ap − a if i = n1 (with notation as in section 1.2.3). The completeness of the filtration shows now that R contains a p-th root of unity, in fact for any a ∈ R with ap ≡ a mod (X) there is a p-th root of unity of the form 1 + aX mod(X 2 ). This shows that Qp (X) contains an isomorphic copy of the field Qp (ζp ) and because both are extensions of degree n = p − 1 of Q they have to be isomorphic. The final statement of the Lemma is now obvious.

On finite resolutions of K(n)-local spheres

153

We choose a primitive p-th root of unity in Qp (X) and we still denote it by ζp thus identifying Qp (X) with Qp (ζp ). We choose our subgroup E ⊂ Sn to be generated by ζp . If n = k(p − 1) then (cf. 1.2.4) the centralizer CDn (E) is a division algebra which is central over Qp (ζp ) of dimension k 2 . In particular, if n = p − 1, then CDn (E) = Qp (ζp ), and CD×n (E) = Qp (ζp )× . We need to know the structure of the group of units in Qp (ζp ) = Qp (X). This is well known in algebraic number theory. For the convenience of the reader we give some details. Consider the filtration by subgroups (7)

0 ⊂ F 2 ⊂ F 1 ⊂ Zp [ζp ]× ⊂ Qp (ζp )× n

n

where F 2 and F 1 are the filtration subgroups of Zp [ζp ]× introduced above. The n n valuation v, normalized by v(p) = 1, is a split epimorphism Qp (ζp )× → n1 Z ∼ =Z with kernel Zp [ζp ]× . Next, the description of the p-th power map given above shows that F 2 is a free Zp -module of rank n and the torsion-subgroup of Zp [ζp ]× n identifies with Zp [ζp ]× /F 2 ∼ = E × Z/n. In particular, we see that the filtration n (7) is split and we obtain an isomorphism (70 ) Qp (ζp )× ∼ = Z × Z/n × Z/p × Zn . p

Furthermore, the discussion above shows that we can choose generators as follows: • The element X satisfies v(X) = n1 and can therefore be chosen as a generator of the factor Z. pn −1 • The subgroup Z/n is the subgroup generated by ω p−1 . • Z/p ∼ = E is the subgroup generated by ζp . • The elements ηj := 1+X j ∈ Zp [ζp ]× ∼ = Zp [X]× , j = 2, . . . , n+1, qualify as a system of topological generators of Znp . We will need to understand the action of the Galois group Gal(Qp (ζp )/Qp ) on Zp [ζp ]× . It is clear that the three factors F 2 , E and Z/n are invariant with n respect to the action of the Galois group and that the action on Z/n ⊂ Z× p is trivial. Furthermore, the Galois group can be canonically identified with the group Aut(E) of automorphisms of E, in other words the action on E is the tautological action. To understand the action on F 2 we observe that the Galois automorphisms n can be realized by the conjugation action (in Dn ) of the quotient of the cyclic pn −1 (p−1)2

group generated by τ := ω elementary calculation shows

by the subgroup generated by τ n . Then an

τ∗ (ηj ) = 1 + ω

j(1−pn ) p−1

Xj ,

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i.e. ηj is an eigenvector in gr j with eigenvalue ω eigenvector with eigenvalue ω

n j(1−pn ) p−1

j(1−pn ) p−1

. To obtain a genuine

in Zp [ζp ]× we note that

n −1) 1 X ij(pp−1 τi ω n i=1

n

ej :=

is an idempotent in Zp [Z/n] which satisfies τ ej = ω by ηj0 := (ej )∗ ηj

j(1−pn ) p−1

ej . If we replace ηj

then the elements ηj0 , j = 2, . . . , n + 1, generate one-dimensional characters, say χ(j), and as a Galois-module F 2 is isomorphic to the direct sum of all the n different one-dimensional characters of Aut(E) over Zp . We note that if we rename χ(n + 1) as χ(1) then we get χ(j) = χ(1)⊗j for j = 2, . . . , n + 1. In the sequel we will interpret χ(j), for any integer j ∈ Z, as the tensor product χ(1)⊗j . After these preparations we will now describe NGn (E). By abuse of notation n ∼ × we continue to denote the image of elements of D× n in Gn = Dn /hS i by the same name. Proposition 20. Let n = p − 1 and p be odd. a) There is an exact sequence 1 → CGn (E) → NGn (E) → Aut(E) → 1 . b) There is an isomorphism CGn (E) ∼ = H × E × Znp where H ∼ = Z/2n × Z/ n2 is generated by X and τ n X 2 , E is generated by ζp , and the elements ηj = 1 + X j , j = 2, . . . , n + 1, form a system of topological generators of Znp . c) The action of Aut(E) leaves H, E and L Znp invariant: the action of Aut(E) n on E is the tautological action while Znp ∼ = j=1 χ(j) and χ(j) is generated by ηj 0 for j = 2, . . . , n and by ηn+1 0 for j = 1. d) The subgroup generated by ζp , H and τ is a finite subgroup F of N of order pn3 which contains H × E as a normal subgroup with quotient Aut(E). Proof. a) We have seen above that there is an exact sequence 1 → CD×n (E) → ND×n (E) → Aut(E) → 1

On finite resolutions of K(n)-local spheres

155

pn −1

where ω (p−1)2 projects to a generator of Aut(E). In particular all of the automorphisms of the group E are realized by conjugation in D× n and hence also × n ∼ in the central quotient Gn = Dn /hS i. This shows (a). b) We claim that the epimorphism from D× n to Gn induces an epimorphism from CD×n (E) to CGn (E) with kernel the central subgroup generated by S n . In fact, if we lift an element g ∈ CGn (E) to an element ge ∈ D× eζp ge−1 = ζp z n then g n with z ∈ hS i. But then we obtain 1 = geζpp ge−1 = (e gζp ge−1 )p = (ζp z)p = z p

and hence z = 1, in other words ge ∈ CD×n (E). Part (b) follows now from pn −1

the observation that S n = p = ω 2 X n , i.e. in terms of the decomposition (7’) of CD×n (E), the element S n has components (n, n2 ) in Z × Z/n and trivial components in Z/p × Znp . (c) and (d) are clear from the discussion above.

Remark The subgroup H is intrinsically defined as the subgroup of the abelian profinite group CGn (E) generated by elements of finite order prime to p. As such it is necessarily invariant with respect to the action of Aut(E). The splitting H ∼ = Z/2n × Z/ n2 , however, is visibly not invariant with respect to this action. 3.6.3.2. In the next step we will construct an explicit F (N )-resolution of the trivial Zp [[N ]]-module Zp . If we set N := N/(H × E) then we have N = Znp o Z/n where the semidirect product is taken with respect to the Z/nmodule structure on Znp given by the direct sum of the n different characters of Z/n over Zp . Proposition 21. a) The trivial Zp [[N ]]-module Zp admits a projective resolution P• : 0 → Pn → . . . → P1 → P0 → P−1 = Zp with Pr ∼ =

M

χ(i1 + . . . + ir ) ↑N Z/n

(i1 ,...,ir )

and where the direct sum is taken over all sequences (i1 , . . . , ir ) with n − 1 ≥ i1 > i2 > . . . > ir ≥ 0. (For r = 0 there is a unique summand 1 ↑N Z/n corresponding to the empty sequence.) b) Considered as a complex of Zp [[N ]]-modules, via the projection N → N , this complex is an F (N )-resolution in which all modules are summands in a finite direct sum of modules Zp ↑N E.

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Proof. a) We start by observing that H

∗

(Znp ; Zp )

∼ = Λ(

n−1 M

χ(i))

i=0

as a module over Zp [Z/n]. The resolution is now constructed step by step as in the proof of Theorem 6. The identity of Zp considered as a map of Zp [Z/n]-modules extends to a Zp [[N ]]-linear homomorphism P0 := Zp ↑N Z/n → Zp . Let N0 be its kernel. Then n we can compute H∗ (Zp , N0 /(p)) and in particular we find H0 (Znp ; N0 /(p))

∼ =

n−1 M

χ(i) ⊗Zp Fp ,

i=0

Ln−1 as a module over Zp [Z/n]. The Zp [Z/n]-module i=0 χ(i) is projective and Ln−1 n hence we can lift the resulting homomorphism i=0 χ(i) → H0 (Zp , N0 /(p)) to N0 and by Nakayama’s Lemma the Zp [[N ]]-linear extension P1 :=

n−1 M

χ(i) ↑N Z/n → N0

i=0

is onto. Then we repeat the game with the kernel N1 of this epimorphism, and so on. When we finally arrive at Nn−1 we see that H0 (Znp , Nn−1 /(p))

∼ = χ(

n−1 X

i) ⊗Zp Fp , Hi (Znp , Nn−1 /(p)) = 0 if i > 0 .

i=0

Pn−1 Then we can construct in the same manner a map Pn := χ( i=0 i) ↑N Z/n → Nn−1 which by Nakayama’s lemma is now even an isomorphism. b) This is an immediate consequence of Lemma 14.

3.6.4. Now we turn towards the problem of realizing the resolution P• conn structed in Proposition 21b, or rather the induced resolution P• ↑G N of the Gn induced module Zp ↑N , by a sequence of maps between spectra. Let F be the subgroup of N of order pn3 described in Proposition 20. The characters χ(i) of the last section can be considered as characters of F via the canonical projection F → F/(H × E) = Aut(E) ∼ = Z/n. We will also need to consider the group F1 := F ∩ Sn which is of order pn2 and is generated by ζp and τ , and the group F2 which is cyclic of order pn and is generated by ζp and τ n.

157

On finite resolutions of K(n)-local spheres Lemma 22. For any i ∈ Z there is an isomorphism of Morava modules (En )∗ (Σ2pni E hF ) ∼ = Homcts (χ(i) ↑Gn , (En )∗ ) . n

F

Proof. For i = 0 this is nothing but 1.3.2. More generally, for every k ∈ Z there is an isomorphism of Morava-modules k n (En )∗ (Σk EnhF ) → Homcts (Zp ↑G F , (En )∗ (S )) .

We will show that there is an invertible element ∆(i) in (En )∗ of degree 2pni on which F acts via χ(i). Then we will get the desired isomorphism by composing with the isomorphism of Morava modules n Homcts (Zp ↑FGn , (En )∗ (S 2pni )) → Homcts (χ(i) ↑G F , (En )∗ )

given by ϕ 7→ g 7→ σ −1 (ϕ(g))g∗ (∆(i))

∼ =

where σ is the suspension isomorphism E∗ = E∗ (S 0 ) −→ E∗+2pni (S 2pni ). p−1

We recall that F is generated by τ , ζp and X = ω 2 S. We need some information about the action of these elements on (En )∗ . For this we recall that the action of an element g ∈ Sn is determined as follows (cf. [DH1]): if we lift g to a power series ge(x) ∈ (En )0 [[x]] then there is a unique continuous ring homomorphism g∗ : (En )0 → (En )0 and a unique ∗ -isomorphism h ∈ fn ) to the formal group law H defined (En )0 [[x]] from the formal group law g∗ (Γ fn (e by H(x, y) = ge−1 Γ g(x), ge(y)) The action of g on u is then given by g∗ (u) = 0 0 ge (0)h (0)u.

In particular, if g(x) = ax with a ∈ F× uller lift of a will, q and if the Teichm¨ by abuse of notation, still be denoted by a then we can take as lift ge(x) = ax and the [p]-series of the formal group law H satisfies [p]H (x) = a−1 ([p]Γe n (ax)) = a−1 ([p]Γe n (ax))

= a−1 p(ax) +Γe n u1 (ax)p +Γe n . . . +Γe n un−1 (ax)p = px +H u1 ap−1 xp +H . . . +H un−1 ap

n−1 −1

xp

n−1

n−1

+Γe n (ax)p n

+H xp .

n

This shows that the ∗-isomorphism h is the identity, i.e. h(x) = x, and that g∗ is given by (8)

i

g∗ (ui ) = ap −1 ui , g∗ (u) = au .

In the case of τ we have a = ω (9)

pn −1 (p−1)2

so that

τ∗ (u) = ω

pn −1 (p−1)2

u.

The action of ζp is more difficult. However, we only need to know that ζp acts trivially modulo the maximal ideal m = (p, u1 , . . . , un−1 ) ⊂ (En )∗ , and in

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particular that (10)

ζp ∗ (u) ≡ u mod m .

In fact, this holds for any element g in the p-Sylow subgroup Sn of Sn . en is already defined over Zp [[u1 , . . . , un−1 ]] and The universal deformation Γ therefore the subgroup of Gn given by Galois automorphisms of Fq acts trivially on the ui and on u. Together with (8) this implies (11)

X∗ (u) = ω

p−1 2

u.

The action of τ and of ζp on (En )∗ is WFq -linear while the action of X is only Zp -linear and satisfies X∗ (wx) = w σ X∗ (x) if w ∈ WFq and x ∈ (En )∗ . Now consider the element ∆0 :=

Y

g∗ (u) .

g∈F2

This is clearly fixed by the subgroup F2 . Furthermore by (9) and (10) we have n n Y Y i(pn −1) p(p−1) pn −1 τ∗in (up ) = ω p p−1 up = ω p 2 p−1 upn = −upn mod m ∆0 ≡ i=1

i=1

so that ∆ is an invertible element of degree −2pn. Now F2 is normal in F and the quotient F/F2 is isomorphic to Z/n × Z/n with generators τ and X. This quotient acts on the F2 -fixed points and by (11) we find for any λ ∈ WFq Y Y p−1 p−1 X∗ (λ∆0 ) = λσ X∗ (∆0 ) = λσ g∗ X∗ (u) = λσ g∗ (ω 2 u) = λσ ω 2 pn ∆0 . 0

g∈F2

In particular we get X∗ (ω

− pn 2

g∈F2

∆0 ) = ω

− pn 2

∆0 , and thus pn

∆00 := ω − 2 ∆0 is still an invertible element in (En )−2pn which is fixed by τ n , ζp and X, and which satisfies pn pn Y τ∗ (∆00 ) = ω − 2 τ∗ (∆0 ) = ω − 2 g∗ τ∗ (u) pn

= ω− 2

Y

g∈F2

g∗ (ω

pn −1 (p−1)2

pn

u) = ω − 2 ω

pn −1 p p−1

pn −1

∆0 = ω (p−1) ∆00 .

g∈F2

Therefore, for any i ∈ Z, the class (∆00 )−i is an invertible element in (En )2pni on which F acts via χ(i) (with the convention adopted in the discussion before Proposition 20). We will need some partial information about the homotopy groups of the 2 spectra Σ2p ni EnhG , i ∈ Z, where G runs through various subgroups of F which contain the central subgroup Z ⊂ F generated by τ n .

On finite resolutions of K(n)-local spheres

159

Lemma 23. Suppose G is a subgroup of F which contains Z. a) If G is of order prime to p, then for any i ∈ Z the homotopy fixed point 2 spectral sequence converging to π∗ (Σ2p ni EnhG ) satisfies E2s,t = 0 if s > 0. The spectral sequence collapses at E2 and π∗ (EnhG ) ∼ = (En )G ∗ is concentrated in degrees divisible by 2n. b) If G contains an element of order p then for any i ∈ Z the homotopy fixed 2 point spectral sequence converging to π∗ (Σ2p ni EnhG ) satisfies 0 if q is even, 0 < q < 2n E 0,0 if q = 0 2 2 πq (Σ2p ni EnhG ) ∼ = 0 if q is odd, q 6≡ {1, 3, . . . , 2p − 3} mod 2pn s(q),s(q)+q E2 if q is odd, q ≡ {1, 3, . . . , 2p − 3} mod 2p2 n

where s(2q + 1) = 2(p − 2 − q 0 ) + 1 if 2q + 1 = 2q 0 + 1 + 2p2 nl and q 0 ∈ {0, 1, . . . , p − 2}. Proof. a) If G contains no elements of order p then it is clear that E2s,t = 0 if s > 0 and the spectral sequence collapses. Furthermore, G contains the central subgroup Z and this subgroup acts trivially on (En )0 (by (8)) and because of (τ n )∗ (u) = ω if t 6≡ 0 mod 2n.

pn −1 p−1

u (again by (8)) we see that E20,t = (En )G t is trivial

b) Because G always contains Z and because E is normal in F , the assumption implies that G contains F2 = Z × E. Because the index of F2 in G is prime to p the homotopy fixed point spectrum EnhG is a direct summand in EnhF2 and it is therefore enough to discuss the case G = F2 . In the case of G = F1 the homotopy fixed point spectral sequence has been analyzed by Hopkins and Miller. Their account remains unpublished. A summary of this analysis is given in section 2 of [N]. If p = 3 a rather detailed discussion which includes the case of F2 can be found in [GHM] and [GHMR1]. The approach used in these papers generalizes without much problems to the case of any p > 2. In the following we will describe the E2 -term and the differentials of this spectral sequence. First of all, let ρ be the (p−1)-dimensional WFq [F2 ]-module which restricted to E is the reduced regular representation and on which the central element pn −1 τ n acts by multiplication by ω p−1 . Then, as a graded Zp [F2 ]-algebra, (En )∗ is isomorphic to the completion of S∗ (ρ)[N −1 ] at its maximal ideal, where S∗ (ρ) is the graded Q symmetric algebra on ρ with ρ in degree −2, and we have inverted N := g∈F2 g∗ (e) ∈ S∗ (ρ) where e is a suitable generator of ρ (cf. [GHMR1, Lemma 3.2]). In fact, the isomorphism identifies N , up to a scalar, with the element ∆00 of the proof of Lemma 22. This isomorphism can be

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used to calculate the E2 -term of the homotopy fixed point spectral sequence as follows (cf. section 2 in [N], or Theorem 3.7 in [GHMR1] if p = 3): • The invariants E20,t are trivial unless t ≡ 0 mod 2n and periodic of period 2pn with periodicity generator ∆00 ∈ (En )−2pn . In the sequel we let ∆ := (∆00 )−1 ∈ (En )2pn . • Multiplication with p annihilates E2s,∗ if s > 0. Furthermore, there are 2 elements α ∈ E21,2p−2 and β ∈ E22,2p −2p such that, as a module over Fq [∆±1 ], • E22k,∗ , for k > 0, is free of rank 1 with generator β k , • E22k+1,∗ , for k ≥ 0, is free of rank 1 with generator αβ k . The elements α and β are infinite cycles and represent the images of the elements α1 ∈ π2p−3 (S 0 ), β1 ∈ π2(p2 −p−1) (S 0 ) with respect to the unit S 0 → EnhF2 of the ring spectrum EnhF2 . The only non-trivial differentials in this spectral sequence are d2p−1 and d2n2 +1 . They are forced by Toda’s relations α1 β1p = 0 and β1pn+1 = 0 and are determined by 2

d2p−1 (∆n ) = cαβ p−1 , d2n2 +1 (∆n α) = c0 β n

2 +1

, 2

i.e. d2p−1 (∆) = −c∆1−n αβ p−1 , and d2n2 +1 (∆α) = c0 ∆−p(p−2) β n +1 where c, c0 are suitable units in Fq . 2 Then we end up with the following result which is more precise than Lemma 23 above. Proposition 24 (cf. [N, Proposition 2.1, 2.2]). a) π∗ (EnhF2 ) is periodic of period 2p2 n and with periodicity generator ∆p . s,∗ b) E∞ is trivial if s is even and s > 2n2 , or if s is odd and s > 2n − 1. s,∗ c) If 2n2 ≥ s = 2k > 0 then E∞ is a free module over Fq [∆±p ] with generator β k , of total degree 2k(p2 − p − 1). s,∗ d) If 2n − 1 ≥ s = 2k + 1 > 0 then E∞ is a free module over Fq [∆±p ] with generators ∆l β k α, 2 ≤ l ≤ p of total degree 2pnl + 2k(p2 − p − 1) + 2p − 3. 0,t e) E∞ = 0 if t 6≡ 0 mod 2n.

After these preparations we can continue with the proof of part (b) of Lemma 23. First we investigate which of the generators in part (c) and (d) of Proposition 24 can contribute to πq for q as in Lemma 23b. We distinguish two cases according to the parity of q. 2Note

that the element ∆ in [N] corresponds to ∆p−1 = ∆n in this paper.

On finite resolutions of K(n)-local spheres

161

1) If 0 ≤ q = 2q 0 < 2n then it is enough to show that there is no k with 0 < k ≤ n2 such that 2k(p2 − p − 1) ≡ 2q 0 mod 2p2 n . Calculating modulo 2pn gives −2k ≡ 2q 0 mod 2pn and because of 0 < 2k ≤ 2n2 and 0 ≤ 2q 0 < 2n this is clearly impossible. In 0,0 particular, we see that πq = 0 if q is even and 0 < q < 2n, and π0 ∼ = E2 . 2) If q = 2q 0 + 1 then we have to consider the congruence (12)

2pnl + 2k(p2 − p − 1) + 2p − 3 ≡ 2q 0 + 1 mod 2p2 n .

Reducing mod 2pn gives −2k + 2p − 3 ≡ 2q 0 + 1 mod 2pn and thus k ≡ p − 2 − q 0 mod pn. In view of 0 ≤ k ≤ p − 2 this implies that for (12) to have a solution we must have q 0 ∈ {0, 1, . . . , p − 2} modulo pn, i.e. q ∈ {1, 3, . . . , 2p − 3} modulo 2pn. Furthermore, for such a q there is a unique k with 0 ≤ k ≤ p − 2 and a unique l such that ∆l β k α is of total degree q. It remains to check that the elements ∆l β k α with 0 ≤ k ≤ p − 2 which are not permanent cycles cannot be of total degree q ≡ 2q 0 + 1 mod 2p2 n with q 0 ∈ {0, 1, . . . , p − 2}. In fact, not being a permanent cycle is equivalent to l ≡ 1 mod p. Calculating modulo 2p2 n gives 2pnl+2k(p2 −p−1)+2p−3−(2q 0 +1) = 2pn(l+k)+2(−k+p−2−q 0) ≡ 2pn(l+k) and this cannot be 0 modulo 2p2 n if l ≡ 1 mod p and 0 ≤ k ≤ p − 2.

We can finally state and prove the following realization result. Theorem 25. There is a resolution of length n (in the sense of 3.3.1) X• : ∗ → EnhN := X−1 → X0 → . . . → Xn → ∗ such that the complex E∗ (X• ) is isomorphic, as a complex of Morava modules, n to the complex Homcts (P• ↑G N , (En )∗ ) where P• is the complex of Proposition 21. Furthermore, for r > 0 we have _ 2 Xr ' Σ2p n(i1 +...+ir ) EnhF (i1 ,...,ir )

where F is the finite subgroup of N of Proposition 20d and where the wedge is taken over all sequences (i1 , . . . , ir ) with n − 1 ≥ i1 > i2 > . . . > ir ≥ 0. (For r = 0 there is a unique summand EnhF corresponding to the empty sequence.) Proof. Because of χ(pi) ∼ = χ(i) Lemma 22 implies that the spectra Xr realize n the Morava modules Homcts (Pr ↑G N , (En )∗ ). So it remains to realize the maps

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and show that the resulting sequence of maps of spectra can be refined into a resolution in the sense of 3.3.1. 2

Because the index of F2 in F is prime to p, we have, for each i, that Σ2p ni EnhF 2 is a direct wedge summand in Σ2p ni EnhF2 . Furthermore, it follows easily from 2 Proposition 24 that EnhF is periodic of period 2p2 n and thus Σ2p ni EnhF is, for each i, a direct wedge summand in EnhF2 . Therefore, in order to realize the maps it is enough to show that the (En )∗ -Hurewicz homomorphism (13) π0 (F (EnhF2 , EnhF2 )) → Hom(En )∗ [[Gn ]] (En )∗ (EnhF2 ), (En )∗ (EnhF2 ) is an isomorphism (onto the group of degree preserving homomorphisms). By Proposition 3 this happens iff the canonical map F2 (14) π0 (En [[Gn /F2 ]]hF2 ) → (En )0 [[Gn /F2 ]]

is an isomorphism.

Now T we choose a decreasing sequence Uj of open subgroups of Gn with F2 = j Uj . Then En [[Gn /F2 ]] ' holimj En [[Gn /Uj ]] .

By Proposition 2 we have En [[Gn /Uj ]]hF2 '

Y

EnhFx,j

F2 \Gn /Uj

where Fx,j is the isotropy group of the coset xUj with respect to the action of F2 on Gn /Uj . Because Z ⊂ F2 is central each Fx,j contains Z. Then Lemma hF F 23b implies that the canonical maps π0 (En x,j ) → (En )0 x,j are isomorphisms for all x and j and therefore F π0 (En [[Gn /Uj ]]hF2 ) → (En )0 [[Gn /Uj ]] 2

is an isomorphism for each j. Furthermore, it is clear that (En )1 [[Gn /Uj ]] = 0 for all j, and by the remark on the Mittag-Leffler condition following Proposition 2 the relevant lim1 -terms for the homotopy groups of the homotopy limit holimj (En )[[Gn /Uj ]]hF2 are also trivial. Therefore we obtain the desired isomorphism (14) by passing to the limit. We have now proved that all maps Xr → Xr+1 can be (uniquely) realized and the compositions of two successive maps are trivial. It remains to construct the factorizations Xr → Cr → Xr+1 , 0 ≤ r ≤ n − 1, such that Cr−1 → Xr → Cr is a cofibration. This will be done inductively. We note that these factorizations will realize the splitting of the exact complex of Morava modules E∗ (X• ) into the usual short exact sequences. In particular, this will show that Cn ' Xn so that the resolution will be automatically of length n.

On finite resolutions of K(n)-local spheres

163

For r = 0 (where we take C−1 = X−1 ) this is just a consequence of the fact that the composition X−1 → X0 → X1 is null. Now suppose that we have already constructed the factorizations Xr → Cr → Xr+1 , 0 ≤ r ≤ k < n − 1 . We need to show that the composition Ck → Xk+1 → Xk+2 is null so that we can factor it through the cofibre Ck+1 of the map Ck → Xk+1 . For this it is enough to show that the induced map [Ck , Xk+2 ] → [Xk , Xk+2 ] is injective. Now the inductively already constructed part of the resolution ∗ → X−1 → X0 → . . . → Xk → Ck → ∗ can be viewed as a tower of (co)fibrations for Ck which we can use to compute π0 (F (Ck , Xk+2 )). In fact, there is an Adams type spectral sequence associated to this tower which has the form E1p,q =⇒ πq−p (F (Ck , Xk+2 )) with E1p,q

( πq (F (Xk−p, Xk+2 )) ∼ = 0

if 0 ≤ p ≤ k if p > k .

We will use this spectral sequence to show that [Ck , Xk+2 ] ∼ = Ker([Xk , Xk+2 ] → [Xk−1 , Xk+2 ]) thus finishing off the proof. We note that in terms of the spectral sequence this claim says that π0 (F (Ck , Xk+2 )) is isomorphic to the kernel of the differential d1 : E10,0 → E11,0 . It is therefore enough to show that E2q,q = 0 = E2q+1,q for q > 0. Now Proposition 2 (including the remark on the Mittag Leffler condition following it) together with Lemma 23 implies already E1q,q = 0 = E1q+1,q if q > 0 and q is even. W 2 Now let q > 0 be odd. We let Yk+2 := (i1 ,...,ik+2 ) Σ2p n(i1 +...+ik+2 ) En so that Xk+2 = (Yk+2 )hF . We claim that for p = q ≤ k and p = q + 1 ≤ k there are natural isomorphisms E1p,q ∼ = πq (F (Xk−p , (Yk+2 )hF )) ∼ = πq (F (Xk−p, Yk+2 )hF ) ∼ = H s(q),s(q)+q (F, Hom(En )∗ [[Gn ]] (En )∗ (Xk−p ), (En )∗ (Yk+2 ) ∼ = H s(q),s(q)+q F, Hom(E ) Homcts (Pk−p ↑Gn , (En )∗ ), π∗ (Yk+2 ) n ∗

N

where s(q) is as in Lemma 23 and the F -module structures in line 2 and 3 come from the action of F on Yk+2 . If we accept these isomorphisms for the

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n moment then we can finish off the proof because P• ↑G N is split exact as a complex of continuous Zp -modules, and this clearly implies that E2p,q = 0 if p > 0.

It remains to justify the chain of isomorphisms which identify E1p,q . The first two of the claimed isomorphisms are obvious and the last one holds because by 1.3.2 (En )∗ (Yr ) is a coinduced module, i.e. (En )∗ (Yr ) ∼ = Homcts (Zp [[Gn ]], π∗ (Yr )) . To get the third isomorphism it suffices to show that the homotopy fixed point spectral sequence gives, for q odd and 0 < q ≤ k < n, an isomorphism πq (F (EnhF , En )hF ) ∼ = H s(q),s(q)+q F, Hom(En )∗ [[Gn ]] (En )∗ (EnhF ), (En )∗ (En ) . In fact, by Proposition 3 we can identify Hom(En )∗ [[Gn ]] (En )∗ (EnhK ), (En )∗ (En ) with π∗ (F (EnhK , En )) whenever K is a closed subgroup of Gn . Now we replace first EnhF in the source by EnhU where U is an open subgroup of Gn . Then Proposition 2 together with Lemma 23 give the required identification. Finally we write EnhF ' LK(n) hocolimj (En )hUj and pass to the limit once more using the remark on the Mittag-Leffler condition following Proposition 2. 3.6.5. The resolution of LK(n) S 0 . It remains to construct a resolution of LK(n) S 0 . For this we start with an F (Gn )-resolution Q• : 0 → Qm → . . . → Q0 → Q−1 = Zp → 0 of finite length m of the trivial Zp [[Gn ]]-module Zp as given by Proposition 17 and Proposition 21. The modules Qr in this resolution are finitely generated 0 n projectives if r > n, and of the form Q0r ⊕ Pr ↑G N with Qr finitely generated projective and Pr as in Proposition 21, if r ≤ n. We can realize these modules by spectra Zr which are summands in a finite wedge of En ’s if r > n, and of the form Zr0 ∨ Xr with Zr0 a summand in a finite wedge of En ’s and Xr as in Theorem 25. Then the strategy of the proof of Theorem 25 can be applied to this situation. The Zp [[Gn ]]-linear maps Qr → Qr−1 can be uniquely realized by maps Zr−1 → Zr of spectra and the resulting sequence of spectra is a resolution of finite length in the sense of 3.3.1. In fact, the factorizations Zr → Cr → Zr+1 can be constructed just as in the proof of Theorem 25, if 0 ≤ r ≤ n, resp. as in the proof of Theorem 4, if r > n, by using that Q• is split exact as a complex of continuous Zp -modules. Theorem 26. Let p be odd and n = p − 1. Suppose Q• is an F (Gn )-resolution of length m of the trivial Zp [[Gn ]]-module Zp such that Qr is a finitely generated 0 n projective Zp [[Gn ]]-module if r > n while Qr ∼ = Q0r ⊕ Pr ↑G N with Qr finitely generated projective and Pr as in Theorem 25 resp. Proposition 21 if 0 ≤ r ≤ n.

On finite resolutions of K(n)-local spheres

165

Then there is a resolution of length m (in the sense of 3.3.1) Z• : ∗ → LK(n) S 0 := Z−1 → Z0 → . . . → Zn → . . . → Zm → ∗ such that the complex E∗ (Z• ) is isomorphic as a complex of Morava modules to the complex Homcts (Q• , (En )∗ ). Furthermore, Zr is a direct summand in a finite wedge of En ’s if r > n while for 0 ≤ r ≤ n _ 2 Zr ' Zr0 ∨ Σ2p n(i1 +...+ir ) EnhF (i1 ,...,ir )

where the wedge is taken over all sequences (i1 , . . . , ir ) with n − 1 ≥ i1 > i2 > . . . > ir ≥ 0 and Zr0 is a direct summand in a finite wedge of En ’s. 3.6.6. Proof of Theorem 10 and Theorem 11. We note that for p = 3 and n = 2 the group F of Theorem 26 (see also 3.6.4) is equal to the group G24 of Theorem 10. Likewise, the character χ(1) of Proposition 21 is equal to the character χ e of G24 . By Theorem 26 and the splitting on En discussed in the appendix below it is therefore enough to prove Theorem 10 and for this we 2 only need to show that the kernel K of the augmentation Z3 ↑G N → Z3 admits a projective resolution of the form G1

G1

G1

G1

2 2 2 2 →K→0. → χ ↑SD → (λ2 ⊕ χ) ↑SD → (Z3 ⊕ λ2 ) ↑SD 0 → Z3 ↑SD 16 16 16 16

In fact, because of the isomorphism G2 ∼ = G12 ×Z3 (cf. [GHMR1]) it is enough to G1 show that the kernel K 1 of the augmentation Z3 ↑N21 → Z3 with N 1 = NG12 (E) admits a projective resolution of the form G1

G1

G1

2 2 2 → K1 → 0 . → χ ↑SD → λ2 ↑SD 0 → Z3 ↑SD 16 16 16

From [He] we know that ExtiZ3 [[S 1 ]] (K 1 , F3 ) can be identified with the cokernel 2 of the map 1 Y ∗ 1 H (S2 ; F3 ) → H ∗ (ω i CS21 (E)ω −i ; F3 ) i=0

given by the inclusions ω CS21 (E)ω −i → S21 , i = 0, 1. Furthermore, this map is Z3 [SD16 ]-linear and from the explicit description of this map in Theorem 4.4 of [GHMR1] it is straightforward to see that, as Z3 [SD16 ]-modules, we obtain χ ⊗ Z 3 F3 if i = 0 λ ⊗ F if i = 1 2 Z3 3 ExtiZ3 [[S1 ]] (K 1 , F3 ) ∼ = 2 Z3 ⊗Z3 F3 if i = 2 0 if i > 2 . i

The resolution of K 1 is then constructed as in the proof of Theorem 6.

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Appendix: Splitting En with respect to the action of Fn(q−1) A.1. Let p be an odd prime, q = pn and F := Fn(q−1) = Cq−1 o Gal(Fq /Fp ) be as in section 2.3. The homotopy groups of En can be described as π∗ (En ) ∼ = WFq [v1 , . . . , vn−1 ][u±1 ]b, i

i

with vi := ui u1−p , i.e. ui = vi up −1 , and where b means completion with respect to the ideal (u1 , . . . , un−1 ). The elements vi are invariant with respect to the action of F and the action of a generator ω of the cyclic subgroup Cq−1 on u is given by multiplication with ω, i.e. ω∗ (u) = ωu (cf. formula (8) in the proof of Lemma 22), while the Galois group acts on the coefficients WFq only via Frobenius. In particular, the homotopy groups of the homotopy fixed points are given by the appropriate completion π∗ (EnhF ) ∼ = Zp [v1 , . . . , vn−1 ][vn±1 ]b, with vn = u−q . Furthermore we have an isomorphism of π∗ (EnhF )[F ]-modules (15)

π∗ (En ) ∼ =

q−1 M

π∗ (EnhF ) ⊗ WFq ui

i=0

and the action on the modules λi := WFq ui is as described above. We are interested in the decomposition of En obtained from a splitting of the group algebra Zp [F ] as a module over itself. The decomposition suggested by (15) is close but not quite equal to the decomposition obtained from such a splitting. A.2. From now on we will restrict attention to the case n = 2 and p odd which we have used in section 2.3, section 3.3 and 3.3.6. If i is divisible by p + 1 then ω i belongs to Zp and therefore the action of ω on λi commutes with the action of Frobenius. Therefore, for such i, λi splits into 2 one-dimensional pieces λi,+ and λi,− , with Frobenius acting trivially on λi,+ and by multiplication by −1 on λi,− . We claim that there is a direct sum decomposition of Zp [F ] into a direct sum of Zp [F ]-modules M M (16) Zp [F ] ∼ λi ⊕ λi,+ ⊕ λi,− . = i6≡0(p+1)

i≡0(p+1)

We leave it to the reader to check that the modules on the right hand side of this isomorphism are all indecomposable and that the only repetition in the decomposition comes from the isomorphisms λi ∼ = λpi , i 6≡ 0 mod p + 1, which are induced by Frobenius.

On finite resolutions of K(n)-local spheres

167

At the request of the referee we outline a direct construction of the decomposition given in (16): our identification of the roots of unity of WFq with Cq−1 specifies a character χ1 of Cq−1 defined over WFq . Let χi = χ1 ⊗i . Then the elements X 1 ei = χi (g −1 )g (q − 1) g∈C q−1

belong to WFq [Cq−1 ] and they are easily checked to be orthogonal idempotents which sum up to the element 1 ∈ WFq [Cq−1 ]. The elements ei form a basis of WFq [Cq−1 ] as a WFq -module and each ei generates a one-dimensional representation over WFq on which Cq−1 acts via χi . For i ≡ 0 mod (p + 1) the element ei lives already in Zp [Cq−1 ], while for p+1 i 6≡ 0 mod (p + 1) the elements ei + epi and ω 2 (ei − epi ) belong to Zp [Cq−1 ] and together they form a basis of Zp [Cq−1 ] as a Zp -module. Furthermore, the Zp -module generated by ei , i ≡ 0 mod (p + 1), is a Zp [Cq−1 ]-module on which Cq−1 acts via χi , and the Zp -submodule generated by δi := ei + epi p+1 and εi := ω 2 (ei − epi ) is also a Zp [Cq−1 ]-module which is isomorphic to λi restricted to Cq−1 . Now consider the isomorphism of Zp [Cq−1 ]-modules Zp [F ] ∼ = Zp [Cq−1 ] ⊕ Zp [Cq−1 ]σ where σ is Frobenius considered as an element of F . Then there is a Zp -basis of Zp [F ] given by ei ± σei , if i ≡ 0 mod(p + 1), and δi ± δi σ, εi ± εi σ if i 6≡ 0 mod(p + 1). In WFq [F ] we have σei = epi σ, in particular σei = ei σ if i ≡ 0 mod(p + 1), and therefore σ(ei ± ei σ) = ±(ei ± ei σ) if i ≡ 0 mod(p + 1) i.e. (ei ± ei σ) generates a direct summand isomorphic to λi,± . Furthermore we have σδi = δi σ, σεi = −εi σ which shows that the Zp -submodule generated by δi + δi σ and εi + εi σ is a Zp [F ]-module, and likewise the Zp -submodule generated by δi − δi σ and εi − εi σ is a Zp [F ]-module. Both of these modules are isomorphic to λi (or λpi ). A.3. Consequently, by the elementary theory of stable splittings we find that E2 splits, with respect to the F - action, into a direct sum of E2hF -module spectra whose homotopy groups are given by HomZp [F ] (λi,± ,

q−1 M

WFq uj ) ⊗ π∗ (E2hF ) ∼ = (WFq ui )± ⊗ π∗ (E2hF )

j=0

resp. HomZp [F ](λi ,

q−1 M j=0

WFq uj ) ⊗ π∗ (E2hF ) ∼ = Zp ui ⊕ Zp upi ⊗ π∗ (E2hF )

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where (WFq ui )± is the ±-eigenspace of the action of Frobenius on WFq ui . In the first case, the corresponding summand of E2 can be identified as E2hF module spectrum with Σ2i E2hF , in the second case with Σ2i E2hF ∨ Σ2pi E2hF and the multiplicity of this spectrum in a splitting of E2 (constructed via the action of F ) is 2.

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´matique Avanc´ Institut de Recherche Mathe ee, C.N.R.S. - Universit´ e Louis Pasteur, 7 rue Ren´ e Descartes, F-67084 Strasbourg, France

CHROMATIC PHENOMENA IN THE ALGEBRA OF BP∗ BP -COMODULES MARK HOVEY Abstract. We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗ BP comodules. We show, following [HS03a], that the category of E(n)∗ E(n)comodules is a localization, in the abelian sense, of the category of BP∗ BP comodules. This gives analogues of the usual structure theorems, such as the Landweber filtration theorem, for E(n)∗ E(n)-comodules. We recall the work of [Hov02a], where an improved version Stable(Γ) of the derived category of comodules over a well-behaved Hopf algebroid (A, Γ) is constructed. The main new result of the paper is that Stable(E(n)∗ E(n)) is a Bousfield localization of Stable(BP∗ BP ), in analogy to the abelian case.

Introduction The object of this paper is to describe some of the author’s recent work, much of it joint with Neil Strickland, on comodules over BP∗ BP and related Hopf algebroids. The basic idea of this work is to realize the chromatic approach to stable homotopy theory in the algebraic world of comodules. This means, in particular, constructing and understanding the localization functor Ln , or the associated finite localization functor Lfn , in the abelian category BP∗ BP -comod of graded BP∗ BP -comodules. It also means constructing Ln and Lfn in some kind of associated derived category Stable(BP∗ BP ) of chain complexes of BP∗ BP -comodules. Ultimately, we would like to understand LK(n) in the algebraic setting as well, and this should involve the Morava stabilizer groups. In the present paper, we confine ourselves to Ln and Lfn . Topologically, Lfn is localization away from a finite spectrum of type n + 1, and Ln is localization with respect to the homology theory E(n). These functors are probably different in the ordinary stable homotopy category (because Ravenel’s telescope conjecture [Rav84] is widely expected to be false), but they turn out to agree on the abelian category of BP∗ BP -comodules. We then have the following theorem. Theorem A. Let Ln denote localization away from BP∗ /In+1 in the category of BP∗ BP -comodules. Then there is an equivalence of categories between E(n)∗ E(n)-comodules and Ln -local BP∗ BP -comodules. Date: October 26, 2006.

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Note that this is localization in an abelian sense. The localization Ln turns out to be the localization functor that inverts all maps whose kernel and cokernel are vn -torsion. This theorem is really a special case of a more general theorem proved in [HS03a]. In general, if (A, Γ) is a flat Hopf algebroid and B is a Landweber exact A-algebra, there is an induced flat Hopf algebroid (B, ΓB ). We prove in [HS03a] that, in this situation, the category of ΓB -comodules is always equivalent to some localization of the category of Γ-comodules. In the case of BP∗ BP , we give a partial classification of such localizations. In particular, E(n)∗ E(n) can be replaced by E∗ E for any Landweber exact commutative ring spectrum E with E∗ /In 6= 0 and E∗ /In+1 = 0. This tells us that all such theories E have equivalent categories of E∗ E-comodules, even though the categories of E∗ -modules can be drastically different. It also leads to structural results about E∗ E-comodules analogous to those of Landweber [Lan76] for BP∗ BP -comodules. To extend this result to the derived category setting, we first must decide what we mean by the derived category. The ordinary derived category, obtained by inverting homology isomorphisms, is usually badly behaved. For example, the analogue of S 0 in the derived category of BP∗ BP -comodules is BP∗ thought of as a complex concentrated in degree 0, but this is not a small object (see the introduction to Section 3). The following is a corollary of the main result of [Hov02a]. Theorem B. Suppose E is a commutative ring spectrum that is Landweber exact over BP . Then there is a bigraded monogenic stable homotopy category Stable(E∗ E) such that π∗∗ S 0 ∼ = Ext∗∗ E∗ E (E∗ , E∗ ). In particular, the sphere, which is E∗ concentrated in degree 0, is a small object of Stable(E∗ E). This stable homotopy category is the homotopy category of a model structure on chain complexes of E∗ E-comodules. The weak equivalences are homotopy isomorphisms, where homotopy is suitably defined. Every cofibrant object is dimensionwise projective over E∗ and every complex of relatively injective comodules is fibrant. Just as in the ordinary stable homotopy category, there exist interesting nontrivial complexes with no homology. On Stable(BP∗ BP ), we define Lfn to be finite localization away from BP∗ /In+1 , which turns out to be a small object in Stable(BP∗ BP ). We define Ln to be Bousfield localization with respect to the homology theory corresponding to E(n)∗ . In Stable(BP∗ BP ), the homology functor H is represented by BP∗ BP , so is somewhat analogous to the BP -homology of a spectrum. Then the homology theory corresponding to E(n)∗ is in fact HE(n), ordinary homology with coefficients in E(n)∗ . Because of this, some of the things one would expect to be true about Ln are false. For example, the two localizations

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Lfn and Ln on Stable(BP∗ BP ) are definitely different in general, so the most naive version of the telescope conjecture is false in Stable(BP∗ BP ). Also, Ln is not a smashing localization, though Lfn is. Thus Ln is less important in Stable(BP∗ BP ) then Lfn . These statements are also true in the category Stable(E(n)∗ E(n)), where f Ln is the identity functor (as it is localization away from E(n)∗ /In+1 = 0), and Ln is localization with respect to ordinary homology (which already has coefficients in E(n)∗ ). Thus Ln Stable(E(n)∗ E(n)) is the classical unbounded derived category of E(n)∗ E(n)-comodules. The main new result of this paper is then the following theorem. Theorem C. There is an equivalence of stable homotopy categories between the localization Lfn Stable(BP∗ BP ) and Stable(E(n)∗ E(n)). As above in Theorem A, we can replace E(n) in Theorem C by any Landweber exact commutative ring spectrum E with E∗ /In 6= 0 and E∗ /In+1 = 0. Theorem C also implies a similar equivalence between Ln Stable(BP∗ BP ) and Ln Stable(E(n)∗ E(n)). From a computational point of view, Theorem C gives rise to the following change of rings theorem. Theorem D. Suppose that M is a finitely presented BP∗ BP -comodule and N is a BP∗ BP -comodule such that N = Ln N and the right derived functors Lin N are 0 for i > 0. Then there is a change of rings isomorphism Ext∗∗ (M, N ) ∼ = ExtE(n) E(n) (E(n)∗ ⊗BP M, E(n)∗ ⊗BP N ). BP∗ BP

∗

∗

∗

This change of rings theorem includes the Miller-Ravenel change of rings theorem [MR77] and the change of rings theorem of the author and Sadofsky [HS99] as special cases. Also, E(n) can be replaced by any Landweber exact commutative ring spectrum E with E∗ /In 6= 0 and E∗ /In+1 = 0. Because the basic structure of the abelian category of comodules over a flat Hopf algebroid is not as well known as it should be, we first summarize this in Section 1. The results in this section were mostly proved in [Hov02a]. We then describe our proof of Theorem A and related results about E(n)∗ E(n)-comodules in Section 2. Further details can be found in [HS03a]. We introduce the stable homotopy category of comodules in Section 3, describing the proof of Theorem B and looking at some particular features of Stable(BP∗ BP ) and Stable(E(n)∗ E(n)). Some of the material in this section can be found in [Hov02a], but some of it is new. We discuss the relation between Stable(BP∗ BP ) and Stable(E(n)∗ E(n)) in Section 4, where we prove Theorems C and D. All of the results in this section are new. It is a pleasure to thank my coauthor Neil Strickland. Many of the theorems in this paper are joint work with him, much of it done at the Isaac Newton Institute for Mathematical Sciences in Fall 2002. I would like to thank the Universitat de Barcelona, the Universitat Aut`onoma de Barcelona, the Centre de Recerca Matematica, and the Isaac Newton Institute for Mathematical

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Sciences for their support during our collaboration. I would also like to thank John Greenlees for many discussions about the material in this paper. 1. Comodules The object of this section is to give an overview of the structural properties of the category Γ-comod. We begin by recalling the structure maps of a Hopf algebroid (A, Γ). A Hopf algebroid (A, Γ) has the following structure maps, which are all maps of commutative rings. • • • •

The counit : Γ → − A, corepresenting the identity map of an object. The left unit ηL : A → − Γ, corepresenting the source of a morphism. The right unit ηR : A → − Γ, corepresenting the target of a morphism. The diagonal ∆ : Γ → − Γ ⊗A Γ, corepresenting the composite of two composable morphisms. Note that this is a tensor product of Abimodules, with the left A-module structure given by ηL and the right A-module structure given by ηR . • The conjugation χ : Γ → − Γ, corepresenting the inverse of a morphism.

There are many relations between these structure maps, but they are all easily obtained from corresponding facts about groupoids. For example, the source and target of the identity morphism at x are both x, so ηL = ηR = 1. e where Note that the conjugation is best thought of as a map χ : Γ → − Γ, e denotes Γ with the opposite A-bimodule structure, so that A acts on the Γ e by ηR . Similarly, the multiplication map is best thought of as left on Γ e→ e ⊗A Γ. µ : Γ ⊗A Γ − Γ, although we could also think of it as having domain Γ With this convention, the fact that composition of a map with its inverse is the identity gives the following commutative diagram, 1⊗χ

∆

e Γ −−−→ Γ ⊗A Γ −−−→ Γ ⊗A Γ

µ

y

Γ −−−→

A

−−−→

Γ

ηL

and a similar diagram involving χ ⊗ 1 and ηR . This is a great deal simpler than the corresponding diagram in [Rav86, Definition A1.1.1(f)]. We recall that a Γ-comodule is a left A-module M together with a counital and coassociative coaction map ψ : M → − Γ ⊗A M of left A-modules, where again Γ is an A-bimodule. A map of comodules is a map of A-modules that preserves the coaction, so we get a category Γ-comod of Γ-comodules. We then have the following proposition [Rav86, Proposition 2.2.8], which explains the importance of Hopf algebroids and comodules in algebraic topology.

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Proposition 1.1. Suppose E is a ring spectrum such that E∗ E is a commutative ring that is flat over E∗ . Then (E∗ , E∗ E) is a Hopf algebroid, and E∗ X is naturally an E∗ E-comodule for X a spectrum. Other good examples of Hopf algebroids include Hopf algebras, which are just Hopf algebroids where ηL = ηR . In particular, a commutative ring A can be thought of as the discrete Hopf algebroid (A, A); a comodule over a discrete Hopf algebroid is just an A-module. Also, if G acts on a commutative ring R by ring automorphisms, the ring of R-valued functions on G is a Hopf algebroid. The right unit is defined by ηR (r)(g) = g(r). This is dual to the twisted group ring R[G]. We will summarize the properties of Γ-comod in the following theorem, but for it to make sense we need to recall some definitions. Definition 1.2. (a) A Hopf algebroid (A, Γ) is flat if ηR makes Γ into a flat A-module. The conjugation shows that it is equivalent to assume ηL is flat. Since is a right inverse for both ηL and ηR , both of these maps are then faithfully flat. (b) A category is complete if it has all small limits, and cocomplete if it has all small colimits. It is bicomplete if it is both complete and cocomplete. (c) Given a regular cardinal λ, a category I is said to be λ-filtered if every subcategory J of I with fewer than λ morphisms has an upper bound in I; that is, there is an object C in I and a natural transformation from the inclusion functor J → − I to the constant J -diagram at C. An object M of a cocomplete category C is said to be λ-presented if C(M.−) commutes with λ-filtered colimits. (d) Suppose C is a closed symmetric monoidal category with monoidal structure X∧Y , unit A, and closed structure F (X, Y ). An object M in C is said to be dualizable if the natural map F (M, A)∧X → − F (M, X) is an isomorphism for all X ∈ C. An object M is said to be invertible if there is an N and an isomorphism M ∧ N ∼ = A. Theorem 1.3. Suppose (A, Γ) is a flat Hopf algebroid. Then Γ-comod is a bicomplete, closed symmetric monoidal abelian category. We also have: (a) Filtered colimits are exact. (b) Given a regular cardinal λ and a comodule M, M is λ-presented if and only if M is λ-presented as an A-module. (c) For any comodule M, there is a cardinal λ such that M is λ-presented. (d) A comodule M is dualizable if and only if it is projective and finitely generated over A. (e) A comodule M is invertible under the symmetric monoidal product if and only if it is invertible as an A-module. We denote the symmetric monoidal structure by M ∧ N with unit A and the closed structure by F (M, N ).

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This theorem is a summary of the results of [Hov02a, Section 1]. We will just discuss some of the issues that arise. First of all, left adjoints are generally easy to construct, since the forgetful functor from Γ-comod to A-mod is itself a left adjoint (its right adjoint is the extended comodule functor discussed in the following paragraph). Thus, one generally forms the left adjoint in A-mod and notices that it has a natural comodule structure. This is true for colimits and for the symmetric monoidal structure M ∧ N . This is defined to be M ⊗A N , the tensor product of left A-modules, with the coaction given as the composite ψ⊗ψ

g

M ⊗A N −−→ (Γ ⊗A M) ⊗A (Γ ⊗A N ) → − Γ ⊗ A M ⊗A N where g(x ⊗ m ⊗ y ⊗ n) = xy ⊗ m ⊗ n. The key to constructing right adjoints is the extended comodule functor from A-mod to Γ-comod that takes M to Γ ⊗A M, with coaction ∆ ⊗ 1. This is the right adjoint to the forgetful functor. As such, it is generally easy to define a desired right adjoint R on extended comodules. For example, one can easily see that we must define the product of extended comodules by Γ Y

(Γ ⊗A Mi ) ∼ = Γ ⊗A

Y

Mi ,

and the closed structure with target an extended comodule by F (M, Γ ⊗A N ) ∼ = Γ ⊗A HomA (M, N ). It is less easy to see how one defines these right adjoints on maps between extended comodules that are not necessarily extended maps, but this can generally be done. Having done this, we use the exact sequence of comodules ψ

ψg

0→ − M− → Γ ⊗A M −→ Γ ⊗A N, where g : Γ⊗A M → − N is the cokernel of ψ, to define RM = ker R(ψg). Since R is supposed to be a right adjoint, it must be left exact, so we must define R in this way. Note that if M is an A-module and N is a Γ-comodule, we have the two tensor products Γ ⊗A (M ⊗A N ) and (Γ ⊗A M) ∧ N . It is useful to know that these are the same [Hov02a]. Lemma 1.4. Suppose (A, Γ) is a flat Hopf algebroid, M is an A-module and N is a Γ-comodule. Then there is a natural isomorphism of comodules (Γ ⊗A M) ∧ N → − Γ ⊗A (M ⊗A N ). Although Theorem 1.3 indicates that the category of Γ-comodules is a very well-behaved abelian category, one obvious property is missing, and that is the existence of a set of generators. Recall that a set of objects G is said to generate an abelian category C if, whenever f is a nonzero map in C, there exists an object G ∈ G such that C(G, f ) is also nonzero. For example, A is a generator of A-mod. This issue of generators is already complicated for Hopf

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algebras; for a finite group G and a field k, the natural generators for the category of k[G]-modules (which is isomorphic to the category of comodules over the ring of k-valued functions on G) are the simple k[G]-modules. There is no canonical description of these in general. However, any simple k[G]module is finitely generated, and of course projective, over k. Referring to part (d) of Theorem 1.3, one might then expect that the set of isomorphism classes of dualizable comodules forms a set of generators for Γ-comod. Sadly, this appears to be false in general, so we need a hypothesis. Definition 1.5. A Hopf algebroid (A, Γ) is called an Adams Hopf algebroid when Γ is a filtered colimit of dualizable comodules. This hypothesis is really due to Adams [Ada74, Section III.13], who used it for the Hopf algebroid (E∗ , E∗ E) to set up universal coefficient spectral sequences. We learned it from [GH00], as well as the following lemma. Lemma 1.6. Suppose (A, Γ) is an Adams Hopf algebroid. Then it is flat, and the dualizable comodules generate the category of Γ-comodules. In categorical language, the category of comodules over an Adams Hopf algebroid is a locally finitely presentable Grothendieck category. All of the Hopf algebroids that commonly arise in algebraic topology, as well as all Hopf algebras over fields, are known to be Adams [Hov02a, Section 1.4]. Note that it may be that one can take smaller sets of generators than all of the dualizable comodules. For example, the set {BP∗ Xn } will serve as a set of generators for BP∗ BP -comodules, where Xn is the 2n-skeleton of BP . However, BP∗ by itself is definitely not a generator for the category of BP∗ BP -comodules. To see this, let P M be the set of primitives in a BP∗ BP comodule M. Then one can easily check that if BP∗ is a generator of the category of BP∗ BP -comodules then any comodule map that is surjective after applying P is in fact surjective. In particular, the map M f Σ|x| BP∗ → − M x∈P M

would be surjective. This is easily seen to be false for M = BP∗ (CP 2 ), for example. Remark 1.7. Theorem 1.3 lists the good points of the category Γ-comod; we now list some of the bad points of Γ-comod. (a) The forgetful functor from Γ-comod to A-modules, or even down to abelian groups, does NOT have a left adjoint; there is no free comodule functor. (b) Γ-comod does not, in general, have enough projectives. If we take (A, Γ) = (Fp , A), where A denotes the dual Steenrod algebra, it is generally believed that there are no nonzero projective comodules.

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(c) If (A, Γ) is not a Hopf algebra, that is if ηL and ηR are not equal, then the forgetful functor from Γ-comod to A-mod is not in general surjective on objects. For example, there is no BP∗ BP -comodule structure on vn−1 BP∗ for n > 0 [JY80, Proposition 2.9]. (d) Products are not in general exact. Hence the inverse limit functor limΓ on sequences ··· → − Mn → − ··· → − M1 → − M0 may have nonzero derived functors limiΓ for all i > 0. Because there are not enough projective comodules, the homological algebra of comodules always involves injectives, or, better, relative injectives. Because the forgetful functor is exact, if I is an injective A-module, then Γ ⊗A I is an injective Γ-comodule. From this it is easy to check that there are enough injectives. However, injective A-modules are complicated, whereas relative injectives have much better properties. A comodule I is defined to be relatively injective if Γ-comod(−, I) takes A-split short exact sequences of comodules to short exact sequences. The following proposition sums up the properties of relative injectives and is wellknown; details can be found in [Hov02a, Section 3.1]. Proposition 1.8. Suppose (A, Γ) is a flat Hopf algebroid. (a) The relatively injective Γ-comodules are the retracts of extended comodules. (b) The coaction ψ : M → − Γ ⊗A M defines an A-split embedding of M into a relatively injective comodule. (c) Relatively injective comodules are closed under coproducts and products. (d) If I is relatively injective, so is I ∧ M and F (M, I) for all comodules I. (e) If P is a comodule that is projective over A, and I is relatively injective, then ExtnΓ (P, I) = 0 for all n > 0. We take this proposition to mean that, to understand Ext∗Γ (M, N ), we must simultaneously resolve M by comodules that are projective over A and N by relative injectives. We return to this point in Section 3. We close this section with a brief description of naturality. There is, of course, a natural notion of a map Φ : (A, Γ) → − (B, Σ) of Hopf algebroids. The map Φ corepresents a natural functor of groupoids, so consists of ring maps Φ0 : A → − B and Φ1 : Γ → − Σ satisfying certain conditions. Such a map induces a symmetric monoidal functor Φ∗ : Γ-comod → − Σ-comod that takes M to B ⊗A M, with comodule structure given by the composite 1⊗ψ g⊗1 B ⊗A M −−→ B ⊗A Γ ⊗A M −−→ Σ ⊗A M ∼ = Σ ⊗B (B ⊗A M)

where g(b ⊗ x) = bΦ1 (x). It is clear that Φ∗ preserves colimits, so should have a right adjoint Φ∗ . It does, but, as usual, Φ∗ is hard to define. We define

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Φ∗ (Σ ⊗B M) = Γ ⊗A M, and then extend this definition to all Σ-comodules in the same way we did for the product of comodules. An important new feature that arises in the study of Hopf algebroids is the notion of weak equivalence. Definition 1.9. A map Φ of Hopf algebroids is defined to be a weak equivalence if Φ∗ is an equivalence of categories. If Φ is a weak equivalence between discrete Hopf algebroids, then Φ is an isomorphism, but a central point of the author’s work on Hopf algebroids is that there are important non-trivial weak equivalences of Hopf algebroids that are not discrete. In general, we have the following characterization of weak equivalences. Theorem 1.10. The map Φ of Hopf algebroids is a weak equivalence if and only if the composite ηR Φ ⊗1 A −→ Γ −−0−→ B ⊗A Γ is a faithfully flat ring extension and the map B ⊗ A Γ ⊗A B → − Σ that takes b ⊗ x ⊗ b0 to ηL (b)Φ1 (x)ηR (b0 ) is a ring isomorphism. The “if”half of this theorem is the main result of [Hov02b]. The “only if” half is much easier and was proven in [HS03a]. This theorem has a better formulation. Hollander [Hol01] has constructed a model structure on presheaves of groupoids on a Grothendieck topology C; the fibrant objects are stacks; see also [Jar01]. In particular, we can take our Grothendieck site to be the flat topology on Aff, the opposite category of commutative rings (with a cardinality bound so we get a small category). In this topology, a cover of R is a finite collection of flat extensions Si of R Q such that Si is faithfully flat over R. Any Hopf algebroid (A, Γ) defines a presheaf of groupoids Spec(A, Γ) by definition; this presheaf is in fact a sheaf in the flat topology by faithfully flat descent [Hov02b]. We can then rephrase Theorem 1.10 as follows. Corollary 1.11. A map Φ of Hopf algebroids is a weak equivalence if and only if Spec Φ is a weak equivalence of sheaves of groupoids in the flat topology. This point of view suggests that one should reconsider the results of this section for quasi-coherent sheaves over a sheaf of groupoids, since a quasicoherent sheaf over Spec(A, Γ) is the same thing as a Γ-comodule [Hov02b]. We have not carried out this program. One reason for this is that we don’t see any clear applications. But another reason is that we do not know whether the Adams condition is invariant under weak equivalence. The difficulty is that, while dualizable comodules and filtered colimits are preserved by weak equivalences, Γ is not. One could simply demand that dualizable sheaves generate the category of quasi-coherent sheaves, but again we do not know

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if this is sufficient to provide a useful theory, or even whether it holds for interesting non-affine sheaves of groupoids. It would be interesting to know if there are equivalences of categories of comodules that are not given by maps of Hopf algebroids, as occurs in Morita theory. Since Hopf algebroids are a generalization of commutative rings, and there are no non-trivial Morita equivalences of commutative rings, it is reasonable to guess that every equivalence of categories of comodules is a zig-zag of weak equivalences. 2. Landweber exact algebras The object of this section is to study the relation between Γ-comodules and ΓB -comodules, where B is a Landweber exact A-algebra. The main application is to the relation between BP∗ BP -comodules and E(n)∗ E(n)comodules. In particular, we sketch the proof of Theorem A and its corollaries in this section. More details can be found in [HS03a]. Given a Hopf algebroid (A, Γ), an A-algebra B is said to be Landweber exact over A if B ⊗A (−) takes exact sequences of Γ-comodules to exact sequences of B-modules. This is called Landweber exactness because Landweber gave a characterization of Landweber exact BP∗ -algebras in his famous Landweber exact functor theorem [Lan76]. One can check that B is Landweber exact over A if and only if the composite ηR

A −→ Γ → − B ⊗A Γ is a flat ring extension. This condition is reminiscent of the characterization of weak equivalences given in Theorem 1.10. We can make it even more so by defining ΓB = B ⊗A Γ ⊗A B. We then have the following lemma, which is easy to prove but can also be found in [HS03a]. Lemma 2.1. Suppose (A, Γ) is a Hopf algebroid and B is an A-algebra. Then (B, ΓB ) is a Hopf algebroid, and the evident map (A, Γ) → − (B, ΓB ) is a map of Hopf algebroids. If (A, Γ) is flat and B is Landweber exact, then (B, ΓB ) is a flat Hopf algebroid. Thus, if B is Landweber exact over A, the map Φ : (A, Γ) → − (B, ΓB ) is almost a weak equivalence, in that ηR

A −→ Γ → − B ⊗A Γ is flat, and B ⊗ A Γ ⊗A B → − ΓB is an isomorphism. The only thing stopping Φ from being a weak equivalence is that B ⊗A (−) may not be faithful on the category of Γ-comodules. The

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idea of the following theorem, proved in [HS03a], is that we can force Φ to be faithful by localizing the category of Γ-comodules. Theorem 2.2. Suppose (A, Γ) is a flat Hopf algebroid, and B is Landweber exact over A. Then the map Φ : (A, Γ) → − (B, ΓB ) yields an equivalence Φ∗ : LT (Γ-comod) → − ΓB -comod where LT (Γ-comod) is the localization of Γ-comod with respect to the hereditary torsion theory T consisting of all Γ-comodules M such that B ⊗ A M = 0. A hereditary torsion theory is just a full subcategory closed under subobjects, quotient objects, extensions, and arbitrary direct sums. The localization LT is obtained by inverting all maps f of Γ-comodules whose kernel and cokernel are in T . Note that the Hopf algebroids that arise in algebraic topology are graded, so B will be a graded A-algebra, and our hereditary torsion theories will also be graded, in the sense that M is in T if and only if all shifts of M are in T . Because it is so surprisingly easy, we will give the proof of Theorem 2.2. Proof. Consider the natural transformation M : Φ∗ Φ∗ M → − M. We claim that this map is a natural isomorphism. One can check this by calculation for extended Σ-comodules M. Since is a natural transformation of left exact functors (because B is Landweber exact), and every Σ-comodule is the kernel of a map of extended comodules, M is an isomorphism for all M. After this, the rest of the proof of Theorem 2.2 is purely formal. A priori, the category LT (Γ-comod) may not be an actual category, since it may not have small Hom sets, always a danger with localization. However, it does exist in a higher universe. The natural transformation ηM : M → − Φ ∗ Φ∗ M becomes an isomorphism upon applying Φ∗ , and therefore, since Φ∗ is exact, the kernel and cokernel of ηM are in T . This gives us the desired equivalence, and incidentally shows that LT (Γ-comod) actually does exist as an honest category, since it is equivalent to ΓB -comod. Theorem 2.2 gives the following corollary, which it is difficult to imagine proving directly. Corollary 2.3. Suppose (A, Γ) is a flat Hopf algebroid, B is Landweber exact over A, and every nonzero Γ-comodule has a primitive. Then every nonzero ΓB -comodule has a primitive. In particular, every E(n)∗ E(n)-comodule has a primitive.

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This corollary is immediate, as LT (Γ-comod) is the full subcategory of Γ-comod consisting of the local objects. To get further information, we need to identify the hereditary torsion theories that can arise. Let Tn denote the collection of all vn -torsion BP∗ BP comodules, so that T0 is the collection of all p-torsion comodules, and T−1 is the collection of all comodules. One can easily check that Tn is a hereditary torsion theory. It is less obvious, but true, that Tn is the smallest graded hereditary torsion theory containing BP∗ /In+1 . Theorem 2.4. Let T be a graded hereditary torsion theory of BP∗ BP -comodules. If T contains a nonzero finitely presented comodule, then T = Tn for some n. This theorem is proved in [HS03a], using the ideas behind the Landweber filtration theorem. Note that this theorem explains why Ln and Lfn agree on the category of BP∗ BP -comodules. The only reasonable definition of Ln is localization with respect to the hereditary torsion theory of all comodules M such that E(n)∗ ⊗BP∗ M = 0, and Lfn is localization with respect to the hereditary torsion theory generated by BP∗ /In+1 . Both of these torsion theories are Tn . Because of this theorem, it is natural to make the following definition. Definition 2.5. Suppose B is a BP∗ -algebra. Define the height of B to be the largest integer n such that B/In is nonzero. If there is no such n, define the height of B to be infinite. From a formal group law point of view, the height of B is the largest possible height of any specialization of the formal group law of B. So the height of E(n) is n, and the height of BP itself is ∞. Here is the main theorem of [HS03a]. Theorem 2.6. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B is a graded Landweber exact A-algebra of height n ≤ ∞. Then the functor M 7→ B ⊗ A M defines an equivalence of categories Ln (Γ-comod) → − ΓB -comod, where Ln is localization with respect to Tn for n < ∞ and L∞ is the identity localization. This theorem is almost a corollary of Theorem 2.2 and Theorem 2.4, except for the infinite height case. We do not have a classification of graded hereditary torsion theories of BP∗ BP -comodules that do not contain a nonzero finitely presented comodule. There are probably uncountably many such torsion theories. However, if B is Landweber exact and B ⊗A M = 0 for some nonzero M, then B/I∞ B = 0. Indeed, M must have a nonzero primitive, and so we conclude by Landweber exactness that B/IB = 0 for some proper invariant ideal I. Since I ⊆ I∞ , it follows that B/I∞ B = 0. But this means

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that 1 ∈ I∞ B, so 1 ∈ In B for some n. Thus B ⊗A A/In = 0. This proves that if B has infinite height, then B ⊗A (−) does not kill any nonzero BP∗ BP comodules. The following corollary is immediate. Corollary 2.7. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B and B 0 are both Landweber exact A-algebras of the same height. Then the category of ΓB -comodules is equivalent to the category of ΓB 0 -comodules. In particular, the category of E(n)∗ E(n)-comodules is equivalent to the category of vn−1 BP∗ (vn−1 BP )-comodules, even though the category of E(n)∗ modules is very different from the category of vn−1 BP∗ -modules. One way to think of the Miller-Ravenel change of rings theorem [MR77, Theorem 3.10] as an isomorphism of certain Ext groups in these two categories. This is now obvious; the Ext groups are isomorphic because the categories they are taken in are equivalent. We point out that the equivalence of categories of comodules in Corollary 2.7 is in fact induced by a zig-zag of weak equivalences of Hopf algebroids. Any map B → − B 0 of Landweber exact BP∗ -algebras of the same height induces a weak equivalence of Hopf algebroids (B, ΓB ) → − (B 0 , ΓB 0 ), where we are still denoting BP∗ BP by Γ. If B and B 0 are Landweber exact BP∗ -algebras of the same height, there may not be a map of BP∗ -algebras between them. However, if we let C = B ⊗BP∗ Γ ⊗BP∗ B 0 , then C has a left and right BP∗ -algebra structure, which we denote by CL and CR . There are maps of BP∗ -algebras B → − CL and B 0 → − CR , and conjugation induces an isomorphism CL → − CR . Since CL is also Landweber exact, of the same height as B and B 0 , this yields the desired zig-zag of weak equivalences. To further understand the structure of the category of E(n)∗ E(n)-comodules, we would like to understand the localization functor Ln better. The following theorem is a summary of the results of [HS03b], and is joint work of the author and Strickland. Theorem 2.8.

(a) A comodule M is Ln -local if and only if

Hom∗BP∗ BP (BP∗ /In+1 , M) = Ext1,∗ BP∗ BP (BP∗ /In+1 , M) = 0, which is true if and only if Hom∗BP∗ (BP∗ /In+1 , M) = Ext1,∗ BP∗ (BP∗ /In+1 , M) = 0, (b) Ln , thought of as an endofunctor of the category of BP∗ BP -comodules, is left exact and preserves finite limits, filtered colimits, and arbitrary direct sums. It has right derived functors which we denote Lin . (c) For i > 0, Lin (M) is isomorphic to the i + 1st local cohomology group of the BP∗ -module M with respect to In+1 . In particular, Lin (M) = 0 for i > n.

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(d) Suppose m < n and M is a vm−1 -torsion comodule on which (vm , vm+1 ) is a regular sequence. Then M is Ln -local. (e) If vm acts invertibly on a comodule M for some m ≤ n, then M is Ln -local and Lin M = 0 for all i > 0. (f) If a comodule M is vn−1 -torsion, then Ln M = vn−1 M. (g) We have k < n, BP∗ /Ik −1 Ln (BP∗ /Ik ) = vn BP∗ /In k = n, 0 k > n. and, for i, n > 0, ( BP∗ /(p, v1 , . . . , vk−1 , vk∞ , . . . , vn∞ ) i = n − k > 0, i Ln (BP∗ /Ik ) = 0 otherwise.

These derived functors Lin can be used to compute BP∗ (Ln X) from BP∗ X by means of a spectral sequence. Theorem 2.9. Let X be a spectrum. There is a natural spectral sequence E∗∗∗ (X) with dr : Ers,t → − Ers+t,t+r−1 and E2 -term E2s,t (X) ∼ = (Lsn BP∗ X)t , converging to BPt−s (Ln X). This is a spectral sequence of BP∗ BP -comodules, in the sense that Ers,∗ is a graded BP∗ BP -comodule for all r ≥ 2 and dr : Ers,∗ → − s+r,∗ Er is a BP∗ BP -comodule map of degree r−1. Furthermore, every element in E20,∗ that comes from BP∗ X is a permanent cycle. This theorem is proved in [HS03b]. It is very closely related to the local cohomology spectral sequence of Greenlees [Gre93] and Greenlees and May [GM95]. One way of putting it is that we show that the Greenlees spectral sequence is a spectral sequence of comodules in this case. When X = S 0 , this implies that the spectral sequence collapses with no extensions, and we recover Ravenel’s computation [Rav84] of BP∗ Ln S 0 . We now derive some corollaries of Theorem B, proved in [HS03a]. Corollary 2.10. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B is a Landweber exact A-algebra of height n. Then Z(p) n>m=0 Q n=m=0 ΓB -comod(B, B/Im ) = Fp [vm ] n>m>0 −1 Fp [vm , vm ] n = m > 0. Proof. Let Φ : (A, Γ) → − (B, ΓB ) be the evident map of Hopf algebroids, so that Ln = Φ∗ Φ∗ by Theorem 2.6. Then we have ΓB -comod(B, B/Im ) ∼ = Γ-comod(A, Φ∗ (B/Im )) ∼ = Γ-comod(A, Ln (A/Im )),

so the result follows from Theorem 2.8 and the analogous calculation for B = BP∗ [Rav86, Theorem 4.3.2].

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This corollary in turn gives rise to the expected structural results about ΓB -comodules, proved in [HS03a]. Theorem 2.11. Let (A, Γ) = (BP∗ , BP∗ BP ), and suppose B is a Landweber exact A-algebra of height n. (a) If I is an invariant radical ideal in B, then I = Im for some m ≤ n. (b) If M is a finitely presented ΓB -comodule, then there is a filtration 0 = M 0 ⊆ M1 ⊆ · · · ⊆ M t = M of M by subcomodules such that, for each s ≤ t, there is a k ≤ n and an r such that Ms /Ms−1 ∼ = sr B/Ik B. Proof. Note that the theorem is invariant under the equivalences of categories of Theorem 2.6. Thus we might as well assume that B = E(n)∗ or BP∗ , and in case B = BP∗ the result is due to Landweber [Lan76]. So we assume B = E(n)∗ . Now, for part (a), suppose I is an invariant radical ideal, and choose the largest k such that Ik ⊆ I. If Ik 6= I, the comodule I/Ik must have a nonzero primitive y. This primitive must also be a primitive in E(n)∗ /Ik . By Corollary 2.10, it must be a power of vk . Since I is radical, this means that Ik+1 ⊆ I. This contradication implies that Ik = I. For part (b), we construct the filtration Mi by induction on i, taking M0 = 0. Having built Mi , if Mi 6= M, we choose a nonzero primitive y in M/Mi . We claim that some multiple z of y is a primitive whose annihilator is Ik for some k ≤ n. Indeed, if y is p-torsion, then we can multiply y by a power of p to obtain a nonzero primitive y1 with py1 = 0. Since v1 is a primitive mod p, if y1 is v1 -torsion we can multiply y1 by power of v1 to obtain a nonzero primitive y2 with py2 = v1 y2 = 0. Continuing in this fashion, we end up with a primitive z such that Ik z = 0 and z is not vk -torsion. Corollary 2.10 implies that Ann z = Ik , as required. We now choose an element w in M whose image in M/Mi is z, and let Mi+1 denote the subcomodule generated by Mi and w. Then Mi+1 /Mi ∼ = sr E(n)∗ /Ik , where r is the degree of z. Since M is finitely presented over the Noetherian ring E(n)∗ , this process must stop, and so Mt = M for some t. 3. The stable homotopy category Stable(Γ) The object of this section is to discuss the construction and basic properties of the stable homotopy category Stable(Γ) associated to an Adams Hopf algebroid (A, Γ). In practice, we are most interested in Stable(E∗ E) for E a commutative Landweber exact ring spectrum. This means our Hopf algebroids should be graded, and the stable homotopy category Stable(Γ) should be bigraded. However, the grading just adds unnecessary complexity to the notation, so we will forget about it for most of this section.

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Construction of Stable(Γ). The usual derived category D(Γ) of Γ-comodules is obtained from Ch(Γ), the category of unbounded chain complexes of Γcomodules, by inverting the homology isomorphisms. This is not the right thing to do to form Stable(Γ). To see this, note that there is a model structure on Ch(Γ) whose homotopy category is the derived category, as there is on Ch(A) for any Grothendieck category A [Bek00]. The cofibrations in this model structure are the monomorphisms, the fibrations are the epimorphisms with DG-injective kernel, and the weak equivalences are the homology isomorphisms. Here a complex X is DG-injective if each Xn is injective and every map from an exact complex to X is chain homotopic to 0. In particular, bounded above complexes of injectives are DG-injective, from which it follows that D(Γ)(A, A)∗ = Ext∗Γ (A, A). Here we are thinking of A as a complex concentrated in degree 0. It is the analog of the sphere S, since it is the unit of ∧. Now, there are often nonnilpotent elements in ExtiΓ (A, A) for i > 0. For example, the well-known element β1 is non-nilpotent in Ext∗BP∗ BP (BP∗ , BP∗ ) for p > 2. Since β1 corresponds to a self-map of BP∗ in D(BP∗ BP ), we can form β1−1 BP∗ . But this complex has trivial homology, so is 0 in D(BP∗ BP ) even though β1 is not nilpotent. This is not good for several reasons; we should be able to see β1−1 BP∗ because it is an important object, and the fact that we can’t also implies that A is not a small object of D(BP∗ BP ). We should be inverting homotopy isomorphisms, not homology isomorphisms. To do this, we need to define the homotopy groups. Definition 3.1. Suppose (A, Γ) is an Adams Hopf algebroid, X ∈ Ch(Γ), and P is a dualizable Γ-comodule. We define the homotopy groups of X with coefficients in P by πnP (X) = H−n (Γ-comod(P, LA ∧ X)), where LA is the cobar resolution of A. We define a map f of complexes to be a homotopy isomorphism if πnP (f ) is an isomorphism for all n and all dualizable comodules P . The stable homotopy category of Γ, Stable(Γ) is defined to be the category obtained from Ch(Γ) by inverting the homotopy isomorphisms. The reason for the sign in the definition of πnP is so that πnP (M) ∼ = ExtnΓ (P, M) for a comodule M. Homotopy groups and homotopy isomorphisms satisfy the expected properties [Hov02a]. Proposition 3.2. Suppose (A, Γ) is an Adams Hopf algebroid, and P is a dualizable comodule.

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(a) A short exact sequence of complexes induces a long exact sequence in the homotopy groups π∗P (−). Note that the boundary map raises dimension by one, because of the sign in the definition of πnP (−). (b) Homotopy groups commute with filtered colimits, so homotopy isomorphisms are closed under filtered colimits. (c) Every chain homotopy equivalence is a homotopy isomorphism, and every homotopy isomorphism is a homology isomorphism. (d) The natural map X → − LA ∧ X is a homotopy isomorphism for all X. To understand Stable(Γ), and for that matter to even see that it is a category at all, we need a model structure on Ch(Γ) in which the weak equivalences are the homotopy isomorphisms. This is the main goal of [Hov02a], where the following theorem is proved. Theorem 3.3. Suppose (A, Γ) is an Adams Hopf algebroid. Then there is a proper symmetric monoidal model structure on Ch(Γ) in which the weak equivalences are the homotopy isomorphisms. Furthermore, if X is cofibrant, then X ∧ (−) preserves homotopy isomorphisms. We call this model structure the homotopy model structure. The cofibrations in the homotopy model structure are dimensionwise split monomorphisms with cofibrant cokernel. If X is cofibrant, then each Xn is projective over A. More precisely, if X is cofibrant, then X is a retract of a complex Y that admits a filtration Y i such that each map Y i → − Y i+1 is a dimensionwise split monomorphism and the quotient Y i+1 /Y i is a complex of relatively projective comodules with trivial differential. Here a comodule is relatively projective if it is a retract of a direct sum of dualizable comodules. A characterization of the fibrations is given in [Hov02a]. Fibrations are of course surjective. Every complex of relative injectives is fibrant, and every fibrant complex is equivalent in a precise sense to a complex of relative injectives. A fibrant replacement of X is given by LB ∧ X. The following theorem is also proved in [Hov02a]. Theorem 3.4. The homotopy model structure is natural, in the sense that a map Φ : (A, Γ) → − (B, Σ) induces a left Quillen functor Φ∗ : Ch(Γ) → − Ch(Σ) of the homotopy model structures. Furthermore, if Φ is a weak equivalence, then Φ∗ is a strong Quillen equivalence, in the sense that both Φ∗ and Φ∗ preserve and reflect homotopy isomorphisms. Global properties of Stable(Γ). We now establish some of the essential properties of Stable(Γ). At this point, we begin to use some of the standard notational conventions of ordinary stable homotopy. Thus, we will begin using S for the image of A in Stable(Γ), thinking of it as analogous to the usual zero-sphere. Similarly, we will sometimes use [X, Y ]∗ for graded maps in Stable(Γ). In practice, Γ is usually graded and so Stable(Γ) is bigraded. However, the internal suspension in the category of Γ-comodules is usually not relevant, so we tend to omit it from the notation.

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Theorem 3.5. Suppose (A, Γ) is an Adams Hopf algebroid. The category Stable(Γ) is a closed symmetric monoidal triangulated category. The dualizable comodules form a set of small, dualizable, weak generators for Stable(Γ). This theorem is really a corollary of Theorem 3.3 and general facts about model categories. It is proved in [Hov02a]; another way to say it is that Stable(Γ) is a unital algebraic stable homotopy category in the sense of [HPS97]. One drawback of Stable(Γ) is that it is not in general monogenic. That is, A and its suspensions are not generally enough to generate the whole category. This is unavoidable even for Hopf algebras. Indeed, if G is a finite group and k is a field, the stable homotopy category of the Hopf algebra of functions from G to k is closely related to the stable module category much studied in modular representation theory [Ben98], as explained in [HPS97]. If G is a p-group, the stable module category is monogenic, but not in general. However, one certainly expects Stable(BP∗ BP ) and Stable(E(n)∗ E(n)) to be monogenic, so we need a condition on (A, Γ) that will ensure that Stable(Γ) is monogenic. Recall that a full subcategory D of an abelian category is called thick if it is closed under retracts and, whenever two out of three terms in a short exact sequence are in D, so is the third. Proposition 3.6. Suppose (A, Γ) is an Adams Hopf algebroid, and every dualizable comodule is in the thick subcategory generated by A. Then Stable(Γ) is monogenic. This proposition is proved in [Hov02a]. To apply it, we note that the filtration theorem for E∗ E-comodules, part (b) of Theorem 2.11, implies that every finitely presented E∗ E-comodule is in the thick subcategory generated by E∗ when E is a commutative ring spectrum that is Landweber exact over BP∗ . Thus we get the following corollary. Corollary 3.7. Suppose E is a commutative ring spectrum that is Landweber exact over BP . Then Stable(E∗ E) is monogenic in the bigraded sense. In particular, a map f in Ch(E∗ E) is a homotopy isomorphism if and only if k πn,k (f ) = πns E∗ (f ) is an isomorphism for all n and k. The bigrading arises because we can suspend a complex X either internally, by suspending each graded comodule Xn , or externally by suspending the complex X. We would like to understand the relation between a comodule M and its image in Stable(Γ). The following proposition is proved in [Hov02a]. Proposition 3.8. Suppose (A, Γ) is an Adams Hopf algebroid. (a) A short exact sequence of comodules, or even complexes, gives rise to a cofiber sequence in Stable(Γ). (b) If M is in the thick subcategory generated by A, then M is a small object of Stable(Γ).

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Mark Hovey (c) If M and N are comodules, then there is a natural map ExtkΓ (M, N ) → − Stable(Γ)(M, N )k that is an isomorphism for M in the thick subcategory generated by A.

Part (b) is not actually proved in [Hov02a], but follows immediately from part (a). In particular, of course, we have Stable(Γ)(A, A)∗ ∼ = π∗A (A) ∼ = Ext∗Γ (A, A). This is the stable homotopy of the sphere in Stable(Γ). The following point is also valuable. Proposition 3.9. Suppose E is a commutative ring spectrum that is Landweber exact over BP . Then Stable(E∗ E) is a Brown category, so that every homology functor is representable. This proposition follows from Theorem 4.1.5 of [HPS97]. Indeed, we can assume E = E(n) or BP , and then one can easily check using the cobar resolution that Ext∗∗ E∗ E (E∗ , E∗ ) is countable. Ordinary homology. We now describe ordinary homology in Stable(Γ). Since homotopy isomorphisms are in particular homology isomorphisms, the ordinary homology of a chain complex X is a homology theory on Stable(Γ). Proposition 3.10. Let (A, Γ) be an Adams Hopf algebroid. Ordinary homology is represented on Stable(Γ) by Γ itself, as usual thought of as a complex concentrated in degree 0. Because of this proposition, we will sometimes denote Γ by H. Proof. For a complex X, we have Γ∗ (X) ∼ = π∗ (QΓ ∧ QX), where Q denotes cofibrant replacement. Since QX ∧ (−) preserves homotopy isomorphisms by Theorem 3.3, we have π∗ (QΓ ∧ QX) ∼ = π∗ (Γ ∧ QX). Since Γ ∧ QX is already fibrant, as it is a complex of relative injectives, we have π∗ (Γ ∧ QX) ∼ = Ch(Γ)(A, Γ ∧ QX)/ ∼, where ∼ denotes the chain homotopy relation. This is in turn isomorphic to Ch(A)(A, QX)/ ∼∼ = H∗ (QX) ∼ = H∗ X by adjointness.

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Note that the Hopf algebroid (H∗ , H∗ H) associated to homology is isomorphic to (A, Γ) itself, concentrated in degree 0. Thus H∗ X is naturally a graded (A, Γ)-comodule, which is bigraded in case (A, Γ) is graded. We get an Adams-Novikov spectral sequence based on H whose E2 -term is E s,t ∼ = Exts (A, Ht X), 2

Γ

which in good cases will converge to π∗ X. In particular, if X = S, this spectral sequence is concentrated in degrees (s, 0), and so collapses and converges to π∗ S ∼ = Ext∗Γ (A, A). Thus, if we take (A, Γ) = (BP∗ , BP∗ BP ), we have built a stable homotopy category in which the usual Adams-Novikov spectral sequence collapses. Note that ordinary cohomology is somewhat complicated. This is actually already true in the derived category D(A). Indeed, in D(A) we have H ∗ (X) ∼ = Ch(A)(QX, S 0 A)∗ and there is no really convenient interpretation of these groups. Similarly, in Ch(Γ), we have H ∗ (X) ∼ = Ch(Γ)(QX, S 0 Γ)∗ ∼ = Ch(A)(QX, S 0 A)∗ . The ordinary derived category of Γ, obtained by inverting the homology isomorphisms, is the Bousfield localization of Stable(Γ) with respect to H. As we have said before, this is a non-trivial localization. Indeed, suppose x is a non-nilpotent class in Exts (A, A) with s > 0. Then x corresponds to a self-map S −s → − S, which is necessarily 0 on homology. Hence the −1 telescope x S will have no homology, but will be nonzero. In particular, if (A, Γ) = (BP∗ , BP∗ BP ), there are non-nilpotent classes in Ext. For example, α1 is non-nilpotent when p = 2 by [Rav86, Theorem 4.4.37]. Homology with coefficients. We now consider ordinary homology with coefficients in an A-module B. Proposition 3.11. Let (A, Γ) be an Adams Hopf algebroid, and let B be an A-module. Then Γ ⊗A B represents the homology theory HB on Stable(Γ) defined by (HB)∗ (X) ∼ = H∗ (B ⊗A QX). If B is Landweber exact over A, then (HB)∗ (X) ∼ = H∗ (B ⊗A X) ∼ = B ⊗A H∗ X. In particular, in this case the Hopf algebroid (HB∗ , HB∗ HB) is isomorphic to (B, ΓB ) concentrated in degree 0. Proof. For an object X of Stable(Γ), we have (HB)∗ (X) ∼ = π∗ (Γ ⊗A B) ∧ QX) ∼ = H∗ (B ⊗A QX),

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where we have used Lemma 1.4 to manipulate the tensor product. In particular, HB∗ (S) ∼ = B concentrated in degree 0. If B is Landweber exact over A, then B ⊗A (−) will preserve homology isomorphisms of complexes of comodules, so (HB)∗ (X) ∼ = H∗ (B ⊗A X) ∼ = B ⊗A H∗ X. In particular,

(HB)∗ (HB) ∼ = B ⊗ A Γ ⊗A B ∼ = ΓB . Thus (HB)∗ X is naturally a graded comodule over (B, ΓB ), when B is Landweber exact over A. Thus we get theories HE(n) when (A, Γ) = (BP∗ , BP∗ BP ) and B = E(n)∗ . The Adams-Novikov spectral sequence based on HB when B is Landweber exact will then have E2 -term E s,t ∼ = Exts (B ⊗A Ht X, B ⊗A Ht Y ). 2

ΓB

In particular, when X = Y = S, this spectral sequence must collapse, since the E2 -term is concentrated where t = 0. However, it is not entirely clear to what it converges. The obvious guess is π∗ LHB S, where LHB denotes Bousfield localization with respect to HB. Bousfield’s convergence results [Bou79] should be re-examined to see if they apply in a more general setting to answer this question. Note that if B is an A-algebra that is also a field, then HB will be a field object of Stable(Γ). In particular, if p is a prime ideal in A with residue field kp , then we can form Hkp . If we apply this to the case (A, Γ) = (BP∗ , BP∗ BP ), we get field spectra HK(n) corresponding to the Morava K-theories, but we also get many other field spectra, including HFp corresponding to the prime ideal I∞ . Note that the objects HK(n) do not detect nilpotence in Stable(BP∗ BP ), since there are non-nilpotent self-maps of S that are zero on homology with any coefficients. 4. Landweber exactness and the stable homotopy category Recall that in Section 2 we showed that the abelian category of E(n)∗ E(n)comodules is a localization of the abelian category of BP∗ BP -comodules. In Section 3, we introduced stable homotopy categories of E(n)∗ E(n) and BP∗ BP -comodules. It is therefore natural to conjecture that Stable(E(n)∗ E(n)) is a Bousfield localization of Stable(BP∗ BP ). The goal of this section is to prove this conjecture, thereby proving Theorem C. The functor Φ∗ . Throughout this section, we let (A, Γ) = (BP∗ , BP∗ BP ), B = E(n)∗ , and we let Φ : (A, Γ) → − (B, ΓB ) be the induced map of Hopf algebroids. The map of Hopf algebroids Φ induces a functor Φ∗ : Γ-comod → − ΓB -comod

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and a left Quillen functor Φ∗ : Ch(Γ) → − Ch(ΓB ) by Theorem 3.4. To prove Theorem C, we must show that Φ∗ induces a Quillen equivalence upon suitably localizing Ch(Γ). The object of the present section is to prove the following theorem. Theorem 4.1. The functor Φ∗ : Ch(Γ) → − Ch(ΓB ) preserves weak equiva∗ lences. Its right adjoint Φ reflects weak equivalences. We prove this theorem in a series of propositions. Proposition 4.2. Let D denote the class of all X ∈ Ch(Γ) such that the map Φ∗ QX → − Φ∗ X is a weak equivalence in Ch(ΓB ), where Q is a cofibrant replacement functor in Ch(Γ). Then D is a thick subcategory. Proof. Note that D is obviously closed under retracts. To see that D is thick, suppose we have a short exact sequence X0 → − X→ − X 00 in Ch(Γ) such that two out of three terms are in D. By Proposition 3.8(a), this is a cofiber sequence in Stable(Γ). Since Φ∗ Q is the total left derived functor of the left Quillen functor Φ∗ , we conclude that Φ∗ QX 0 → − Φ∗ QX → − Φ∗ QX 00 is a cofiber sequence in Stable(ΓB ). On the other hand, because Φ∗ is exact, the sequence Φ∗ X 0 → − Φ∗ X → − Φ∗ X 00 is a short exact sequence in Ch(ΓB ), and hence, applying Proposition 3.8(a) again, is also a cofiber sequence in Stable(ΓB ). There is a map from the first of these cofiber sequences to the second, and by assumption it is an isomorphism on two out of three terms. Since Stable(ΓB ) is a triangulated category, we conclude that it is also an isomorphism on the third term, and so D is thick. Our next goal is to show that D is closed under filtered colimits. For this we need to recall some standard model category theory. Suppose I is a small category, and M is a cofibrantly generated model category, such as Ch(Γ). Then there is a cofibrantly generated model category structure on the diagram category MI [Hir03, Theorem 12.7.1] in which the weak equivalences and fibrations are taken objectwise. Furthermore, the cofibrations in MI are in particular objectwise cofibrations [Hir03, Proposition 12.7.3]. Proposition 4.3. The class D of Proposition 4.2 is closed under filtered colimits.

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Proof. Suppose F : I → − Ch(Γ) is a functor from a filtered small category I such that F (i) ∈ D for all i ∈ I. We must show that colim F (i) ∈ D. Let QF be a cofibrant replacement of F in the model category on Ch(Γ)I discussed prior to this proposition. Because the constant diagram functor obviously preserves fibrations and trivial fibrations, the colimit is a left Quillen functor [Hir03, Theorem 12.7.9]. Hence colim QF is cofibrant. Furthermore, each map QF (i) → − F (i) is a homotopy isomorphism, and so, since homotopy commutes with filtered colimits, we conclude that the map colim QF → − colim F is a weak equivalence. Therefore, colim QF is a cofibrant replacement of colim F . To show that colim F ∈ D, then, we need only show that the map Φ∗ (colim QF ) → − Φ∗ (colim F ) is a homotopy isomorphism. Since Φ∗ itself commutes with colimits, this is equivalent to showing that the map colim Φ∗ QF → − colim Φ∗ F is a homotopy isomorphism. Since QF is cofibrant, and cofibrations of diagrams are in particular objectwise cofibrations, we conclude that QF (i) is a cofibrant replacement for F (i) for all i ∈ I. Since F (i) ∈ D, then, each map Φ∗ QF (i) → − Φ∗ F (i) is a homotopy isomorphism. Hence, again using the fact that homotopy commutes with filtered colimits, colim Φ∗ QF → − colim Φ∗ F is a homotopy isomorphism, so colim F ∈ D. We now know that D is a thick subcategory that is closed under filtered colimits and (obviously) contains all the cofibrant objects of Ch(Γ). This should mean that it has to be all of Ch(Γ), and that is what we now prove. Proposition 4.4. If X ∈ Ch(Γ), then the map Φ∗ QX → − Φ∗ X is a weak equivalence. Proof. The proposition is just saying that the class D of Propositions 4.2 and 4.3 is all of Ch(Γ). We prove this in three steps. We first show that the complexes S n M are in D, where M is a finitely presented Γ-comodule and S n M denotes the complex whose only non-zero entry is M in degree n. We then show that all finitely presented complexes are in D, and finally, we show that every complex is a filtered colimit of finitely presented complexes, so is in D by Proposition 4.3. For the first step, it is clear that S n A is in D since it is cofibrant. The collection of all M such that S n M is in D is a thick subcategory by Proposition 4.2; by induction, therefore, it contains A/Ik for all k. The Landweber filtration theorem then implies that it contains all finitely presented M. Now suppose X is a finitely presented complex. For the purposes of the present proof, we take this to mean that Xn is finitely presented for all n and 0 for almost all n; this is in fact equivalent to X being a finitely presented object of Ch(Γ) in the categorical sense. We easily prove by induction on the nunber of non-zero entries in X that X ∈ D. Indeed, the base case of one non-zero entry is handled in the preceding paragraph. For the induction step, let X 0 be the subcomplex of X obtained by removing the non-zero entry in

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the largest possible degree. Then X 0 ∈ D by the induction hypothesis, and the quotient X/X 0 ∈ D by the preceding paragraph. Since D is thick, X ∈ D. Now suppose X is an arbitrary complex. Let F /X denote the category of all maps F → − X, where F is a finitely presented complex. This is easily seen to have a small skeleton and to be a filtered category. There is an obvious inclusion functor i : F /X → − Ch(Γ), and an obvious map f : colim i → − X. We claim that f is an isomorphism. To see that f is surjective, choose x ∈ Xn . Since the comodule Xn is a filtered colimit of finitely presented comodules, there is a finitely presented comodule F and a map F → − Xn whose image n contains x. This gives a map of complexes D F → − X whose image contains x, and so f is surjective. To see that f is injective, suppose j : F → − X is an object of F /X and x ∈ Fn has jx = 0. Let K denote the kernel of the map j, so that x ∈ Kn . Now K may not be finitely presented, but at least there is a map F 0 → − Kn from a finitely presented comodule whose image contains x. This corresponds to a map D n F 0 → − K of complexes, which induces an object F/D n F 0 → − X of F /X and a map F → − F/D n F 0 in F /X that sends x to 0. Thus x is 0 in colim i and so f is injective. We can now give the proof of Theorem 4.1. Proof of Theorem 4.1. Suppose f : X → − Y is a weak equivalence in Ch(Γ). Then Qf is a weak equivalence between cofibrant objects, so Φ∗ Qf is a weak equivalence since Φ∗ is a left Quillen functor. But we have a commutative square Φ∗ Qf

Φ∗ QX −−−→ Φ∗ QY y y Φ∗ X −−−→ Φ∗ Y Φ∗ f

where the vertical maps are weak equivalences, by Proposition 4.4. Hence Φ∗ f is a weak equivalence as well. To prove the second part of Theorem 4.1, suppose f is a map in Ch(ΓB ) such that Φ∗ f is a weak equivalence in Ch(Γ). Then Φ∗ Φ∗ f is a weak equivalence in Ch(Γ) by what we have just proved. But f is naturally isomorphic to Φ∗ Φ∗ f by Theorem 2.2. Localization. We have just seen that Φ∗ : Ch(Γ) → − Ch(ΓB ) preserves weak equivalences. But of course it does not reflect weak equivalences, since Φ∗ (A/In+1 ) = 0. We dealt with this problem already in the abelian category world by localizing Γ-comod so as to force 0 → − A/In+1 to be an isomorphism. We now want to do the same thing for chain complexes. More precisely, we define Lfn Ch(Γ) to be the category Ch(Γ) equipped with the model structure that is the Bousfield localization of the homotopy model

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structure with respect to the maps 0 → − sk A/In+1 for all k, where sk A/In+1 denotes the complex which is A/In+1 in degree k and 0 elsewhere. Recall from [Hir03] that this means that a left Quillen functor F : Ch(Γ) → − M defines a left Quillen functor F : Lfn Ch(Γ) → − M if and only if 0 → − (LF )(sk A/In+1 ) is an isomorphism in ho M, where LF denotes the total left derived functor of F . The homotopy category of Lfn Ch(Γ) is the finite localization Lfn Stable(Γ) in the sense of Miller [Mil92] of Stable(Γ) away from A/In+1 . The total left derived functor of the identity, thought of as a functor from Ch(Γ) to Lfn Ch(Γ), is the finite localization functor Lfn on Stable(Γ). Proposition 4.5. The Quillen functor Φ∗ : Ch(Γ) → − Ch(ΓB ) induces a left Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ). Proof. We need to show that (LΦ∗ )(A/In+1 ) = 0. In light of Proposition 4.4, (LΦ∗ )X is naturally isomorphic to Φ∗ X for any X ∈ Ch(Γ). Thus (LΦ∗ )(A/In+1 ) ∼ = Φ∗ (A/In+1 ) = 0. Note that Φ∗ still preserves weak equivalences when thought as a functor from Lfn Ch(Γ), Proposition 4.6. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) preserves weak equivalences, and its right adjoint Φ∗ reflects weak equivalences. Proof. Suppose f : X → − Y is a weak equivalence. Factor f = pi, where i is a trivial cofibration and p is a trivial fibration. Then Φ∗ i is a weak equivalence since Φ∗ is a left Quillen functor. On the other hand, since the trivial fibrations do not change under Bousfield localization, p is a weak equivalence in Ch(Γ). Thus Theorem 4.1 implies that Φ∗ p is a weak equivalence. Hence Φ∗ f = (Φ∗ p)(Φ∗ i) is a weak equivalence, as required. The proof that Φ∗ reflects weak equivalences is the same as the proof of the corresponding part of Theorem 4.1. To prove Theorem C, we will show that Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence. To do so, we will use the following lemma, which is proved in [Hov99, Corollary 1.3.16].

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Lemma 4.7. Suppose F : C → − D is a left Quillen functor of model categories, with right adjoint U . Then F is a Quillen equivalence if and only if the following two conditions hold. (a) U reflects weak equivalences between fibrant objects. (b) The map X → − U RF X is a weak equivalence for all cofibrant X in C, where R denotes fibrant replacement and the map is induced by the unit of the adjunction. We have already seen in Proposition 4.6 that Φ∗ reflects all weak equivalences. We point out that there is a much simpler proof that Φ∗ reflects weak equivalences between fibrant objects; if X is fibrant in Ch(ΓB ), then adjointness implies that π∗ (Φ∗ X) ∼ = π∗ X. The other condition of Lemma 4.7 is harder to check. Here are the main points of the argument. (a) We first show, using the fact that Φ∗ preserves filtered colimits, that it suffices to show that A → − Φ∗ (LB) is an Lfn -equivalence, where LB denotes the cobar resolution of B. (b) A Bousfield class argument that shows that it suffices to prove that − vk−1 A/Ik ∧ QΦ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism for all 0 ≤ k ≤ n, where Q denotes cofibrant replacement. (c) We show that, although Φ∗ (LB) is not cofibrant, it is still nice enough that it suffices to check that − vk−1 A/Ik ∧ Φ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism. (d) It was proved in [Hov02b] that the Hopf algebroid (vk−1 A/Ik , vk−1 Γ/Ik ) is weakly equivalent to (vk−1 B/Ik , vk−1 ΓB /Ik ). Hence it suffices to show that v −1 B/Ik → − v −1 B/Ik ∧ Φ∗ Φ∗ (LB) ∼ = v −1 B/Ik ∧ LB k

k

k

is a homotopy isomorphism, and this is clear. We now fill in the details of this argument, beginning with Step (a). Proposition 4.8. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence if and only if the map A→ − Φ∗ (LB) is an Lfn -equivalence. Recall that LB denotes the cobar resolution of B as a ΓB -comodule. Before proving this proposition, we need a lemma.

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Lemma 4.9. The total right derived functor RΦ∗ : Stable(ΓB ) → − Lfn Stable(Γ) preserves coproducts. Proof. First note that because Lfn is a finite localization, it is in particular smashing [Mil92]. Thus the coproduct in Lfn Stable(Γ) is the same as the coproduct in Stable(Γ). Also note that a fibrant replacement in Stable(ΓB ) is given by LB ∧ (−), which clearly preserves coproducts. Hence, it suffices to show that Φ∗ : ΓB -comod → − Γ-comod ∗ preserves coproducts. Since Φ certainly preserves finite coproducts, and any coproduct is a filtered colimit of finite coproducts, it suffices to show that Φ∗ preserves all filtered colimits. This follows from the fact that Ln = Φ∗ Φ∗ preserves filtered colimits (Theorem 2.8); more details can be found in [HS03b]. Proof of Proposition 4.8. Lemma 4.7 tells us that if Φ∗ is a Quillen equivalence, then A → − Φ∗ RΦ∗ A must be a weak equivalence in Lfn Ch(Γ). Since Φ∗ A = B, and since LB is a fibrant replacement for B in Ch(ΓB ), we conclude that A → − Φ∗ (LB) is an Lfn -equivalence. Conversely, suppose A → − Φ∗ (LB) is an Lfn -equivalence. By Lemma 4.7 and Proposition 4.6, it suffices to show that X → − Φ∗ RΦ∗ X is an Lfn -equivalence for all cofibrant X. This is equivalent to proving that η

X X −→ (RΦ∗ )(LΦ∗ )X

is an isomorphism in Lfn Stable(Γ) for all X, where RΦ∗ denotes the total right derived functor of Φ∗ and LΦ∗ denotes the total left derived functor of Φ∗ . Let D denote the full subcategory of Lfn Stable(Γ) of those X such that ηX is an isomorphism. By hypothesis, D contains Lfn A. Since LΦ∗ and RΦ∗ both preserve exact triangles, D is a thick subcategory. As a left adjoint, LΦ∗ preserves coproducts, and Lemma 4.9 assures us that RΦ∗ also preserves coproducts. Thus D is a localizing subcategory. In any monogenic stable homotopy category, the only localizing subcategory that contains the unit is the whole category. We are now reduced to showing that A → − Φ∗ (LB) is an Lfn -equivalence. The theory of Bousfield classes gives us the following lemma. Lemma 4.10. A map f of cofibrant objects in Ch(Γ) is an Lfn -equivalence if and only if vk−1 A/Ik ∧ f is a weak equivalence for all k with 0 ≤ k ≤ n. Proof. As usual, let < X > denote the Bousfield class of X in Stable(Γ), thought of as the collection of all Y in Stable(Γ) such that X ∧ Y = 0. The cofiber sequences vk A/Ik −→ A/Ik → − A/Ik+1

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imply that < A/Ik >=< vk−1 A/Ik > ∨ < A/Ik+1 >, by [Rav84, Lemma 1.34]. Thus, we find n _ < A >= < vk−1 A/Ik > ∨ < A/In+1 > . k=0

As in the usual stable category, this implies that Lfn is localization Ln homotopy −1 with respect to k=0 vk A/Ik . It follows that f is an Lfn -equivalence if and only if vk−1 A/Ik ∧L f is a weak equivalence for all k with 0 ≤ k ≤ n, where ∧L denotes the total left derived functor of ∧. Recall from Theorem 3.3 that if X is cofibrant, then X ∧ (−) preserves homotopy isomorphisms. It follows that, if X is cofibrant, (−) ∧L X ∼ = (−) ∧ X in Stable(Γ). Hence, if f is a map of cofibrant objects, then f is an Lfn -equivalence if and only if vk−1 A/Ik ∧ f is a weak equivalence for all k with 0 ≤ k ≤ n. By combining Proposition 4.8 with Lemma 4.10, we get the following corollary, which is Step (b) of the argument on page 195. Corollary 4.11. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence if and only if the map − vk−1 A/Ik ∧ QΦ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism for all 0 ≤ k ≤ n, where Q denotes a cofibrant replacement functor in Ch(Γ). To accomplish Step (c) of the argument on page 195, we need to know something about Φ∗ (LB). Lemma 4.12. We have TorjA (A/Ik , Φ∗ (LB)m ) = 0 for all j > 0, k ≥ 0, and m ∈ Z. Proof. Recall that ⊗B m

(LB)−m = ΓB ⊗B ΓB for m ≥ 0, and is 0 otherwise. Here ΓB is the cokernel of the left unit ηL : B → − ΓB . Since this map is split as a map of B-modules, ΓB is a flat ⊗B m B-module. Therefore ΓB is also a flat B-module, and hence a filtered colimit of projective B-modules. Now, we have ⊗B m Φ∗ (LB)−m = Γ ⊗A ΓB . j Since TorA (A/Ik , −) commutes with filtered colimits, it suffices to show that TorjA (A/Ik , Γ ⊗A M) = 0

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for all j > 0, all k, and all projective B-modules M. But then we can easily reduce to the case M = B, so we must show that TorjA (A/Ik , Γ ⊗A B) = 0 for all j > 0 and all k. We prove this by induction on k, using the exact sequences vk 0→ − A/Ik −→ A/Ik → − A/Ik+1 → − 0. We are reduced to showing that vk is not a zero-divisor on (Γ ⊗A B)/Ik . Since Ik is invariant, this is the same as showing that vk is not a zero-divisor on Γ ⊗A (B/Ik ). Since vk is itself invariant modulo Ik and Γ is flat over A, this is in turn equivalent to showing that vk is not a zero-divisor on B/Ik . This follows because B is Landweber exact. With this lemma in hand, we can carry out Step (c) of our argument. Proposition 4.13. The map vk−1 A/Ik ∧ QΦ∗ (LB) → − vk−1 A/Ik ∧ Φ∗ (LB) is a homotopy isomorphism in Ch(Γ). In particular, Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence if and only if − vk−1 A/Ik ∧ Φ∗ (LB) vk−1 A/Ik → is a homotopy isomorphism for all 0 ≤ k ≤ n. Proof. Let q denote the map q : QΦ∗ (LB) → − Φ∗ (LB). Because homotopy commutes with filtered colimits, it suffices to show that A/Ik ∧ q is a homotopy isomorphism for all k. Now, q is a trivial fibration in the homotopy model structure, so we have a short exact sequence of complexes q

0→ − K→ − QΦ∗ (LB) → − Φ∗ (LB) → − 0. The long exact sequence in homotopy implies that πtA (K) = 0 for all n. Lemma 4.12 implies that we have a short exact sequence 0→ − A/Ik ∧ K → − A/Ik ∧ QΦ∗ (LB) → − A/Ik ∧ Φ∗ (LB) → − 0 for all k. The long exact sequence in homotopy implies that we need only check that πtA (A/Ik ∧ K) = 0 for all n. To se this, note that the short exact sequence defining K realizes Km as the first syzygy of Φ∗ (LB)m , since (QΦ∗ (LB))m is projective over A. Hence Lemma 4.12 implies that TorjA (A/Ik , Km ) = 0

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for all j > 0. Hence we have short exact sequences 0→ − A/Ik ∧ K → − A/Ik ∧ K → − A/Ik+1 ∧ K → − 0 for all k. The long exact sequence in homotopy and induction on k now complete the proof. The final step of the argument on page 195 requires us to know more about the Hopf algebroids (vk−1 A/Ik , vk−1 Γ/Ik ) and (vk−1 B/Ik , vk−1 ΓB /Ik ). Lemma 4.14. Let (C, Σ) denote (vk−1 A/Ik , vk−1 Γ/Ik ), and let (CB , ΣB ) denote (vk−1 B/Ik , vk−1 Γ/Ik ). Then: (a) Both (C, Σ) and (CB , ΣB ) are Adams Hopf algebroids; (b) The stable homotopy categories Stable(Σ) and Stable(ΣB ) are monogenic; (c) If 0 ≤ k ≤ n, Φ induces a weak equivalence of Hopf algebroids Φ : (C, Σ) → − (CB , ΣB ). Proof. For part (a), we know that Γ is a filtered colimit of comodules that are finitely generated projective A-modules. By tensoring with vk−1 A/Ik , we se that Σ is a filtered colimit of comodules that are finitely generated projective C-modules, and so (C, Σ) is an Adams Hopf algebroid. Similarly, (CB , ΣB ) is an Adams Hopf algebroid. For part (b), the proof is again the same for (C, Σ) and (CB , ΣB ), so we concentrate on (C, Σ). We will use Proposition 3.6, so we need to show that every dualizable Σ-comodule is in the thick subcategory generated by C. We will do this by showing that every finitely presented Σ-comodule has a Landweber filtration. To do so, we will use the Hopf algebroid (vk−1 A, vk−1 Γvk−1 ), obtained by inverting vk and ηR vk . A Σ-comodule M is just a (vk−1 Γvk−1 )-comodule on which Ik acts trivially. Since Ik is finitely generated, M is finitely presented if and only if it is finitely presented as a vk−1 Γvk−1 -comodule. Since vk−1 A is Landweber exact of height k, Theorem 2.11 gives us a Landweber filtration of M as a vk−1 Γvk−1 -comodule in which each filtration quotient is isomorphic to vk−1 A/Ij for some j ≤ k. Since M is killed by Ik , in fact each filtration quotient must be isomorphic to vk−1 A/Ik , giving us our Landweber filtration of M as a Σ-comodule. Part (c) is a special case of Theorem E of [Hov02b]. Lemma 4.14 allows us to carry out the final step of our argument. Proposition 4.15. Suppose f is a map in Ch(Γ), and 0 ≤ k ≤ n. Then vk−1 A/Ik ∧f is a homotopy isomorphism in Ch(Γ) if and only if vk−1 B/Ik ∧Φ∗ f is a homotopy isomorphism in Ch(ΓB ). Proof. Let C = vk−1 A/Ik and let Σ = vk−1 Γ/Ik , as in Lemma 4.14. The category of Σ-comodules is just the full subcategory of Γ-comodules on which Ik acts trivially and vk acts invertibly. This follows from the fact that Ik is

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invariant and vk is primitive modulo Ik . Thus, if X is a complex in Ch(Σ), we can also think of X as a complex in Ch(Γ). As such, X has homotopy groups π∗C (X) and π∗A (X). We claim that these are naturally isomorphic. Indeed, one can easily check that LC, the cobar complex of (C, Σ), is just vk−1 LA/Ik . Hence LC ∧ X is naturally isomorphic to LA ∧ X. From this, one can easily check the desired isomorphism. Therefore, using parts (a) and (b) of Lemma 4.14, we conclude that vk−1 A/Ik ∧ f is a homotopy isomorphism in Ch(Γ) if and only if it is a weak equivalence in Ch(Σ). Similarly, let CB = vk−1 B/Ik and ΣB = vk−1 ΓB /Ik . We find that vk−1 B/Ik ∧ Φ∗ f is a homotopy isomorphism in Ch(ΓB ) if and only if it is a weak equivalence in Ch(ΣB ). Now, use part (c) of Lemma 4.14 and Theorem 3.4 to conclude that vk−1 A/Ik ∧ f is a weak equivalence in Ch(Σ) if and only if v −1 B/Ik ⊗ −1 (v −1 A/Ik ∧ f ) ∼ = v −1 B/Ik ∧ Φ∗ f k

vk A/Ik

k

k

is a weak equivalence in Ch(ΣB ).

We can now complete the proof of Theorem C, which we first restate in stronger form. Theorem 4.16. The Quillen functor Φ∗ : Lfn Ch(Γ) → − Ch(ΓB ) is a Quillen equivalence. Furthermore, Φ∗ and its right adjoint Φ∗ preserve and reflect weak equivalences. Proof. Proposition 4.13 and Proposition 4.15 imply that, to check that Φ∗ is a Quillen equivalence, we only need to check that the map − vk−1 B/Ik ∧ Φ∗ Φ∗ (LB) vk−1 B/Ik → is a homotopy isomorphism for 0 ≤ k ≤ n. But Φ∗ Φ∗ is natually isomorphic to the identity functor by Theorem 2.2. Hence we need only check that the map − vk−1 B/Ik ∧ LB vk−1 B/Ik → is a homotopy isomorphism, which follows from Proposition 3.2(d). We have already seen that Φ∗ preserves weak equivalences and that Φ∗ reflects them in Proposition 4.6. Suppose f is a map in Ch(Γ) such that Φ∗ f is a weak equivalence. Since Φ∗ preserves weak equivalences, it follows that Φ∗ Qf is a weak equivalence. But, since Φ∗ is a Quillen equivalence, it must reflect weak equivalences between cofibrant objects by [Hov99, Corollary 1.3.16]. Hence Qf is a weak equivalence, and so f is a weak equivalence. This proves that Φ∗ reflects weak equivalences. Now suppose g is a weak equivalence in Ch(ΓB ). By Theorem 2.2, g is naturally isomorphic to Φ∗ Φ∗ g. Since Φ∗ reflects weak equivalences, we conclude that Φ∗ g is a weak equivalence.

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We point out that it is possible to further localize the Quillen equivalence in Theorem 4.16 to obtain a Quillen equivalence Φ∗ : Ln Ch(Γ) → − Ln Ch(ΓB ) where Ln is Bousfield localization with respect to HE(n) is the first case, and ordinary homology H in the second. The change of rings theorem. In this final part, we show how our work implies the Miller-Ravenel change of rings theorem. To begin with, here is our generic change of rings theorem. Recall our notational conventions: (A, Γ) = (BP∗ , BP∗ BP ), B = E(n)∗ , and ΓB = E(n)∗ E(n). Theorem 4.17. Suppose X ∈ Ch(Γ) and Y is an Lfn -local object of Stable(Γ). Then Stable(Γ)(X, Y )∗ ∼ = Stable(ΓB )(Φ∗ X, Φ∗ Y )∗ . Proof. First of all, since Φ∗ preserves weak equivalences, we have Stable(ΓB )(Φ∗ X, Φ∗ Y ) ∼ = Stable(ΓB )(Φ∗ QX, Φ∗ QY ). Also, by Theorem 4.16, we have Stable(ΓB )(Φ∗ QX, Φ∗ QY ) ∼ = (Lfn Stable(Γ))(X, Y ). Since Y is already Lfn -local, we have (Lfn Stable(Γ))(X, Y ) ∼ = Stable(Γ)(X, Y ), as required.

We claim that this corollary captures the Miller-Ravenel change of rings theorem. To see this, we need the following lemma. Lemma 4.18. Suppose N is a Γ-comodule with Ln N = N and Lin N = 0 for i > 0. Then N is an Lfn -local object of Stable(Γ). Given this lemma, we immediately get the following corollary, which is Theorem D and more like the usual change of rings theorems. Corollary 4.19. Suppose M is a finitely presented BP∗ BP -comodule, and N is a BP∗ BP -comodule such that Ln N = N and Lin N = 0 for all i > 0. Then Ext∗∗ (M, N ) ∼ = ExtE(n) E(n) (E(n)∗ ⊗BP M, E(n)∗ ⊗BP N ). BP∗ BP

∗

∗

∗

This corollary includes both the Miller-Ravenel change of rings theorem [MR77], by taking N with N = vn−1 N , and the change of rings theorem of the author and Sadofsky [HS99], by taking N with vj−1 N = N for some j ≤ n. Here we are using Theorem 2.8(e) to verify the hypothesis of Corollary 4.19. Proof. Simply apply Theorem 4.17, using Lemma 4.18 to see that N is Lfn local, and Proposition 3.8(c) to identify the groups in question as Ext groups.

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We still owe the reader a proof of Lemma 4.18, which, incidentally, is presumably a special case of a spectral sequence that computes H∗ (Lfn X) for X ∈ Stable(Γ) from the derived functors Lin H∗ X, in analogy to the spectral sequence of Theorem 2.9. The converse of Lemma 4.18 is true as well, though we do not need it. Proof of Lemma 4.18. Note that, by definition, N is Lfn -local if and only if Stable(Γ)(A/In+1 , N )∗ = 0. This is equivalent to ExtiΓ (A/In+1 , N ) = 0 for all i, by Proposition 3.8(c). Now ExtiΓ (A/In+1 , N ) = 0 for i = 0, 1 if and only if Ln N = N , by Theorem 2.8(a). Suppose in addition that Lin N = 0 for i > 0. We claim that we can find cosyzygies Cj of N in the category of Γ-comodules such that Ln Cj = Cj and Lin Cj = 0 for i > 0. If we assume this, then for i > 0 we have Exti (A/In+1 , N ) ∼ = Ext1 (A/In+1 , Ci−1 ) = 0. Γ

Γ

We now construct the cosyzygies Cj by induction on j, taking C0 = N . Suppose we have constructed Cj . Since Ln Cj = Cj , Cj has no vn -torsion. It follows that the injective hull, or indeed any essential extension of Cj , has no vn -torsion. Therefore, we can find an short exact sequence 0→ − Cj → − Ij → − Cj+1 → − 0 of Γ-comodules where Ij is an injective comodule with no vn -torsion. In particular, Ln Ij = Ij by Theorem 2.8(a). If we apply Ln to this sequence, we find that Ln Cj+1 = Cj+1 and Lin Cj+1 = 0 for i > 0, completing the induction step. References [Ada74] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill., 1974, Chicago Lectures in Mathematics. MR 53 #6534 [Bek00] Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475. MR 1 780 498 [Ben98] D. J. Benson, Representations and cohomology. I, second ed., Cambridge University Press, Cambridge, 1998, Basic representation theory of finite groups and associative algebras. MR 99f:20001a [Bou79] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281. [GH00] Paul G. Goerss and Michael J. Hopkins, Andr´e-Quillen (co)-homology for simplicial algebras over simplicial operads, Une d´egustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), Amer. Math. Soc., Providence, RI, 2000, pp. 41–85. MR 2001m:18012 [GM95] J. P. C. Greenlees and J. P. May, Completions in algebra and topology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 255–276. MR 96j:55011

Chromatic phenomena in the algebra of BP∗ BP -comodules [Gre93] [Hir03]

[Hol01] [Hov99] [Hov02a] [Hov02b] [HPS97] [HS99]

[HS03a] [HS03b] [Jar01]

[JY80]

[Lan76] [Mil92]

[MR77]

[Rav84] [Rav86]

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J. P. C. Greenlees, K-homology of universal spaces and local cohomology of the representation ring, Topology 32 (1993), no. 2, 295–308. MR 94c:19007 Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1 944 041 Sharon Hollander, Homotopy theory for stacks, Ph.D. thesis, MIT, 2001. Mark Hovey, Model categories, American Mathematical Society, Providence, RI, 1999. MR 99h:55031 Mark Hovey, Homotopy theory of comodules over a Hopf algebroid, preprint, 2002. Mark Hovey, Morita theory for Hopf algebroids and presheaves of groupoids, Amer. J. Math. 124 (2002), 1289–1318. Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 98a:55017 Mark Hovey and Hal Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, J. London Math. Soc. (2) 60 (1999), no. 1, 284–302. MR 2000h:55017 Mark Hovey and Neil P. Strickland, Comodules and Landweber exact homology theories, preprint, 2003. , Local cohomology of BP ∗ BP -comodules, preprint, 2003. J. F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology Homotopy Appl. 3 (2001), no. 2, 361–384 (electronic), Equivariant stable homotopy theory and related areas (Stanford, CA, 2000). MR 1 856 032 David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), no. 1, 117– 136. MR 81b:55010 Peter S. Landweber, Homological properties of comodules over M U ∗ (M U) and BP∗ (BP), Amer. J. Math. 98 (1976), no. 3, 591–610. MR 54 #11311 H. R. Miller, Finite localizations, Boletin de la Sociedad Matematica Mexicana 37 (1992), 383–390, special volume in memory of Jos´e Adem, in book form, edited by Enrique Ram´ırez de Arellano. Haynes R. Miller and Douglas C. Ravenel, Morava stabilizer algebras and the localization of Novikov’s E2 -term, Duke Math. J. 44 (1977), no. 2, 433–447. MR 56 #16613 Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR 85k:55009 , Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press Inc., Orlando, FL, 1986. MR 87j:55003

Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: [email protected]

NUMERICAL POLYNOMIALS AND ENDOMORPHISMS OF FORMAL GROUP LAWS KEITH JOHNSON

Abstract. Endomorphism rings of formal group laws are used to construct families of polynomials which take integer values at the integers. Applications are made to the study of Hopf algebroids of stable cooperations for generalized homology theories and to the study of generalized Bernoulli numbers.

Primary 13B25, 14L05; Secondary 55N20, 11B68

A numerical polynomial is a polynomial with coefficients in Q which takes integer values at the integers, or more generally, with coefficients in a number field K taking values in A, the ring of integers of K, when evaluated at elements of A. The study of such polynomials has a long history and a substantial literature (for example [5], [17], [21], [22]). The connection with formal group laws and topology stems from the results of [1] where the notion of a stably numerical Laurent polynomial was introduced: a Laurent polynomial f (x) ∈ Q[x, x−1 ] is stably numerical if xk f (x) is numerical for some integer k. The main result of [1] is that the Hopf algebra of degree 0 stable cooperations for complex K-theory, K0 K, is isomorphic to the algebra of stably numerical Laurent polynomials for Z. This is essentially a result connecting stably numerical Laurent polynomials with the multiplicative formal group law, and this connection was described explicitly in [9]. The point of the current paper is that this method is applicable to a large class of formal group laws, that its use provides a large supply of stably numerical Laurent polynomials, both for Z and for the integers in other number fields, and that when it is applied to the formal group laws associated to elliptic curves new and useful information about the Hopf algebroid of stable cooperations for elliptic homology results. It also gives information about the generalized Bernoulli numbers introduced in [20]. The paper is organized as follows: In §1 we recall the basic properties of formal group laws that we require and, for a given formal group law, F , construct a sequence of numerical polynomials, {βi }. In §2 we relate these polynomials for certain formal group laws to the set of generators for the Hopf algebroid of stable cooperations for elliptic homology constructed in [12] and so deduce an integrality condition for elements of that algebroid. In §3 we relate generalized Bernoulli numbers to the polynomials constructed 204

Numerical Polynomials and Endomorphisms of Formal Group Laws 205 in §1 and deduce results related to the classical Von Staudt theorem. In the special case of formal group laws associated to elliptic curves we recover some number-theoretic results due to Carlitz ([6], [7], [8]). 1. Formal Group Laws Let R denote a commutative ring with unit. We will make use of the following definitions and terminology concerning formal group laws: Definition 1.1. A formal group law over R is a power series F (x, y) ∈ R[[x, y]] with the properties F (0, x) = F (x, 0) = x, F (x, F (y, z)) = F (F (x, y), z) and F (x, y) = F (y, x) Definition 1.2. If R is torsion-free then the logarithm of a formal group law, F , over R is the power series logF (x) ∈ R ⊗ Q[[x]] given by Z x dt logF (x) = . ∂F 0 ∂y (t, 0) The inverse (with respect to composition) of log F (x) is the exponential function of F , denoted expF (x). These series have the property that F (x, y) = expF (logF (x) + logF (y)). Definition 1.3. A homomorphism between formal group laws, F and G, over R is a power series α(x) ∈ R[[x]] such that α(0) = 0 and α(F (x, y)) = G(α(x), α(y)) Let J(α) denote the coefficient of x in α(x). α is an isomorphism if J(α) is a unit in R and it is a strict isomorphism if J(α) = 1. Definition 1.4. An endomorphism of a formal group law, F , is a homomorphism of F with itself, i.e. a power series α(x) ∈ R[[x]] with the property α(F (x, y)) = F (α(x), α(y)). The set of endomorphisms forms a ring, using F for addition and composition for multiplication, which we denoted EndR (F ). Any formal group law, F , has a family of endomorphisms defined inductively by [1]x = x, [2]x = F (x, x), [n]x = F ([n − 1]x, x) and F ([−n]x, [n]x) = 0. These define a homomorphism Z → EndR (F ). If R is torsion-free then all endomorphisms of F are of the form α(x) = expF (a logF (x))

206

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where a = J(α). In particular if R is torsion-free then [n]x = expF (n logF (x)) Definition 1.5. Let R be an algebra over a ring A. A formal A-module is a formal group law, F , over R together with a homomorphism A → EndR (F ) extending the homomorphism Z → EndR (F ) and whose composition with J gives the A-algebra structure map for R. The most common example of a formal A-module occurs when R is an algebra over Z(p) , the integers localized at p. In this case F is always a formal Z(p) -module. (Since [k](x) ≡ kx mod x2 the series [k](x) has an inverse in Z(p) [[x]] if k is an integer prime to p. Thus F has endomorphisms [1/k](x) and so, for any integer l, endomorphisms [l/k](x) = [l]([1/k](x))). In general if F is a formal A-module we will use [a]x to denote the endomorphism of F associated to a. Two examples of formal group laws which we will be particularly interested in are: (a) The multiplicative formal group law F (x, y) = x + y + xy over R = Z. (b) The Euler formal group law associated to the Jacobi elliptic curve y 2 = 1 − 2δx2 + x4 = Sδ, (x): q q F (x, y) = (x Sδ, (y) + y Sδ, (x))/(1 + x2 y 2 ) over R = Z[1/2][δ, ].

Definition 1.6. Let R be a torsion free A-algebra. The polynomial p(w) ∈ (R ⊗ Q)[w] is (A, R)-numerical if p(a) ∈ R whenever a ∈ A. The Laurent polynomial p(w) ∈ (R⊗Q)[w, w −1 ] is stably (A, R)-numerical if there is an integer k such that w k p(w) is (A, R)-numerical. Similarly p(w1 , . . . , wn ) ∈ (R⊗ Q)[w1 , . . . , wn ] is (A, R)-numerical if p(a1 , . . . , an ) ∈ R whenever a1 , . . . an ∈ A and is stably (A, R)-numerical if there are integers k1 , . . . , kn such that (w1k1 . . . wnkn )p(w1 , . . . , wn ) is (A, R)-numerical. If R = A then we will contract these terms to A-numerical and stably Anumerical respectively, and to simply numerical if A can be inferred from the context. Given a formal group law or A-module, F , defined over a torsion free ring R, we may produce a sequence of (Z, R)-numerical or (A, R)-numerical polynomials by expanding the power series expF (w logF (x)) in powers of x: expF (w logF (x)) =

∞ X i=1

βi xi

Numerical Polynomials and Endomorphisms of Formal Group Laws 207 The βi ’s are easily seen to be polynomials in (R ⊗ Q)[w] (with βi of degree ≤ i) and, in view of the fact that for a ∈ Z (or for a ∈ A if F is a formal A-module) ∞ X βi (a)xi = [a](x) ∈ R[x] i=1

we see that the βi ’s are (Z, R) or (A, R)-numerical. In the case of Lazard’s universal formal group law these polynomials were defined, using the umbral calculus, in [25]. In a similar way we may produce stably (Z, R)-numerical or stably (A, R)numerical polynomials by expanding the series (1/v) expF ((v/u) logF (ux)) : (1/v) exp ((v/u) log (ux)) = F

F

∞ X

αi (u, v)xi

i=1

Since the αi ’s are homogeneous and αi (1, w) = (1/w)βi (w) the αi ’s are clearly stably (Z, R)-numerical or (A, R)-numerical. This may appear to be only a minor variation but it has a feature of particular interest in that the αi ’s are the coefficients of a strict isomorphism i.e. α1 (u, v) = 1). Given a formal group law F over a ring R we may, for any unit u ∈ R ∗ , define a related formal group law Fu by Fu (x, y) =

1 F (ux, uy) u

This formal group law is, of course, isomorphic to F (via α(x) = ux) and so any two such formal group laws Fu and Fv are isomorphic. If the invertible power series ux and vx lie in the endomorphism ring of F (for example if R is a Z(p) -module and u, v are integers prime to p) then among the isomorphisms from Fu to Fv is v v −1 [ ](ux) u which is the composition of isomorphisms from Fu to F and F to Fv with an automorphism of F . This particular isomorphism is the unique strict isomorphism from Fu to Fv and X v v [ ](ux) = αi (u, v)xi+1 u i=1 ∞

−1

A particular case of this (originally appearing in [9](see also [12])), in which the coefficient polynomials, αi , are familiar is that of the multiplicative group law x + y + xy. The logarithm in this case is the analytic log logF (x) = ln(1 + x) =

∞ X (−1)i−1 xi i=1

i

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and so expF (x) = ex − 1. Thus v −1 [v/u](ux) = v −1 expF ((v/u) logF (ux)) = v −1 (e(v/u)ln(1+ux) − 1) = v −1 ((1 + ux)v/u − 1) ∞ X v/u −1 =v (ux)i i i=1 and so αi (u, v) = (v − u)(v − 2u) . . . (v − (i − 1)u)/i! is the i-th homogeneous binomial polynomial. An alternative approach to the integrality and divisibility questions concerning formal group laws considered here is by way of the umbral calculus. A development of this material for similar applications in algebraic topology can be found in [24] and [11]. 2. Stable Cooperations and Elliptic Homology If E∗ ( ) is a complex oriented homology theory then there is associated to it a formal group law, FE over the coefficient ring E∗ . A basic problem is to calculate E∗ E, the Hopf algebroid of stable cooperations for E∗ ( ) which in some cases may be described in terms of FE . If we restrict our attention to those theories possessing a Conner-Floyd isomorphism E∗ ( ) ∼ = E∗ ⊗M U∗ MU∗ ( ) then E∗ E may be identified with the subalgebra of E∗ ⊗ E∗ ⊗ Q generated L over E∗ by the coefficients of the series expLFE (logR FE (x)). Here expFE and logR FE are the exponential and logarithm series of the two formal group laws L FE = ηL∗ FE and FER = ηR∗ FE induced from FE by the left and right counit maps ηL , ηR : E∗ → E∗ E. The Hopf algebroid structure of E∗ ⊗ E∗ ⊗ Q over E∗ is determined by ηL = id ⊗ 1 and ηR = 1 ⊗ id. A more detailed account of this approach to E∗ E can be found in section 1 of [12]. If the homology theory is complex K-theory, K∗ ( ) then K∗ = Z[t, t−1 ], the associated formal group law is the graded multiplicative law FK = x+y +txy, FKL and FKR are Fu and Fv respectively where F is the ungraded multiplicative law and u = ηL (t) and v = ηR (t). It follows that K∗ K is the subalgebra of K∗ ⊗ K∗ ⊗ Q generated over K∗ by the homogeneous binomial polynomials. For elliptic homology, Ell∗ ( ), as defined in [18], Ell∗ = Z[1/2][δ, , ∆−1 ] were ∆ is the discriminant of the associated Jacobi elliptic curve, the associated formal group law is the Euler formal group law of section 1 which we will denote FEll , the logarithm is Z x dt p logEll (x) = Sδ, (t) 0

Numerical Polynomials and Endomorphisms of Formal Group Laws 209 −1 and Ell∗ Ell is a subalgebra of Ell∗ ⊗ Ell∗ ⊗ Q = Q[δL , L , δR , R , ∆−1 L , ∆R ] where δL = ηL (δ) etc. If we form the series

m(x) =

expLEll (logR Ell (x))

=

∞ X

mi (δL , δR , L , R )xi

i=1

then the coefficients, which are polynomials in δL , δR , L , and R , generate Ell∗ Ell over Ell∗ . The Hopf algebroid Ell∗ Ell has been studied in [12], [19] and [3]. To obtain integrality conditions that these polynomials must satisfy we relate them to formal group laws of the form Fu considered in section 1 by defining, for integers k, l, homomorphisms φ : Ell∗ → Z[1/2] and Φ : Ell∗ → Z[1/2][t] by φ(δ) = k, φ() = l, Φ(δ) = kt2 , Φ() = lt4 . These maps have the property Φ∗ FEll = (φ∗ FEll )t Proposition 2.1. For any integers k, l the polynomials mi (ku2 , kv 2 , lu4 , lv 4 ) ∈ Q[u, v] are stably Z[1/2]-numerical. Proof. If we make Q[u, v] into a Hopf algebroid over Q[t] by taking u = ηL (t), v = ηR (t) then Φ∗ extends to a map of Hopf algebroids Φ∗ ⊗ Φ∗ : Ell∗ ⊗ Ell∗ ⊗ Q → Q[u, v] such that mi (ku2 , kv 2 , lu4 , lv 4 ) = (Φ∗ ⊗ Φ∗ )(mi ) We also have ∞ X (Φ∗ ⊗ Φ∗ )(mi )xi = expΦ∗ FL (logΦ∗ FR (x)) i=1

= exp(φ∗ F )v (log(φ∗ F )u (x)) 1 v = expφ∗ F ( logφ∗ F (ux)) v u and so the coefficient polynomials of this series are stably Z[1/2]-numerical by the results of section 1. For special values of k, l we can say more. An elliptic curve having the property that its endomorphism ring, A, is not a subring of Q is said to have complex multiplication. For curves of this sort the associated formal group law will, after extension of its ring of definition, be a formal A-module. There is an extensive literature devoted to this subject including [14], in which the possibilities for the endomorphism ring are enumerated. It is known that in characteristic 0 such curves have endomorphism rings which are orders in imaginary quadratic field extensions of Q, and also that there are only 13 isomorphism classes of such curves whose j-invariants are rational. These are catalogued in [13], p. 376. Since the j-invariant of the curve associated to the Euler formal group law is a rational function of δ and , we may

210

Keith Johnson

enumerate some values of k, l for which we obtain an extension of the previous proposition. Proposition 2.2. For the following values of δ, the endomorphism ring of the Jacobi elliptic curve y 2 = 1 − 2δx2 + x4 is the order Z + τ Z in the field Q(τ ). δ τ j(τ ) 0 3 −3 −21 2 3 6 21 18 7 33 21 513

1 1 −3 −7 2 12 32 448 √ 162 − 90i √ 7 57 − 40 2√ 417 + 240√3 249 − 176 2 √ 129537 + 48960i 7

i i √ (1 + i√3)/2 (1 √+ i 7)/2 i√2 i 3 2i√ i 7√ 1√+ i 7/2 i√2 i 3 2i√ i 7

1728 1728 0 −3375 8000 54000 287496 16581375 −3375 8000 54000 287496 16581375

Corollary 2.3. For k, l taking the values of δ, listed in the previous proposition the polynomials mi (ku2 , kv 2 , lu4 , lv 4 ) ∈ Q[1/2, τ ][u, v] are stably numerical for the ring Z + τ Z. The values of k, l given in corollary 2.3 are not the only ones for which the polynomials mi are stably numerical for a ring larger than Z. There are irrational values of the j-invariant for which the associated Euler curve has complex multiplication and so the same phenomena occurs however these are not so easy to classify. A consequence of this is that the conditions on the polynomials mi we have described cannot completely characterize the Hopf algebroid Ell∗ Ell. There are two other approaches to Ell∗ Ell which do give complete characterizations, however. One, due to Baker ([3]) uses generalized modular forms while the other, due to Laures ([19]) uses 2 variable q-series. This raises the possibility of using these essentially topological results to obtain number theoretic information about elliptic curves by way of further integrality conditions that the polynomials mi might satisfy. 3. Bernoulli Numbers Definition 3.1 ([20]). The Bernoulli numbers of a formal group law F over R are the elements BnF of R defined by ∞ X x xn = BnF expF (x) n=0 n!

Numerical Polynomials and Endomorphisms of Formal Group Laws 211 It follows easily from this definition that these numbers can be computed inductively in terms of the coefficients of the exponential series exp F (x) = P n xn /n! using the recurrence ( n X n+1 1 if n = 0 n−j BjF = j 0 if n > 0 j=0

Arithmetic properties and topological applications of these numbers have been studied in [10], [18], [24]. For the multiplicative formal group law they specialize to the classical Bernoulli numbers which we will denote simply by Bn . In this case a connection between these numbers and numerical polynomials is given by: Proposition 3.2.

Bn (w n 2n

− 1) is stably numerical and not divisible.

One interpretation of this result is as equating the denominator of Bn /2n and the largest integer m(n) (ordered by divisibility) for which (w n −1)/m(n) is stably numerical. The first of these is given by Von Staudt’s theorem while the second can be calculated one prime factor at a time using the structure of (Z/pZ)∗ . The natural generalization of this result requires a preliminary definition: Definition 3.3. If F is a formal group law over the torsion-free ring R, then the algebra of F -numerical polynomials, which we will denote A F , is the subalgebra of (R ⊗ Q)[w] generated by the polynomials {βi (w), i = 1, 2, . . . } associated to F by the construction given in §1. The algebra of stably F numerical polynomials is the subalgebra AF [w −1 ]. ¿From §1 it is clear that elements of AF are numerical for Z or for A if F is a formal A-module. AF need not, however, equal the full algebra of polynomials numerical for Z or A. For example if F is the additive formal group law, then AF contains only constant polynomials. Based on this definition we have the following generalization of Proposition 3.2. In the case of formal group laws of singular elliptic curves an equivalent result may be found in [8]. Proposition 3.4.

F Bn n

(w n − 1) is stably F-numerical.

Proof. First note that, from the definition of the BnF ’s we have: ∞ ∞ w 1 1 X BnF w n xn 1 X BnF xn − = − expF (wx) expF (x) x n=0 n! x n=0 n! =

∞ X BF n

n=1

Next note that

n

(w n − 1)

xn−1 (n − 1)!

exp (x) − w1 expF (wx) 1 w − = 1 F expF (wx) expF (x) expF (x) expF (wx) w

212

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and that the numerator and denominator of this quotient can both be expressed as power series in expF (x) with coefficients in AF [w −1 ], both of which have leading term x. We thus have ∞ X w 1 − = γi (w)(expF (x))i expF (wx) expF (x) i=0

with each γi (w) stably F-numerical. Finally note that for any i if (expF (x))i is expanded in powers of x the result is a series of the form ∞ X xj rj j! j=1 F

with rj ∈ R. Combining these we see that Bnn (w n − 1) can be expressed as an R-linear combination of the γi ’s and so is in AF [w −1 ] The generalization of Von Staudt’s theorem to the BnF ’s has been established by Clarke ([10]), and gives an expression for the denominator of BnF /n in terms of the coefficients of logF (x). If AF is an algebra for which it is possible to determine directly the largest possible denominator, m(n), for which (w n − 1)/m(n) is stably numerical then this proposition can be used to establish divisibility results about the log coefficients. This is the case, for example, with the Euler formal group laws considered in §2. For algebras of numerical polynomials for rings of integers in quadratic number fields the function m(n) can be determined one prime factor at a time as in the rational case. Thus, for values of k, l, for which the the associated curve has complex multiplication, divisibility results about the associated log coefficients, which are special values of the Legendre polynomials, can be deduced. One consequence of this is a proof that such curves have supersingular reduction at rational primes which ramify or are inert in the number field and ordinary reduction at primes which split. References [1] J. F. Adams, A. S. Harris and R. M. Switzer, Hopf algebras of cooperations for real and complex K-theory, Proc. London Math. Soc. (3) 23, (1971), 385–408. [2] A. Baker, Combinatorial and Arithmetic Identities Based on Formal Group Laws, Algebraic Topology, Barcelona, 1986 (J.Aguade and R. Kane, eds.), Lecture Notes in Math. v. 1298, Springer Verlag, New York, 1987, pp. 17–34. [3] A. Baker, Operations and Cooperations in Elliptic Cohomology, Part 1 : Generalized Modular Forms and the Cooperation Algebra, New York Jour. Math. 1 (1995), 39–74. [4] A. Baker, F. W. Clarke, N.Ray, and L. Schwartz, On the Kummer Congruences and the Stable Homotopy of BU , Trans. Am. Math. Soc. 315 (1989), 591–603. [5] P. J. Cahen, Polynomes a Valeurs Entieres, Canad. J. Math. 24 (1972), 747–754. [6] L. Carlitz, The Coefficients of the Reciprocal of a Series, Duke Math. J. 8 (1941), 689–700. [7] L. Carlitz, Congruences Connected With The Power Series Expansions of the Jacobi Elliptic Functions, Duke Math. J. 20 (1953), 1–12.

Numerical Polynomials and Endomorphisms of Formal Group Laws 213 [8] L. Carlitz, The Coefficients of Singular Elliptic Functions, Math. Annalen 127 (1954), 162–169. [9] F. W. Clarke, On the determination of K∗ (K) using the Conner-Floyd isomorphism, preprint, Swansea, 1972. [10] F. W. Clarke, The Universal Von Staudt Theorems, Trans. Am. Math. Soc. 315 (1989), 591–603. [11] F. W. Clarke, J. Hunton and N. Ray, Extensions of Umbral Calculus II: Double Delta Operators, Leibniz Extensions and Hattori-Stong Theorems, Ann. Inst. Fourier 51 (2001), 297–336. [12] F. W. Clarke and K. Johnson, Cooperations in Elliptic Homology, LMS Lecture Notes 176 (1992), 131–143. [13] H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993. [14] M. Deuring, Die Typen der Multiplikatorenringe ellipisher Functionen korper, Abh. Math. Sem. Hamburg 14 (1941), 197–272. [15] L. Euler, De integratione aequationis differentialis √mdx = √ndy 4 , Novi Comm. 1−x4 1−y

[16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

Acad. Sci. Petropolitanae 6 (1756/7), 1761, 37–57; Collected Works, first series, Vol. 20, B. G. Teuber, 1912. M. Hazewinkel, Formal Groups and Applications, Academic Press, New York, (1973). D. Hensley, Polynomials which take Gaussian Integer Values at Gaussian Integers, J. Number Theory 9 (1977), 510–524. P. S. Landweber, Elliptic cohomology and modular forms, Elliptic curves and modular forms in topology, Princeton 1986, Lecture Notes in Math., vol. 1326, 1988, pp. 55–68. G. Laures, The Topological q-expansion Principle, Topology 38 (1999), 387–425. H. Miller, Universal Bernoulli Numbers and the S 1 Transfer, Current Trends in Algebraic Topology, CMS-AMS, Providence, R.I., 1982, pp. 437–449. A. Ostrowski, Uber Ganzwertige Polynome in Algebraischen Zahlkorper, J. Reine Angew. Math. (Crelle) 149 (1919), 117–124. G. Poyla, Uber Ganzwertige Polynome in Algebraischen Zahlkorper, J. Reine Angew. Math. (Crelle) 149 (1919), 97–116. D. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, New York, 1986. N. Ray, Extensions of Umbral Calculus: Penumbral Coalgebras and Generalized Bernoulli Numbers, Adv. in Math. 61 (1986), 49–100. N. Ray, Loops on the 3-sphere and Umbral Calculus, Contemp. Math. 96 (1989), 297–302.

Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 E-mail address: [email protected]

THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS NITU KITCHLOO AND JACK MORAVA Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new localization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.

Introduction If P → M is a principal bundle with structure group G then LP → LM is a principal bundle with structure group LG = Maps(S 1 , G), We will assume in the rest of the paper that the frame bundle of M has been refined to have structure group G, and that the tangent bundle of M is thus defined via some representation V of G. It follows that the tangent bundle of LM is defined by the representation LV of LG. The circle group T acts on all these spaces by rotation of loops. This is a report on the beginnings of a theory of differential topology for such objects. Note that if we want the structure group LG to be connected, we need G to be 1-connected; thus SU (n) is preferable to U (n). This helps explain why Calabi-Yau manifolds are so central in string theory, and this note is written assuming this simplifying hypothesis. Alternately, we could work over the universal cover of LM; then π2 (M ) would act on everything by decktranslations, and our topological invariants become modules over the Novikov ring Z[H2 (M )]. ¿From the point of view we’re developing, these translations may be relevant to modularity, but this issue, like several others, will be backgrounded here. Date: 9 March 2005. 1991 Mathematics Subject Classification. 22E6, 55N34, 55P35. NK is partially supported by NSF grant DMS 0436600, JM by DMS 0406461. 214

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The circle action on the free loopspace defines a structure much closer to classical differential geometry than one finds on more general (eg) Hilbert manifolds; this action defines something like a Fourier filtration on the tangent space of this infinite-dimensional manifold, which is in some sense locally finite. This leads to a host of new kinds of geometric invariants, such as the Witten genus; but this filtration is unfamiliar, and has been difficult to work with [17]. The main conceptual result of this note [which was motivated by ideas of Cohen, Godin, and Segal] is the definition of a canonical ‘thickening’ L† M of a free loopspace, with the same equivariant homotopy type, but better differential-geometric and analytic properties: the usual tangent bundle to the free loopspace, pulled back over this ‘dressed’ model, admits a canonical filtration by finite-dimensional equivariant bundles. This thickening involves a contractible LG-space called the affine Tits building A(LG), which occurs under various guises in nature: it is a homotopy colimit of homogeneous spaces with respect to a finite collection of compact Lie subgroups of LG, and it is also the affine space of principal G-connections on the trivial bundle over S 1 ; we explore its structure in more detail in the appendix. The thickening L† M seems very natural, from a physical point of view: its elements are not the raw geometric loops of pure mathematics, but are rather loops endowed with a choice of connection (depending on the structure group G of interest). This is a conceptual distinction which means little in pure mathematics, but perhaps a lot in physics. In the first section below, we recall why the Spanier-Whitehead dual of a finite CW-space is a ring-spectrum, and sketch the construction (due to Milnor and Spanier, and Atiyah) of a model for that dual, when the space is a smooth compact manifold. Our goal is to produce an analog of this construction for a free loopspace, which captures as much as possible of its string-topological algebraic structure. In the second section, we introduce the technology used in our construction: pro-spectra associated to filtered infinite-dimensional vector bundles, and the topological Tits building which leads to the construction of such a filtration for the tangent bundle. In §3 we observe that recent work of Freed, Hopkins, and Teleman on the Verlinde algebra can be reformulated as a conjectural duality between LG-equivariant K-theory of a certain dualizing spectrum for LG constructed from its Tits building, and positive-energy representations of LG. In §4 we use a new strong localization theorem to study the equivariant K-theory of our construction, and we show how this recovers the Witten genus from a string-topological point of view. We plan to discuss actions of various string-topological operads [15] on our construction in a later paper; that work is in progress.

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We would like to thank R. Cohen, V. Godin, S. Stolz, and A. Stacey for many helpful conversations, and we would also like to acknowledge work of G. Segal and S. Mitchell as motivation for many of the ideas in this paper. 1. The Atiyah dual of a manifold If X is a finite complex, then the function spectrum F (X, S 0 ) is a ringspectrum (because S 0 is). If X is a manifold M, Spanier-Whitehead duality says that F (M+ , S 0 ) ∼ M −T M . If E → X is a vector bundle over a compact space, we can define its Thom space to be the one-point compactification X E := E+ . There is always a vector bundle E⊥ over X such that E ⊕ E⊥ ∼ = 1N is trivial, and following Atiyah, we write X −E := S −N X E⊥ . With this notation, the Thom collapse map for an embedding M ⊂ RN is a map ν N S N = RN M −T M , + → M = S Moreover, the Thom collapse for the diagonal embedding of M into the zero section of M+ ∧ M ν gives us N N M+ ∧ M ν → M ν⊕T M = M+ ∧ RN + → R+ = S

defining the equivalence with the functional dual. More generally, a smooth map f : M → N of compact closed orientable manifolds has a PontrjaginThom dual map fP T : N −T N → M −T M of spectra; in particular, the map S 0 → M −T M dual to the projection to a point defines a kind of fundamental class, and the dual to the diagonal of M makes M −T M into a ring-spectrum. The ring structure of M −T M has been studied by various authors (see for example [13], [25]). Prospectus: Chas and Sullivan [11] have constructed a very interesting product on the homology of a free loopspace, suitably desuspended, motivated by string theory. Cohen and Jones [15] saw that this product comes from a ring-spectrum structure on LM −T M := LM −e where e : LM → M

∗T M

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is the evaluation map at 1 ∈ S 1 . Unfortunately this evaluation map is not T-equivariant, so the Chas-Sullivan Cohen-Jones spectrum is not in general a T-spectrum. The full Atiyah dual constructed below promises to capture some of this equivariant structure. The Chas-Sullivan Cohen-Johes spectrum and the full Atiyah dual live in rather different worlds: our prospectrum is an equivariant object, whose multiplicative properties are not year clear, while the CSCJ spectrum has good multiplicative properties, but it is not a Tspectrum. In some vague sense our object resembles a kind of center for the Chas-Sullivan-Cohen-Jones spectrum, and we hope that a better understanding of the relation between open and closed strings will make it possible to say something more explict about this.

2. Problems & Solutions For our constructions, we need two pieces of technology: Cohen, Jones, and Segal [16](appendix) associate to a filtration E : · · · ⊂ Ei ⊂ Ei+1 ⊂ . . . of an infinite-dimensional vector bundle over X, a pro-object X −E : · · · → X −Ei+1 → X −Ei → . . . in the category of spectra. [A rigid model for such an object can be constructed by taking E to be a bundle of Hilbert spaces, which are trivializable by Kuiper’s theorem. Choose a trivialization E ∼ = H × X and an exhaustive filtration {Hk } of H by finite-dimensional vector spaces; then we can define ⊥

X −Ei = lim S −Hk X Hk ∩Ei , with Ei⊥ the orthogonal complement of Ei in the trivialized bundle E.] This pro-object will, in general, depend on the choice of filtration. We will be interested in the direct systems associated to such a pro-object by a cohomology theory; of course in general the colimit of this system can be very different from the cohomology of the limit of the system of pro-objects. Example 2.1. If X = CP∞ , η is the Hopf bundle, and E is ∞η : · · · ⊂ (k − 1)η ⊂ kη ⊂ (k + 1)η ⊂ . . . then the induced maps of cohomology groups are multiplication by the Euler classes of the bundles Ei+1 /Ei , so −∞η H ∗ (CP∞ , Z) := colim{Z[t], t − mult} = Z[t, t−1 ] .

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We would like to apply such a construction to the tangent bundle of a free loopspace. Unfortunately, these tangent bundles do not, in general, possess any such nice filtration by finite-dimensional (T-equivariant) subbundles [17]! However, such a splitting does exist in a neighborhood of the constant loops: M = LM T ⊂ LM has normal bundle ν(M ⊂ LM) = T M ⊗C (C[q, q −1 ]/C) (at least, up to completions; and assuming things complex for convenience). Here small perturbations of a constant loop are identified with their Fourier expansions X an q n , n∈Z

with q = eiθ . The related fact, that T LM is defined by the representation LV of LG looped up from the finite-dimensional representation V of G, will be important below: for LV is not a positive-energy representation of LG. The main step toward our resolution of this problem depends on the following result, proved in §7 below. Such constructions were first studied by Quillen, and were explored further by S. Mitchell [26]. The first author has studied these buildings for a general Kac-Moody group [23]; most of the properties of the affine building used below hold for this larger class.

Theorem 2.2. There exist a certain finite set of compact ‘parabolic’ subgroups HI of LG (see 7.2), such that the topological affine Tits building A(LG) := hocolimI LG/HI ˜ of LG is T×LG-equivariantly contractible. In other words, given any com˜ pact subgroup K ⊂ T×LG, the fixed point space A(LG)K is contractible. Remark 2.3. The group LG admits a universal central extension LG by a circle group C. The natural action of the rotation group T on LG lifts to LG, and the T-action preserves the subgroups HI . Hence A(LG) admits an ˜ action of T×LG, with the center acting trivially. We can therefore express A(LG) as A(LG) = hocolimI LG/HI where HI is the induced central extension of HI .

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Other descriptions of A(LG) This Tits building has other descriptions as well. For example: 1. A(LG) can be seen as the classifying space for proper actions with respect ˜ to the class of compact Lie subgroups of T×LG. That means that the space ˜ A(LG) is a T×LG-CW complex with the isotropy groups of the action being ˜ compact Lie groups. Moreover, given any compact Lie group K ⊂ T×LG, the K fixed point space A(LG) is contractible. It is not hard to show that these ˜ properties uniquely determine A(LG) upto T×LG-homotopy equivalence. 2. It also admits a more differential-geometric description as the smooth infinite dimensional manifold of holonomies on S 1 × G (see Appendix): Let S denote the subset of the space of smooth maps from R to G given by S = {g : R → G, g(0) = 1, g(t + 1) = g(t) · g(1)} ; then S is homeomorphic to A(LG). The action of h(t) ∈ LG on g(t) is given by hg(t) = h(t) · g(t) · h(0)−1 , where we identify the circle with R/Z. The action of x ∈ R/Z = T is given by xg(t) = g(t + x) · g(x)−1 . 3. The description given above shows that A(LG) is equivalent to the affine space A(S 1 × G) of connections on the trivial G-bundle S 1 × G. This identification associates to the function f (t) ∈ S, the connection f 0 (t)f (t)−1 . Conversely, the connection ∇t on S 1 × G defines the function f (t) given by transporting the element (0, 1) ∈ R × G to the point (t, f (t)) ∈ R × G using the connection ∇t pulled back to the trivial bundle R × G. Remark 2.4. These equivalent descriptions have various useful consequences. For example, the model given by the space S of holonomies says that given a finite cyclic group H ⊂ T, the fixed point space S H is homeomorphic to S. Moreover, this is a homeomorphism of LG-spaces, where we consider S H as an LG-space and identify LG with LGH in the obvious way. Notice also that S T is G-homeomorphic to the model of the adjoint representation of G defined by Hom(R, G). Similarly, the map S → G given by evaluation at t = 1 is a principal ΩG bundle, and the action of G = LG/ΩG on the base G is given by conjugation. This allows us to relate our work to that of Freed, Hopkins and Teleman in the following section. Finally, the description of A(LG) as the affine space A(S 1 × G) implies that the fixed point space A(LG)K is contractible for any compact subgroup ˜ K ⊆ T×LG. If E → B is a principal bundle with structure group LG, then (motivated by ideas of [14]) we construct a ‘thickening’ of B:

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Definition 2.5. The thickening of B associated to the bundle E is the space B † (E) = E ×LG A(LG) = hocolimI E/HI . We will omit E from the notation, when the defining bundle is clear from context. Remark 2.6. If P → M is a principal G bundle, then LP → LM is a principal LG bundle. In this case, the description above gives L† M := LM † (LP ) a smooth structure: L† M = {(γ, ω) | γ ∈ LM,

ω ∈ A(γ ∗ (P ))}

where A(γ ∗ (P )) is the space of connections on the pullback bundle γ ∗ (P ). ˜ Let T×LG be the extension of the central extension of LG by T, acting as rotations, and let U be a unitary representation of this group, of finite type. ˜ We propose to construct a Thom T×LG-prospectrum A(LG)−U , as a substitute for the nonexistent equivariant prospectrum defined by −U , regarded as a vector bundle over a point. The central extension of LG splits when restricted to the constant loops, ˜ so we have a torus T := T × C × T ⊂ T×LG, where T is a maximal torus of G, and C is the circle of the central extension; T is in fact a common ˇ be the character group maximal torus for the family HI of parabolics. Let T ˇ of this torus, and let Z[[T]] be its completed group ring. On restriction to ˜ I, the subgroup T×H ∼ U |T×H ˜ I = ⊕ UI (α) decomposes into a sum of finite dimensional representations; because U is of finite type, the isotypical summands appear only finitely often. Let ˇ char U |T×H ∈ Z[[T]] ˜ I

be its character. ˇ of characters, let For any finite subset R ⊂ T UI (R) = ⊕ {UI (α) | char UI (α) ⊂ ZhRi} ⊂ UI be the subresentation of UI with ‘support in R’, and let SI−U to be the Thom ˜ I -prospectrum associated to the filtered (equivariant) vector bundle R 7→ T×H UI (R) over a point. If I ⊂ J then HI maps naturally to HJ , and there is a corresponding morphism UJ → UI of filtered vector bundles, given by inclusions UJ (R) → UI (R). This defines a filtered system of vector bundles over the small category or diagram defined by inclusions of parabolics.

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˜ Example 2.7. Consider the standard representation of T×LSU (2) on LC2 . Let L1 and L2 denote the canonical coordinate summands of C2 , and let q denote the character of the rotation group T. The parabolic subgroups HI ⊂ LSU (2) in this case are: H1 = SU (2) ⊂ LSU (2) given by constant loops, and a bz −1 a b H0 = { } where ∈ SU (2) ; cz d c d

T = H∅ = H0 ∩H1 is the maximal torus of SU (2). Notice that the representation LC2 of the loop group is certainly not of positive energy. However, when ˜ I , it decomposes as a sum of representations: restricted to the subgroups T×H M LC2 = (L1 ⊕ L2 )q k k∈Z

˜ 1 , T×H ˜ ∅ and as for T×H

M

(L1 ⊕ qL2 )q k

k∈Z

˜ 0 . The resulting filtered vector bundle is in this case just a pushout. for T×H ˜ Definition 2.8. We define A(LG)−U to be the T×LG-prospectrum A(LG)−U = hocolimI LG+ ∧HI SI−U , where LG+ denotes LG, with a disjoint basepoint. Homotopy colimits in the category of prospectra can be defined in general, using the model category structure of [12]. Remark 2.9. Given any principal LG-bundle E → B, and a representation U of LG, we define the Thom prospectrum of the virtual bundle associated to the representation −U to be B!−U = E+ ∧LG A(LG)−U = hocolimI E+ ∧HI SI−U . In particular, if P is the refinement of the frame bundle of M via a representation V of G, then the tangent bundle of LM is defined by the representation LV of LG. Definition 2.10. The Atiyah dual LM −TLM of LM is the T-prospectrum L† M −LV . We will explore this object further in §6. Note that its underlying nonequivariant object maps naturally to (a thickening of) the Chas-Sullivan spectrum.

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Nitu Kitchloo and Jack Morava 3. The dualizing spectrum of LG

The dualizing spectrum of a topological group K is defined [24] as the K-homotopy fixed point spectrum: DK = K+hK = F (EK+ , K+ )K where K+ is the suspension spectrum of the space K+ , endowed with a right K-action. The dualizing spectrum DK admits a K-action given by the residual left K-action on K+ . If K is a compact Lie group, then it is known [24] that DK is the one point compactification of the adjoint representation Ad(K)+ . It is also known that there is a K × K-equivariant homotopy equivalence K+ ∼ = F (K+ , DK ) . It follows from the compactness of K+ that for any free K+ -spectrum E, we have the K-equivariant homotopy equivalence E∼ = F (K+ , E ∧K DK ) . +

It is our plan to understand the dualizing spectrum for the (central extension of the) loop group. Theorem 3.1. There is an equivalence DLG ∼ = holimI LG+ ∧HI Ad(HI )+ of left LG-spectra. Proof. We have the sequence of equivalences: DLG = F (ELG+ , LG+ )LG ∼ = F (ELG+ ∧ A(LG)+ , LG+ )LG . The final space may be written as holimI F (ELG+ ∧HI LG+ , LG+ )LG = holimI F (ELG+ , LG+ )HI . Now recall the equivalence of HI × HI -spectra: (1) LG+ ∼ = F (HI + , LG+ ∧HI DHI ) . Taking HI -homotopy fixed points implies a left HI -equivalence F (ELG+ , LG+ )HI = (LG+ )hHI ∼ = LG+ ∧HI Ad(HI )+ ; where we have used equation (1) at the end. Replacing this term into the homotopy limit completes the proof. Similarly, we have: Theorem 3.2. There is an equivalence DLG ∼ = holimI LG+ ∧HI Ad(HI )+ of left LG-spectra.

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Remark 3.3. The diagram underlying DLG or DLG can be constructed in the category of spaces. Given an inclusion I ⊆ J, the orbit of a suitable element in Ad(HJ ) gives an embedding HJ /HI ⊂ Ad(HJ ), and the Pontrjagin-Thom construction for this embedding defines an HJ -equivariant map Ad(HJ )+ −→ HJ + ∧HI Ad(HI )+ which extends to the map LG+ ∧HJ Ad(HJ )+ −→ LG+ ∧HI Ad(HI )+ required for the diagram. Moreover, composites of these maps can be made compatible up to homotopy. A conjectural relationship with the work of Freed, Hopkins and Teleman The discussion below assumes the existence of LG-equivariant K-theory, as defined in [19] (see Appendix). Given a space X with a proper LGaction, Freed-Hopkins-Teleman define an LG-equivariant spectrum over X. The equivariant K-theory groups are defined as the homotopy groups of the space of sections of this spectrum. In this section, we will assume that these K-theory groups can be defined for spectra X with proper LG-action. We will primarily be interested with LG-CW spectra with proper isotropy. This hypothesis provides us with a convenient language. We expect to return to the underlying technical issues in a later paper. The center of LG acts trivially on DLG , defining a second grading on ∗ KLG (DLG ); we will use a formal variable z to keep track of the grading, so M ∗,n ∗ KLG (DLG ) = KLG (DLG )z n . n

The spectral sequence for the cohomology of a cosimplicial spectrum, in the ∗ case of KLG (DLG ), has E2i,j = colimiI KHj I (Ad(HI )+ ) .

This spectral sequence respects the second grading given by powers of z. As before, we may decompose KH∗ I (Ad(HI )+ ) under the action of the center of LG (contained in HI ) M ∗,n KH∗ I (Ad(HI )+ ) = KHI (Ad(HI )+ )z n n

∗,n We therefore have a spectral sequence converging to KLG (DLG ), with

E2i,j,n = colimiI KHj,nI (Ad(HI )+ ) . For n > 0, we will show presently that this spectral sequence collapses to give ∗,n KLG (DLG ) = colimI KH∗,n (Ad(HI )+ ) I

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Therefore, this group admits a natural Thom class given by the system {Ad(HI )+ }, of spinor bundles for the adjoint representations of the parabolics HI . We therefore have a (global) Thom isomorphism: K ∗+r+1,n (DLG ) ∼ = colimI Rep∗,n (HI ) . LG

where Rep (HI ) is the subgroup of Rep∗ (HI ) corresponding to KH∗,n I (Ad(HI )+ ) under the (local) Thom isomorphism. We will also show that the above colimit may naturally be identified with the free abelian group generated by the regular, dominant characters of level n. This free abelian group can further be identified with the Grothendieck group of level n−h positive energy representations of LG, where h denotes the dual Coxeter number. This Grothendieck group is known as the Verlinde algebra (of level n − h). ∗+r+1,n Therefore, we get an (abstract) isomorphism between KLG (DLG ) and the Verlinde algebra of level n − h. In [19], Freed-Hopkins-Teleman give a geometric meaning to the Thom isomorphism above, which explains the shift in level by showing that the Thom class has internal level h. ∗,n

Under the assumptions on KLG given above, we get: ∗,n Theorem 3.4. For n > 0, the groups KLG (DLG ) are two-periodic. We have a (Thom) isomorphism of groups: r+1,n (DLG ) Vn−h ∼ =K LG

where Vk is the Verlinde algebra of level k, h is the dual Coxeter number of r,n G, and r is its rank. Moreover, KLG (DLG ) = 0. Before we prove the collapse of the spectral sequence, let us consider an example: Example 3.5. To illustrate this in an example, recall the case of G = SU (2). In this case r = 1, h(G) = 2. Here the groups HI are given by H0 = SU (2) × S 1 ,

H1 = S 1 × SU (2),

H 0 ∩ H1 = T = S 1 × S 1 .

The respective representation rings may be identified by restriction with subalgebras of KT (pt) = Z[u±1 , z ±1 ] : KH0 (pt) = Z[u + u−1 , (z/u)±1 ],

KH1 (pt) = Z[z ±1 , u + u−1 ] .

Now consider the two pushforward maps involved in the colimit: ϕ0 : KT (pt) → KH0 (pt),

ϕ1 : KT (pt) → KH1 (pt)

A quick calculation shows that for k > 0, we have ( (z/u)k Symk (u + u−1 ), j = 0 ϕj (z k ) = zk , j=1 ( (z/u)k Symk−1 (u + u−1 ), j = 0 k −1 ϕj (z u ) = 0, j=1,

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where Symk (V ) denotes the k-th symmetric power of the representation V , e.g. Symk (u + u−1 ) = uk + · · · + u−k . The colimit is the cokernel of ϕ1 ⊕ ϕ0 : KT (pt) −→ KH1 (pt) ⊕ KH0 (pt) restricted to positive powers of z. Now consider the decomposition Z[u±1 , z] = Z[u + u−1 , z] ⊕ u−1 Z[u + u−1 , z] . It is easy to check from this that the cokernel for nontrivial powers of z is isomorphic to the cokernel of ϕ0 restricted to u−1 Z[u + u−1 , z] and hence is M Z[u + u−1 ] (z/u)k+2 k+1 (u + u−1 )i hSym k≥0 which agrees with the classical result [18].

We now get to the collapse of the spectral sequence. It is sufficient to establish: Proposition 3.6. Assume n > 0, then colimiI KHr+1,n (Ad(HI )+ ) is trivial if I i > 0. For i = 0, this group is isomorphic to the free abelian group generated by regular, dominant, level n characters of LG. Proof. To simplify the notation, we will abbreviate KHr+1,n (Ad(HI )+ ) by K(I), I hence the letter K denotes the functor I 7→ K(I). The strategy in proving the above proposition L is to show that the functor K decomposes into a sum of functors K = KJ , where all the functors KJ have trivial higher derived colimits. We then show that colimKJ is also trivial for all but one of the functors, and we identify its colimit as the free abelian group generated by the dominant regular characters of level n. Let R(T )(n) denote the free abelian group generated by characters of LG of level n. Recall that the fundamental domain for the action of the affine ˜ on the set of characters of level n, is the set of characters Weyl group W in the affine alcove ∆ (at height n). The affine alcove is an affine r-simplex in the (dual) Lie algebra of the maximal torus of LG. Hence we may index the walls of ∆ by the category C of proper subsets of the r + 1-element set {0, . . . , r}, where ∆I ⊆ ∆J if J ⊂ I. We get a corresponding decomposition: M R(T )(n) = RJ (T )(n) J∈C

where RJ (T )(n) is the free abelian group generated by the characters that ˜ -translates of characters in the interior of the face ∆I . Let RJ (T )(n) are W denote the free abelian group generated by the characters in the interior of the face ∆J , so we have an isomorphism: −1 ˜ /WJ ] ∼ RJ (T )(n) ⊗ Z[W = RJ (T )(n), eλ ⊗ w 7→ ew λ

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where WJ is the isotropy of the wall ∆J (and is also the Weyl group of HJ ). Consider the push forward map πI : R(T )(n) → K(I). By pushforward, we ˆ mean with respect to the A-orientation. Therefore, the character of πI (eλ ) is given by: P sgn(w) w(λ) Y e w∈WI (−1) , A(I) = (eα/2 − e−α/2 ), A(I) the product in the Weyl denominator A(I) being taken over the positive roots of HI . The map πI is surjective, and one may define KJ to be the functor I 7→ πI (RJ (T )(n)). It is not hard to see that direct sum decomposition of R(T )(n) remains direct when we push forward to K(I), hence we get a direct sum decomposition of functors: M K= KJ J∈C

We now proceed to calculate colim KJ by splitting it out of another functor LJ . We define the functor I 7→ LJ (I) by ˜ /WI ] LJ (I) = RJ (T )(n) ⊗ Z(sgn) ⊗Z[W ] Z[W i

J

where Z(sgn) denotes the sign representation of Z[WJ ]. One may check that the following map is a retraction of functors: LJ (I) → KJ (I),

eλ ⊗ 1 ⊗ w 7→ (−1)w πI (ew

−1 λ

)

where (−1)w denotes the sign of the element w. Hence, the groups colimi KJ are retracts of the homology of the complex RJ (T )(n) ⊗ Z(sgn) ⊗Z[WJ ] C∗ , where C∗ is the simplicial chain complex of the affine hyperplane given by ˜ orbit of ∆. Since WJ is a finite subgroup, this simplicial complex is the W WJ -equivariantly contractible. Thus C∗ is WJ -equivariantly equivalent to the constant complex Z in dimension zero. It follows that colimi KJ = 0 if i > 0, and colim KJ is a retract of RJ (T )(n) ⊗ Z(sgn) ⊗Z[WJ ] Z. Furthermore, if J is nonempty, then the groups RJ (T )(n)⊗Z(sgn)⊗Z[WJ ] Z are two torsion, hence the map to KJ is trivial. This shows that colim KJ = 0 if J is not the empty set. Finally, to complete the proof, one simply observes that for the empty set, the above retraction is an isomorphism, and hence colim Kφ = Rφ (T )(n), which is the free abelian group generated by the level n dominant regular weights of LG. Remark 3.7. We can calculate the equivariant K-homology KLG∗ (A(LG)) using the same spectral sequence. This establishes an isomorphism ∗ between KLG (DLG ) and KLG ∗ (A(LG)). Results above imply that the latter group calculates the Verlinde algebra. Infact, the results of [19] may be stated in terms of K-homology, and our spectral sequence argument may be seen as an alternate proof of their results. Recall also that A(LG) is the classifying space for proper actions (i.e. with compact isotropy) so these results also bear an interesting relationship to the Baum-Connes conjecture [7]

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Question. For the manifold LM, with frame bundle LP , we can construct a spectrum DLM := holimI LP+ ∧HI Ad(HI )+ It would be very interesting to understand something about KT (DLM ). 4. Localization Theorems If E is a T-equivariant complex-oriented multiplicative cohomology theory, and X is a T-space, we have contravariant (j ∗ ) and covariant (j ! ) homomorphisms associated to the fixedpoint inclusion j : XT ⊂ X , satisfying j ∗ j ! (x) = x · eT (ν) ; if the Euler class of the normal bundle ν is invertible, this leads to a close relation between the cohomology of X and X T . More generally, if f : M → N is an equivariant map, then its PontrjaginThom transfer is related to the analogous transfer defined by its restriction fT : MT → NT to the fixedpoint spaces, by a ‘clean intersection’ formula: ∗ ∗ jN ◦ f ! (−) = f T! (jM (−) · eT (ν(f )|M T )) .

Definition 4.1. The fixed-point orientation defined by the Thom class Th† (ν(f T )) = Th(ν(f T )) · eT (ν(f )|M T for the normal bundle of the inclusion of fixed-point spaces is the product of the usual Thom class with the equivariant Euler class of the full normal bundle restricted to the fixed-point space. Recall that we have the formula f T! (−) = fPT∗T (− · Th(ν(f T ))), where fPTT is the Thom collapse map: fPTT : Rk+ ∧ N+ → (M T )ν(f

T)

corresponding to f T (having first replaced f T by an embedding of M T into N T × Rk for large k). In our new notation the clean intersection formula becomes ∗ ∗ jN ◦ f ! = f T† ◦ jM with a new Pontrjagin-Thom transfer f T† (−) = fPT∗T (− · Th† (ν(f T ))) . In the case of most interest to us (free loopspaces), we identified the normal bundle above, in §2; using that description, we have Y eT (ν(M ⊂ LM)) = (e(Li ) +E [k](q)), 06=k∈Z;i

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where the Li are the line bundles in a formal decomposition of T M, q is the Euler class of the standard one-dimensional complex representation of T, and +E is the sum with respect to the formal group law defined by the orientation of E. It may not be immediately obvious, but it turns out that such a formula implies that the fixed-point orientation defined above will have good multiplicative properties. Such Weierstrass products sometimes behave better when ‘renormalized’, by dividing by their values on constant bundles [2]. If E is KT with the usual complex (Todd) orientation, we have e(L) +K [k](q) = 1 − q k L ; but for our purposes things turn out better with the Atiyah-Bott-Shapiro spin orientation; in that case the corresponding Euler class is (q k L)1/2 − (q k L)−1/2 . The square roots make sense under the simple-connectivity assumptions on G mentioned in the introduction: To be precise, let V be a representation of LG with an intertwining action of T. We restrict ourselves to finite type representations V which (for lack of a better name) we call symmetric, i.e. such that V is equivalent to the representation of LG obtained by composing V with the involution of LG which reverses the orientation of the loops. The restriction of the representation V ˜ to the constant loops T × G ⊂ T×LG has a decomposition X V = VT⊕ Vk q k k6=0

where Vk are representations of G, and q denotes the fundamental representation of T. Let V (m) be the finite dimensional subrepresentation X V (m) = V T ⊕ Vk q k ; 0 0, α0 (h) ≤ 1} . We may identify this space with ∆ using the roots αi . So, for example, the codimension 1 face ∆i for 1 ≤ i ≤ n is identified with the subset of the alcove {(1, h) | αi(h) = 0}, for i 6= 0, and ∆0 is identified with the subspace {(1, h) | α0(h) = 1}. General facts about Loop groups [22, 26] show that the surjective map Lalg G × ∆ −→ A,

(f (z), y) 7→ Adf (z) (y)

˜ alg Ghas isotropy HI on the subspace ∆I . Hence it factors through a T×L equivariant homeomorphism between A(Lalg G) and the affine space A. No˜ alg G admits a fixed point on tice that any compact subgroup K ⊂ T×L K A(Lalg G). Hence, the space A(Lalg G) is also affine. This completes the proof. Remark 7.6. The affine space A above should be thought of as the space of (algebraic) connections on the trivial G-bundle on S 1 . The action of the group Lalg G then corresponds to the action of the (algebraic) gauge group. This analogy can be taken one step further to define the space of (algebraic) holonomies: Salg = {g : R → G | g(t) = f (e2πit ) · exp(tX); f (z) ∈ Ωalg G, X ∈ Lie(G)} ˜ alg G given topologized as a quotient of Ωalg G × Lie(G), with the action of T×L by left multiplication (see discussion before 2.4). In fact, in [26] Mitchell shows that A(Lalg G) is Lalg G equivariantly homeomorphic to the space Salg (attributing the result to Quillen). We now define the smooth Tits building Definition 7.7. Let A(LG) be the homotopy colimit: A(LG) = hocolimI∈C LG/HI = LG ×Lalg G A(Lalg G) . ˜ Theorem 7.8. The smooth Tits building A(LG) is T×LG-equivariantly contractible.

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Proof. By the above remark, the space A(LG) is LG-equivariant homeomorphic to the space LG ×Lalg G Salg . This is easily seen to be LG-equivariantly homeomorphic to the space of smooth maps: S = {g : R → G | g(t) = f (e2πit ) · exp(tX); f (z) ∈ ΩG, X ∈ Lie(G)} Equivalently, we may describe S as S = {g : R → G, g(0) = 1, g(t + 1) = g(t) · g(1)} ; from which it follows (see 2.4) that S is LG-equivariantly homeomorphic to the space of connections on the trivial G-bundle on S 1 , which we denote by A(S 1 × G). This sequence of LG-equivariant homeomorphisms from A(LG) to A(S 1 × G) is compatible with the action of T. The proof is now complete, ˜ since the space of connections A(S 1 × G) is clearly T×LG-equivariantly contractible. References 1. F. Adams, A variant of E.H. Brown’s representability theorem, Topology, Vol. 10 (1971) 185-198. 2. M. Ando, J. Morava, A Renormalized Riemann-Roch formula and the Thom isomorphism for the free loopspace, in the Milgram Festschrift, Contemp. Math. 279 (2001) 3. ——, ——–, H. Sadofsky, Completions of Z/(p)-Tate cohomology of periodic spectra, Geometry & Topology 2 (1998) 145 - 174 4. —–, M. Hopkins, N. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Inv. Math. 146 (2001) 595 - 687 5. M. Artin, B. Mazur Etale homotopy, Springer LNM 100 (1969) 6. M. Atiyah, I. MacDonald, Commutative Algebra 7. P. Baum, A. Connes, and N. Higson. Classifying space for proper actions and K-theory of group C ∗ -algebras, p. 240 - 291 in C ∗ − algebras: 1943-1993, Contemporary Math 167, AMS (1994) 8. N. Bourbaki, Algebre Commutatif 9. T. Brocker, T. tom Dieck, Representations of Compact Lie groups, Springer GTM 98. 10. JL Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology 29 (1990) 461 - 480 11. M. Chas, D. Sullivan, String topology, available at math.AT/9911159 12. J.D. Christensen, D.C. Isaksen, Duality and prospectra (in progress) 13. R. Cohen, Multiplicative properties of Atiyah duality, available at math.AT/0403486 14. ———, V. Godin, A polarized view of string topology, available at math.AT/0303003 15. ——–, J.D.S Jones, A homotopy-theoretic realization of string topology, available at math.GT/0107187 16. ——–, ——–, G. Segal, Floer’s infinite-dimensional Morse theory and homotopy theory, in The Floer memorial volume, Progr. Math 133, Birkh¨ auser (1995) 17. ——–, A. Stacey, Fourier decomposition of loop bundles, available at math.AT/0210351 18. D. Freed, The Verlinde Algebra is Twisted Equivariant K-Theory, available at math.RT/0101038 19. D. Freed, M. Hopkins, C. Teleman, Twisted K-theory and loop group representations I, available at math.AT/0312155 20. M. Hovey, N.P. Strickland, Morava K-theories and localization, Mem. Amer. Math. Soc. 139 (1999), No.666.

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21. J.D.S Jones, S.B. Petrack, The fixed point theorem in equivariant cohomology, Transactions of the Amer. Math. Soc., Vol. 322, No. 1 (1990) 35-49. 22. V.G. Kac, Infinite dimensional Lie algebras, Cambridge University Press, 1985 23. N. Kitchloo, Topology of Kac-Moody groups, Thesis, M.I.T., 1998. 24. J. Klein, The dualizing spectrum of a topological group, Math. Ann 319 (2001) 421 456 25. G. Lewis, J.P. May, M. Steinberger, Equivariant stable homotopy theory, Springer LNM 1213 (1986) 26. S.A. Mitchell, Quillen’s theorem on buildings and the loops on a symmetric space, Enseign. Math. 34 (1988) 123-166 27. J. Morava, Forms of K-theory, Math Zeits. 201 (1989) 28. I. Rosu, Equivariant K-theory and equivariant cohomology, Math. Zeits. 243 (2003) 423-448. ´ 29. G. Segal, Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) 129-151. 30. S. Stolz, P. Teichner, What is an elliptic object? available at math.ucsd.edu 31. T. Torii, On degeneration of one-dimensional formal group laws and stable homotopy theory, AJM 125 (2003) 1037-1077 32. D. Zagier, Note on the Landweber-Stong elliptic genus, in Elliptic curves and modular forms in algebraic topology, ed. P. Landweber, Springer 1326 (1988) Department of Mathematics, UCSD, LaJolla, CA 92093. Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218. E-mail address: [email protected], [email protected]

RATIONAL VERTEX OPERATOR ALGEBRAS GEOFFREY MASON UNIVERSITY OF CALIFORNIA, SANTA CRUZ Abstract. We discuss the foundations of vertex operator algebras and their representations, concentrating on rational vertex operator algebras. We use the Witt-Grothendieck group of conformal field theories as a vehicle to describe results, in particular the connections with modular forms.

§1. Introduction Vertex operator algebras suggest themselves as objects that could play a rˆole in a geometric description of elliptic cohomology and related topics. As linear spaces with lots of symmetry they can participate in K-theoretic type constructions, and many (but not all) vertex operator algebras are endowed with a natural modular form as part of their structure. The incorporation of vertex operator algebras into topology is well under way (e.g. [Bo], [MS], [MSV], [T]), but it seems true to say that the underlying algebraic theory is not well understood by many potential users of the subject. What follows is an attempt to convey some of the basic ideas about vertex operator algebras, in particular the rapidly advancing theory of rational vertex operator algebras. These are the algebras most naturally associated to modular forms, and in the guise of RCFT (rational conformal field theory) they constitute an active area of research in physics ([FMS]) as well as mathematics. For more information and background the reader may refer to the following: [DM4], [FLM], [G], [K1], [KR], [QFS]. Additional references will be mentioned below. In my talk at the Newton Institute I emphasized questions about group actions (orbifold theory) and in particular how one can get information about finite group cohomology (e.g., for the Monster group) by looking at maps of equivariant Witt-Grothendieck groups of vertex operator algebras into group cohomology. In order to limit the length of the present exposition, however, we make no mention of groups or orbifolds here. §2. Vertex Operator Algebras and Locality Fix a complex linear space V , the Fock space. Elements of V are called states. A (quantum) field on V is an element X a(n)z −n−1 ∈ End(V)[[z, z −1 ]] a(z) = n∈Z

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with the property that v ∈ V ⇒ a(n)v = 0 for all large enough integers n. P (z is a formal variable.) One writes a(z)v = a(n)vz −n−1 , where only finitely many negative powers of z occur for a field a(z). The space of all fields on V is denoted by F (V ). The endomorphisms a(n) are the (Fourier) modes of the field a(z). A pair of fields a(z), b(z) ∈ F (V ) are mutually local in case the following holds: there is k ≥ 0 such that (y − z)k [a(y), b(z)] = 0 where y, z are independent variables. We write a(z) ∼ b(z) if a(z) and b(z) are mutually local fields and a(z) ∼k b(z) if one wishes to stress the degree of locality k, which depends on a(z) and b(z). Examples of Quantum Fields a. (Derivative of a field): If a(z) ∈ F (V ) then X ∂a(z) = a0 (z) = (−n − 1)a(n)z −n−2 ∈ F (V ), moreover b(z) ∈ F (V ) and a(z) ∼ b(z) ⇒ a0 (z) ∼ b(z).

b. (Tensor Products): Given Fock spaces V, W , there is a natural injection F (V ) ⊗ F (W ) −→ F (V ⊗ W ). P P (The map arises via a(z) ⊗ b(z) 7→ c(n)z −n−1 where c(n) = i∈Z a(i) ⊗ b(n − i − 1). Although this is an infinite sum of operators on V ⊗ W , it is well-defined because a(z), b(z) are fields: a state in V ⊗ W is annihilated by all but a finite number of the summands.) c. (Virasoro Algebra): The Virasoro algebra is the Lie algebra M V ir = CL(n) ⊕ Ck n∈Z

with relations [V ir, k] = 0 and

m3 − m δm+n,0 k. 12 Denote by V ir + the Lie subalgebra spanned by k together with the L(n), n ≥ 0. Let C0 be the 1-dimensional V ir + -module annihilated by all L(n), n > 0, and on which k, L(0) operate as prescribed scalars c, h respectively. Let U (c, h) be the Verma module IndVV ir+ (C0 ), and set X ω(z) = L(n)z −n−2 . [L(m), L(n)] = (m − n)L(m + n) +

n∈Z

(Note the change in grading convention in this case, which amounts to L(n) = ω(n − 1).) Then ω(z) ∈ F (U ), moreover ω(z) ∼4 ω(z).

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We now define a vertex operator algebra. It is a Z-graded Fock space M V = Vn n∈Z

with dimVn < ∞ and Vn = 0 for n δm,−n c. ˆ ∗ ˆ≥ We have the Verma module M(1, λ) = IndH = ˆ ≥ C0 . Here, λ ∈ H , H H H ⊗ C[t] ⊕ Ck and C0 is the 1-dimensional module in which H ⊗ tC[t] acts as 0, k acts as 1, and H = H ⊗ t0 acts through λ. Then M = M(1, 0) has the structure of a vertex operator algebra. We have c = dimH and Y 1 (5) ZM (q) = q −c/24 (1 − q n )−c = η(q)c n≥1

where η(q) is the Dedekind eta function. There C0 = P are identifications −n−1 M0 , H = M1 , and if h ∈ H then Y (h, z) = where h(n) n∈Z h(n)z is the natural action of h ⊗ tn on M. Physically, this theory models c free (noninteracting) bosons. d. Lattice Theories. Here we are given an even lattice L, i.e. a free abelian group L equipped with a positive-definite symmetric bilinear form : L ⊗ L → Z such that < x, x >∈ 2Z. Extend to H = C ⊗Z L and let M be the associated free boson theory. Then there is a vertex operator algebra with Fock space (6)

VL = M ⊗ C[L] = ⊕α∈L C ⊗ eα

where C[L] = ⊕α Ceα is the group algebra of L and eα eβ = eα+β . M = M ⊗e0 is a conformal subalgebra of VL . The partition function is θL (q) (7) ZVL (q) = η(q)rkL P /2 where θL (q) = is the theta-function of L. Physically, this α∈L q theory corresponds to rkL bosons with momenta restricted to lie on L. e. Tensor products. Given vertex operator algebras (V i , Yi ) of central charge ci , i = 1, 2, their tensor product is a vertex operator algebra (V 1 ⊗ V 2 , Y1 ⊗ Y2 ) with vacuum 11 ⊗ 12 , conformal vector ω1 ⊗ 12 + 11 ⊗ ω2 , central charge c1 + c2 and partition function ZV 1 ⊗V 2 (q) ∼ = ZV 1 (q)ZV 2 (q). For example, if M1 and M2 are free bosonic theories of central charge c1 , c2 respectively, then M1 ⊗ M2 is the free bosonic theory of central charge c1 + c2 . Similarly, given a pair of even lattices L1 , L2 , we have (8) V L ⊗ VL ∼ = VL ⊥L 1

2

where L1 ⊥ L2 is orthogonal direct sum.

1

2

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f. Moonshine Module. The most famous vertex operator algebra, it is tradition to follow Frenkel-Lepowsky-Meurman and denote it by V \ . We have c = 24 and ZV \ (q) = J(q) = q −1 + 196884q + 21493760q 2 + ... We give some different types of examples concerning important structural features of a vertex operator algebra. g. Connection with commutative algebras. Suppose that V has no negative weight spaces: Vn = 0 for n < 0 (a ubiquitous condition which does not, however, always hold). Then by skew-symmetry and associativity it follows that V0 admits the structure of a commutative, associative algebra with product u.v = u(−1)v, u, v ∈ V0 and identity 1. There are interesting examples in which V0 is the cohomology ring of a manifold (cf. Gorbounov’s lecture at this conference). h. Connection with affine Lie algebras. Suppose that V has a non-degenerate vacuum, that is V0 = C1. Then V has no negative weight spaces, and V1 carries the structure of Lie algebra via [u, v] = u(0)v, u, v ∈ V1 . Moreover V1 comes equipped with an invariant bilinear form : V1 ⊗ V1 → C where u(1)v =< u, v > 1. Fourier modes of the fields Y (u, z) for u ∈ V1 close on the corresponding affine Lie algebra (use the commutator formula). i. Connection with non-associative algebras. Suppose that V has a nondegenerate vacuum and that V1 = 0. Then V2 carries the structure of commutative, non-associative algebra via u.v = u(1)v, u, v ∈ V2 with identity ω/2. There is also an invariant form as in Example h. In the case of V \ the corresponding algebra is the famous Griess algebra with automorphism group the Monster [Gr] . We have turned history around here: it was Griess’s algebra and the numerical observations of Conway-Norton [CN] that led to the discovery of vertex operator algebras and construction of the Moonshine module at the hands of Borcherds [B] and Frenkel-Lepowsky-Meurman [FLM]. j. Connection with Poisson-Lie algebras. Define C2 (V ) = {u(−2)v|u, v ∈ V }. The quotient space P (V ) = V /C2 (V ) carries a natural Poisson-Lie algebra structure with multiplication and bracket induced by the same operations as Examples g and h, which are thus both incorporated into the larger structure. The reader will readily identify P (M), where M is a Heisenberg vertex operator algebra, with a well-known Poisson-Lie algebra. For the Moonshine module, P (V \ ) is a finite-dimensional Poisson-Lie algebra which admits the Monster simple group as automorphisms. This and similar examples seem not to have been closely studied thus far. (For example, I do not know the dimension of P (V \ )). Zhu’s influential paper [Z] was the first to make serious use of the finite-dimensionality of P (V ) (the so-called C2 -cofiniteness condition).

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See [DLM3] for more on the Poisson-Lie algebra-vertex operator algebra connection, and [ABD], [Bu], [GN] for results related to C2 -cofiniteness. §3 Representations of Vertex Operator Algebras There is a natural notion of module over a vertex operator algebra V , and the family of V -modules forms an abelian category V − mod. Representation theory is a fundamental part of vertex operator algebra theory, and an active subject of research at this time. Describing V −mod for a given V is generally a difficult task. A V -module is a pair (M, YM ) where M is a complex linear space M (9) M= Mn n∈C

graded into finite-dimensional pieces Mn indexed by a subset of C, together with a linear map YM : V → F (M), v 7→ YM (v, z).

There is a finiteness condition on the grading: if t ∈ C then Mt+n = 0 for all integers n k. 0, if n > k. Triviality (Corollary of Theorem 4.1) (Ravenel [Rav84]) • X: p-completed • X: p-completed harmonic: • #{n ∈ N | πn X 6= 0} < +∞, bounded below harmonic: • #{n ∈ N | K(n)∗ X 6= 0} < =⇒ X = ∗. +∞, =⇒ X = ∗. Vanishing (Corollary of Theorem 4.1) (Ravenel [Rav84]) • X: p-completed • X: p-completed harmonic: bounded below harmonic: • #{n ∈ N | πn Y 6= 0} < +∞, • #{n ∈ N | K(n)∗ Y 6= 0} < =⇒ [Y, X] = 0. +∞, =⇒ [Y, X] = 0. Recovery (Corollary of Theorem 4.1) (Ravenel [Rav84]) X: harmonic, X: bounded below harmonic, =⇒ The p-completion Xp may be re=⇒ The p-completion Xp may be covered from X(k,+∞) for any k = 0: recovered from Ck X for any k = 0: Xp = holimn L∞ X(k,+∞) ∧ M(pn ) Xp = L∞\{0,1,··· ,k} Ck X. where M(pn ) is the mod-pn Moore spectrum. However, this analogy is somewhat unusual because of the inclusion: {bounded below harmonic spectra} j {harmonic spectra} Thus, we are naturally led to: Question 4.3. Is there any relation between the chromatic tower and the Postnikov tower for bounded below harmonic spectra? Warning 4.4. The suspension behaves differently. More precisely, whereas the suspension does not change the chromatic filtration, the suspension shifts

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the Postnikov filtration: Lk (ΣX) = Σ(Lk X),

(ΣX)(−∞,k] = Σ X(−∞,k−1] .

This suggests we must fix some unstable condition to obtain such a relation. For this purpose, we restrict our attention to the Kriz’s case [Kri94]: Let X = T (f ∗ γ), the Thom spectrum of a virtual bundle f ∗ γ over a space S with classifying map f f ∗ γ −−−→ γ y y f

S −−−→ BU, where γ is the universal bundle over BU with dim γ = 0 so that T (γ) = MU f

(if p > 2, consider also S → BSO). Then we offer our speculation: Question . For any X = T (f ∗ γ), as above, is there an inclusion Ker(π∗ X → π∗ Lk X) j Ker(π∗ X → π∗ X(−∞,k] ) for any k ∈ Z≥0 ? In other words, although there is no commutative diagram

Lk X

X ? ?

6∃

?? ?? ?? ?? / X(−∞,k] ,

we speculate the injectivity of πk X → πk Lk X

k ∈ Z≥0 .

The Question is true for the simplest case X = S 0 = T (t), where t is the trivial map t : {a point} → BU. Proposition 4.5. πk S 0 → πk Lk S 0 is injective. Actually, this follows easily from [MRW77] and [Rav84] (see also [HS99] and [Min03] for related techniques to slightly extend Proposition 4.5). However, a very interesting special case is given by ∗ X = T (πBU γ) = MU ∧ S+ ,

where S is a space and πBU is the projection map πBU : BU × S → BU. In fact, for this case, the question (conjecture) essentially claims the injectivity of BPk S+ → (Lk BP )k S+ , and we easily arrive at the following:

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Proposition 4.6. Suppose the Johnson question is true for S, i.e. suppose the canonical map BPk S+ → vk−1 BPk S+ is injective. Then Question is true for X = MU ∧ S+ . Of course, this immediately follows from the commutative diagram: BPk S+ oo ooo o o w o

(Lk BP )k S+

OOO OOO O' / v −1 BP k

k S+

Although we can not prove the converse, it appears that the Question for the special case X = MU ∧ S+ asks almost as much as the Johnson question for the case S. In this way, the Question might be regarded as a conceptual explanation (and even an approach) for the Johnson question. For special cases and related matters concerning the Johnson question, consult e.g. [RW80, JW73, JWY94, HRW98]. References [Bou79a] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281. [Bou99] A. K. Bousfield, On K(n)-equivalence of spaces, Contemp. Math. 239 (1999), 85–89. [Dev98] E. S. Devinatz, A counterexample to a BP -analogue of the chromatic splitting conjecture, Proc. Amer. Math. Soc., 126 (1998) 907–911. [DHS88] E. Devinatz, M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory I, Annals of Math. 128 (1988) 207–241. [HRW98] M. J. Hopkins, D. C. Ravenel and W. S. Wilson, Morava Hopf algebras and spaces K(n)-equivalent to finite Postnikov systems, W. G. Dwyer, et. al., editors, Stable and Unstable Homotopy, Fields Institute Communications, 19 (1998) 137–163. [HS98] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II, Annals of Math. 148 (1998) 1–49. [Hov95] M. Hovey, Bousfield localization functors and Hopkins’ chromatic splitting conjecture, Contemp. Math. 181 (1995) 225–250. [HS99] M. Hovey and H. Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, J. London Math. Soc. 60 (1999), 284–302. [JW73] D. C. Johnson and W. S. Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353. [JWY94] D. C. Johnson, W. S. Wilson, and D. Y. Yan, Brown-Peterson homology of elementary p-groups, II, Topology and its applications, 59 (1994) 117–136. [JY80] D. C. Johnson and Z. Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980) 117–136. [Kri94] I. Kriz, All complex Thom spectra are harmonic, Contemp. Math. 158 (1994) 127– 134. [Kri97] I. Kriz, Morava K-theory of classifying spaces: some calculations, Topology 36 (1997) 1247–1273. [KL00] I. Kriz and K. P. Lee, Odd-degree elements in the Morava K(n)-cohomology of finite groups, Topology Appl. 103 (2000) 229–241.

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[MRW77] H. R. Miller, D. C. Ravenel and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Annals of Math. 106 (1977) 469–516. [Min02] N. Minami, From K(n+1)∗ (X) to K(n)∗ (X), Proc. Amer. Math. Soc. 130 (2002) 1557–1562. [Min03] N. Minami, On the chromatic tower, Amer. J. Math. 125 (2003), 449–473. [Min-a] N. Minami, On the horizontal vanishing line of the E(n)-based modified AdamsNovikov spectral sequence via injective resolutions, preprint. [Min-b] N. Minami, Some chromatic phenomena of bounded below harmonic spectra, preprint. [Mit90] S. A. Mitchell, The Morava K-Theory of Algebraic K-Theory Spectra, K-Theory 3 (1990) 607–626. [Mit92] S. A. Mitchell, Harmonic Localization of Algebraic K-Theory Spectra, Trans. Amer. Math. Soc., 332 (1992) 823–837. [Mor89] J. Morava, Forms of K-theory, Math. Z. 201 (1989) 401–428. [Rav84] D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math., 106 (1984), 351–414. [Rav87] D. C. Ravenel, The geometric realization of the chromatic resolution, Ann. of Math. Stud. 128, Princeton University Press, Princeton, (1987), 168–179. [Rav92] D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Stud. 128, Princeton University Press, Princeton, 1992. [RW80] D. C. Ravenel and W. S. Wilson, The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980) 691–748. [RWY98] D. C. Ravenel, W. S. Wilson and N. Yagita, Brown-Peterson Cohomology from Morava K-Theory, K-Theory 15 (1998), 147–199. [SY95] K. Shimomura and A. Yabe, The homotopy groups of π∗ (L2 S 0 ), Topology, 34 (1995) 261–289. [SW] K. Shimomura and X. Wang, The homotopy groups π∗ (L2 S 0 ) at the prime 3, preprint. [Wil99] W. S. Wilson, K(n + 1) equivalence implies K(n) equivalence, Contemp. Math. 239 (1999), 375–376. Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya 466-8555, JAPAN E-mail address: [email protected]

THE MOTIVIC THOM ISOMORPHISM JACK MORAVA Abstract. The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Deligne’s ideas on motivic Galois groups.

1. Introduction This talk is in part a review of some recent developments in Galois theory, and in part conjectural; the latter component attempts to fit some ideas of Kontsevich about deformation quantization and motives into the framework of algebraic topology. I will argue the plausibility of the existence of liftings of (the spectra representing) classical complex cobordism and K-theory to objects in some derived category of mixed motives over Q. In itself this is probably a relatively minor technical question, but it seems to be remarkably consistent with the program of Connes, Kreimer, and others suggesting the existence of a Galois theory of renormalizations. 1.1. One place to start is the genus of complex-oriented manifolds associated to the Hirzebruch power series z = zΓ(z) = Γ(1 + z) exp∞ (z) [25 §4.6]. Its corresponding one-dimensional formal group law is defined over the real numbers, with the entire function exp∞ (z) = Γ(z)−1 : 0 7→ 0 as its exponential. I propose to take seriously the related idea that the Gamma function Γ(z) ≡ z −1 mod R[[z]] defines some kind of universal asymptotic uniformizing parameter, or coordinate, at ∞ on the projective line, analogous to the role played by the exponential at the unit for the multiplicative group, or the identity function at the unit for the additive group. Date: 15 November 2003. 1991 Mathematics Subject Classification. 11G, 19F, 57R, 81T. The author was supported in part by the NSF. 265

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1.2. The second point of reference is a classical conjecture of Galois theory. The cyclotomic closure Qcyc of the rationals, defined by adjoining all roots of unity to Q, is the maximal extension of Q with commutative Galois group; ˆ × of profinite integers, that group, isomorphic to the multiplicative group Z plays an important role in work of Quillen and Sullivan on the Adams conjecture and in differential topology. Shafarevich (cf. [27]) conjectures that the Galois group Gal(Q/Qcyc ) is a free profinite group; in other words, the full Galois group of Q over Q fits in an exact sequence ˆ× → 1 . d → Gal(Q/Q) → Gal(Qcyc /Q) ∼ 1 → Gal(Q/Qcyc ) ∼ = Free =Z

What will be more relevant here is a related conjecture of Deligne [13 §8.9.5], concerning a certain motivic analog of the Galois group which I will denote Galmot (Q/Q), which is not a profinite but rather a proalgebraic groupscheme over Q; it is in some sense a best approximation to the classical Galois group in this category, which should contain the original group as a Zariski-dense subobject. [I should say that calling this object a Galois group is an abuse of terminology; it is more properly described (cf. §4.4) as the motivic Tate Galois group of Spec Z (without reference to Q).] In any case, this motivic group fits in a similar extension 1 → Fodd → Galmot (Q/Q) → Gm → 1 of groupschemes over Q, where Gm is the multiplicative groupscheme, and Fodd is the prounipotent groupscheme defined by a free graded Lie algebra fodd with one generator of each odd degree greater than one; the grading is specified by the action of the multiplicative group on the Lie algebra. The generators of this Lie algebra are thought to correspond with the odd zeta-values (via Hodge realization, [14 §2.14]) which are expected to be transcendental numbers (and thus outside the sphere of influence of Galois groups of the classical kind). Kontsevich introduced his Gamma-genus in the context of the Duflo - Kirillov theorem in representation theory. He argued that it lies in the same orbit, under an action of the motivic Galois group, as the analog of the clasˆ sical A-genus [3 §8.5]. How this group fits in the topological context is less familiar, and to a certain extent this paper is nothing but an attempt to find a place for that group in algebraic topology. The history of this question is intimately connected with Grothendieck’s theory of anabelian geometry, and it enters Kontsevich’s work through a conjectured Galois action on some form of the little disks operad. My impression these ideas are not yet very familiar to topologists, so I have included a very brief account of some of their history, with a few references, as an appendix below. I should acknowledge here that Libgober and Hoffman [25,40] have studied a genus with related, but not identical, properties, and that an attempt to

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understand their work was instrumental in crystallizing the ideas behind this paper. I owe many mathematicians - including G. Carlsson, D. Christensen, F. Cohen, P. Deligne, A. Goncharov R. Jardine, T. Kohno, and T. Wenger thanks for conversations about the material in this paper, and I am particularly indebted to a very knowledgeable and patient referee. In many cases they have saved me from mistakes and overstatements; but other such errors may remain, and those are the solely my responsibility. I also wish to thank the Newton Institute, and the Fields Institute program at Western Ontario, for support during the preparation of this paper. 2. The Gamma-genus 2.1. The Gamma-function is meromorphic, with simple poles at z = 0, −1, −2, . . . ; we might therefore hope for a Weierstrass product of the form Y z Γ(1 + z)−1 ∼ (1 + ) , n n≥1 from which we might hope to derive a power series expansion X X z z (− )k log Γ(1 + z) ∼ − log(1 + ) ∼ n n n≥1 n,k≥1 for its logarithm. Rearranging this carelessly leads to X ζ(k) X (−z)k 1 ∼ (−z)k , k k n k k≥1 n,k≥1

which is unfortunately implausible since ζ(1) diverges. In view of elementary renormalization theory, however, we should not be daunted: we can add ‘counter-terms’ to conclude that Y X ζ(k) z log (1 + ) e−z/n ∼ − (−z)k , n k n≥1 k≥2 and with a little more care we deduce the correct formula X ζ(k) (−z)k ) , Γ(1 + z) = exp(−γz + k k≥2

where γ is Euler’s constant. Reservations about the logic of this argument may perhaps be dispelled by observing that X ζ(2k) Γ(1 + z) Γ(1 − z) = exp( z 2k ) ; k k≥1 Euler’s duplication formula implies that the left-hand side equals πz , zΓ(z) Γ(1 − z) = sin πz

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consistent with the familiar evaluation of ζ at positive even integers in terms of Bernoulli numbers. 2.2. From this perspective, the Hirzebruch series X ζ(odd) πz 1 Γ(1 + z) = ( ) 2 exp(−γz + z odd ) ; sin πz odd for Kontsevich’s genus does in fact look like some kind of deformation of the ˆ A-genus; its values on a complex-oriented manifold will be polynomials in odd zeta-values, with rational coefficients. Similarly, the Witten genus Y x/2 φW (x) = [(1 − q n u)(1 − q n u−1 )]−1 sinh x/2 n≥1 ([57], with u = ex ) can be written in the form X xk exp(−2 gk ) , k! k≥1

where the coefficients gk are modular forms, with godd = 0: it is also a ˆ in another direction. deformation of A, [Behind the apparent discrepancies in these formulae is the issue of complex versus oriented cobordism: there are several possible conventions relating Chern and Pontrjagin classes. Hirzebruch expresses the latter as symmetric functions of indeterminates x2i , and writes the genus associated to the formal Q ˆ series Q(z) as Q(x2i ); thus for the A-genus, 1√ z Q(z) = 2 1 √ . sinh 2 z An alternate convention, used here, writes this symmetric function in the form Y 1 1 −xi /2 xi /2 )2 · ( )2 ) . (( sinh xi /2 sinh(−xi /2) The relation between the indeterminates x and z is a separate issue; I take z to be 2πix.] Kontsevich suggests that the values of the zeta function at odd positive integers (expected to be transcendental) are subject to an action of the moˆ tivic group Galmot (Q/Q), and that the A-genus and his Γ genus lie in the same orbit of this action. One natural way to understand this is to seek an action of that group on genera, and thus on the complex cobordism ring; or, perhaps more naturally, on some form of its representing spectrum. Before confronting this question, it may be useful to present a little more background on these zeta-values.

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3. Symmetric and quasisymmetric functions 3.1. The formula Y

(1 + xk z) =

k≥1

X

ek z k = exp(−

k≥0

X pk k

(−z)k ) ,

k≥1

where ek is the kth elementary symmetric function, and pk is the kth power sum, can be derived by formal manipulations very Q much like those in the preceding section, by expanding the logarithm of (1+ xn z); such arguments go back to Newton. The specialization xk 7→ k −2

(cf. the second edition of MacDonald’s book [43 Ch I §2 ex 21]) leads to Bernoulli numbers, but the map xk 7→ k −1 is trickier, because of convergence problems like those mentioned above; it defines a homomorphism from the ring of symmetric functions to the reals, sending pk to ζ(k) when k > 1, while p1 7→ γ [24]. Under this homomorphism the even power sums p2k take values in the field Q(π). 3.2. The Gamma-genus is thus a specialization of the formal group law with exponential Y X z =z (1 + xk z)−1 = (−1)k hk z k+1 Exp∞ (z) = e(z) k≥1 k≥0 having the complete symmetric functions (up to signs) as its coefficients. This is a group law of additive type: its exponential, and hence its logarithm, are both defined over the ring of polynomials generated by the elements hk . This group law is classical: it is defined by the Boardman-Hurewicz-Quillen complete Chern number homomorphism MU ∗ (X) → H ∗ (X, Z[h∗ ]) defined on coefficients by the homomorphism Lazard → Symm from Lazard’s ring which classifies the universal group law of additive type. The Landweber-Novikov Hopf algebra S∗ = Z[t∗ ] represents the prounipotent groupscheme D0 of formal diffeomorphisms X z 7→ t(z) = z + tk z k+1 k≥1

of the line, with coproduct

∆(t(z)) = (t ⊗ 1)((1 ⊗ t)(z)) ∈ (S∗ ⊗ S∗ )[[z]] .

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The universal group law t−1 (t(X) + t(Y )) of additive type is thus classified by the homomorphism Lazard → Lazard ⊗ S∗ → Z ⊗ S∗ → Symm representing the orbit of the Thom map Lazard → Z (which classifies the additive group law) under the action of D0 . This identifies the algebra S∗ with the ring of symmetric functions by tk 7→ (−1)k hk . 3.3. The symmetric functions are a subring Symm → QSymm of the larger ring of quasisymmetric functions, which is Hopf dual to the universal enveloping algebra ZhZ1 , . . . i of the free graded Lie algebra f∗ with one generator in each positive degree [23], given the cocommutative coproduct X ∆Zi = Zj ⊗ Z k . i=j+k

Standard monomial basis elements for this dual Hopf algebra, under specializations like those discussed above [6 §2.4, 23], map to polyzeta values X 1 ζ(i1 , . . . , ik ) = ik ∈ R ; i1 n · · · n 1 k n1 >···>nk ≥1 note that there are convergence difficulties unless i1 > 1. If we think of the Gamma-genus as taking values in the field Q(ζ) ⊂ R generated by such polyzeta values, then it is the specialization of a homomorphism Lazard → Symm → QSymm representing a morphism from the prounipotent groupscheme F with Lie algebra f∗ to the moduli space of one-dimensional formal group laws. Because we are dealing with group laws of additive type, there seems to be little loss in working systematically over a field of characteristic zero, where Lie-theoretic methods are available. Over such a field any formal group is of additive type: the localization of the map from the Lazard ring to the symmetric functions is an isomorphism. Similarly, over the rationals the Landweber - Novikov algebra is dual to the enveloping algebra of the Lie algebra of vector fields zk = z k+1 ∂/∂z , k ≥ 1 on the line, and the embedding of the symmetric in the quasisymmetric functions sends the free generators Zk to the Virasoro generators zk , corresponding to a group homomorphism F → D0 .

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3.4. It will be useful to summarize a few facts about Malcev completions and pro-unipotent groups [13 §9]. The rational group ring Q[G] of a discrete [ with group G has a natural lower central series filtration; its completion Q[G] respect to that filtration is a topological Hopf algebra, whose continuous dual represents a pro-unipotent groupscheme over Q. Applied to a finitelygenerated free group, for example, this construction yields a Magnus algebra of noncommutative formal power series. There are many variations on this theme: in particular, an action of the multiplicative group Gm defines a grading on a Lie algebra. The action tk 7→ uk tk , (u a unit) on the group of formal diffeomorphisms defines an extension of its Lie algebra by a new Virasoro generator v0 , corresponding to an extension S∗ [t± 0 ] of the Landweber-Novikov algebra. The group of formal diffeomorphisms is pro-unipotent, and this enlarged object is most naturally interpreted as a semidirect product D0 o Gm . Grading the free Lie algebra f similarly extends the homomorphism above to F o Gm → D o G m . 4. Motivic versions of classical K-theory and cobordism 4.1. There are now several (eg [39, 55]) good and probably equivalent constructions of a triangulated category DM(k) of motives over a field k of characteristic zero. The subject is deep and fascinating, and I know at best some of its vague outlines. Since this paper is mostly inspirational, I will not try to provide an account of that category; but as it is after all modelled on spectra, it is perhaps not too much of a reach to think that some of its aspects will look familiar to topologists. One approach to defining a motivic category DM(k) starts from a category whose morphisms are elements of a group of algebraic correspondences. At some later point it becomes useful to tensor these groups with Q, resulting in a category DMQ (k) whose Hom-objects are rational vector spaces. The underlying concern of this paper is the relation of such motivic categories to classical topology; but stable homotopy theory over the rationals is equivalent to the theory of graded vector spaces. This has the advantage of rendering some of the conjectures below almost trivially true – and the disadvantage of making them essentially contentless. Behind these conjectures, however, lies the hope that they might say something before rationalization, and for that reason I have outlined here a rough theory of integral geometric realizations of motives:

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4.2. The category DM(k) contains certain canonical Tate objects Z(n), defined [55 §2.1, but see also 13 §2] as tensor powers of a reduced version Z(1) of the projective line. Grothendieck’s original category of ‘pure’ motives, constructed from smooth projective varieties, is (in some generality [28]) semisimple, but categories of motives built from more general (non-closed) varieties admit nontrivial extensions. The (derived) category DMTQ (k) of mixed Tate motives can be defined as the smallest tensor triangulated subcategory of DMQ (k) containing the Tate objects. In this rationalized category, it is natural to denote the (images of) the generating objects by Q(n); however, I will be most interested here in the case k = Q and in a certain more subtle construction of a (rationalized, though I will now drop the subscript) subcategory DMT (Z) of DMTQ (Q) [14 §1.6], closely related to the motives over Spec Z ‘with integral coefficients’ in the sense of [13 §1.23, 2.1]. This is still a Q-linear category, but its objects have stronger integrality properties than one might naively expect. In particular: one of the foundation-stones of the theory of mixed motives is an isomorphism Ext1 (Q(0), Q(n)) ∼ = K2n−1 (Z) ⊗ Q M T (Z)

(cf. [2, 13 §8.2]. As the referee points out, one has to be careful here; the corresponding description for the category MT (Q) involves the algebraic Ktheory of Q, which is much larger than that of Z). The groups on the right have rank one for odd n > 1, and vanish otherwise, by work of Borel; the theory of regulators says that to some extent the zeta-values (n − 1)! ζ(n) (2πi)n (cf. [13 §3.7]) can be interpreted as natural generators for these groups. This is strikingly reminiscent to a homotopy-theorist of the identification (for n even) of the group Ext1 Adams (K(S 0 ), K(S 2n )) of extensions of modules over the Adams operations, with the cyclic subgroup of Q/Z generated by this zeta-number. These connections between the image of the J-homomorphism and the groups K4k−1 (Z), go back to the earliest days of algebraic K-theory [16, 50]. 4.3. This suggests that there might be some use for a notion of geometric or homotopy-theoretic realization for motives, which manages to retain some integral information. Aside from tradition (algebraic geometers usually work with cycles over Q, and topologists have been neglecting correspondences since Lefschetz), there seems to be no obstacle to the development of such a theory. Indeed, let E be a multiplicative (co)homology functor (ie a ringspectrum), supplied with a natural class of E-orientable manifolds: if E were stable homotopy, for example, we could use stably parallelizeable manifolds.

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In the case of interest below, however, E will be complex K-theory, and the manifolds will be smooth (proper) algebraic varieties over C. Such manifolds (X, Y, . . . ) define an additive category CorrE with E∗ (X × Y ) (suitably graded) as Hom-objects; composition of such morphisms can be defined using Pontrjagin-Thom transfers [45]. It is straightforward to check that X 7→ X+ ∧ E : CorrE → (Spectra) is a functor: the necessary homomorphism E∗ (X × Y ) = [S ∗ , X+ ∧ Y+ ∧ E] → [X+ ∧ E, Y+ ∧ E]∗ of Hom-objects is defined by the adjoint composition X+ ∧ Y+ ∧ E ∧ X+ ∧ E ∼ = X+ ∧ X+ ∧ Y+ ∧ E ∧ E → Y+ ∧ E built from the multiplication map of E and the composition X+ ∧ X+ ∧ E → X+ ∧ E → E of the transfer ∆! associated to the diagonal map, with the projection of X to a point. Following the pattern laid out by Voevodsky, we can now define a category of (topological) ‘E-motives’, and when E = K it is a classical fact [1] that an algebraic cycle defines a nice K-theory class. This allows us to associate to an embedding of k in C, a triangulated ‘realization’ functor Kmot : X 7→ X(C)+ ∧ K : DM(k) → (Spectra) . [Since algebraic cycles are triangulable (by Lojasiewicz), a theorem of Sullivan allows us to define these cycles in connective complex K-theory.] 4.4. When k is a number field, DMTQ (k) possesses a theory of truncations, or t-structures [14, 37], analogous to the Postnikov systems of homotopy theory; the heart of this structure is an abelian tensor category MTQ (k) of mixed Tate motives. Its existence permits us to think of DMTQ (k) as the derived category of MTQ (k); in particular, the (co)homology of an object of the larger category becomes in a natural way [31 §2.4] an object of MTQ (k). Similar considerations hold for the more rigid category DMT (Z), and since we are working in a rational, stable context, I will write π∗ for the homology groups of an object in this category, given this enriched structure. [For the purposes of this presentation I’ve reversed the logical order of construction: in fact in [14] the category MT (Z) is constructed first.] Now under very general conditions (involving a suitably rigid duality), an abelian tensor category with rational Hom-objects can be identified with a category of representations of a certain groupscheme of automorphisms of a suitable forgetful functor on the category; the resulting groupscheme is called a motivic (Galois) group. This theory applies to MT (Z), and as was noted in the introduction, Galmot (Q/Q) is the corresponding groupscheme [13

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§8.9.5, 14 §2; cf. also 2 §5.10]. In the preceding paragraph we constructed a homological functor Kmot := π∗ Kmot from DMQ to Q-vector spaces, together with a preferred lift of the functor to the category of spectra. It is easy to see that πodd Kmot = 0 , while π2n Kmot = Q(n) , as representations of Galmot (Q/Q), and thus that Kmot is represented by the mixed (Bott!)-Tate object ⊕n∈Z Q(n)[2n] = Q[b± ] ; in other words, we have constructed a lifting of the rationalized classical K-theory functor to the category of mixed Tate motives, with an action of Galmot (Q/Q) which factors through the multiplicative quotient. The possible existence of a descent spectral sequence for the automorphisms of the K-theoretic realization functor of §4.3 seems to be an interesting question, especially when restricted to some category of mixed Tate motives. 4.5. The main conjecture of this paper is that, similarly, a rational version of complex cobordism lifts to an object MUmot ∈ DMT (Z), with an action of Galmot (Q/Q) on π∗ MUmot defined by the obvious embedding Fodd o Gm → F o Gm followed by a homomorphism from the latter group to the diffeomorphisms of the formal line, cf. §2.4 above. This is a conjecture about an object characterized by its universal properties, so it can be reformulated in terms of the structures thus classified. The theory of Chern classes is founded on Grothendieck’s calculation of the cohomology of the projectification P (V ) of a vector bundle V over a scheme X. It follows immediately from his result that the cohomology of Atiyah’s model X V := P (V ⊕ 1)/P (V ) for a Thom space as a relative motive is free on one generator over that of X. Such a generator is a Thom class for V , but in the motivic context there seems to be no natural way to construct such a thing; this is related to the inconvenient nonexistence of abundantly many sections of vector bundles in the algebraic category. For a systematic theory of Thom classes it is enough, according to the splitting principle, to work with line bundles L, and in this context it is relevant that the Thom complexes −1 X L = P (L ⊕ 1)/P (L) ∼ = P (1 ⊕ L−1 )/P (L−1 ) = X L of a line bundle and its reciprocal are isomorphic objects. The conjecture about MUmot can be thus reformulated in terms of a theory of motivic

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Thom and Euler classes Umot (L), e(L) for line bundles L, satisfying a motivic Thom axiom Umot (L−1 ) = −Umot (L) , e(L−1 ) = −e(L) ; the conjecture is then the assertion that an element σ ∈ Galmot (Q/Q) sends Umot (L) to another Thom class X σ(Umot (L)) = [1 + σk e(L)k ] · Umot (L) k>0

for L, with coefficients σk depending only on σ. Since σ(Umot (L−1 )) = −σ(Umot (L)) , it follows that X X [1 + σk (−e(L))k ] · (−Umot (L)) = −[1 + σk e(L)k ] · Umot (L) , k>0

k>0

which entails that the classes σodd = 0, distinguishing the Hopf subalgebra Sev = Z[t2k | k > 0] which represents the group of odd diffeomorphisms of the formal line [46 §3.3]. Away from the prime two, classical complex cobordism is a kind of base extension MU[1/2] ∼ SO/SU ∧ MSO[1/2] of oriented cobordism, and I’m suggesting the existence of a similar splitting for the hypothetical motivic lift of complex cobordism. 4.6. After this paper had been submitted for publication, I became aware of the very elegant recent work of Levine and Morel [38], where an algebraic cobordism functor is characterized as a universal cohomology theory on the category of schemes, endowed with pullback and pushforward transformations satisfying certain natural axioms pf compatibility. [Voevodsky [56] has also considered a motivic version of the cobordism spectrum; its relation with their work is discussed briefly in the introduction to their paper.] I believe their work is fundamentally compatible with the conjectures made here, given a slight difference in framework and emphasis: they suppose a Thom isomorphism (or, equivalently, a system of covariant transfers) is to be given as part of the structure of a cohomology theory on schemes, while the spectrum hypothesized here is merely a ringspectrum, with no preferred choice of orientation.

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4.7. I should note that the trivial action of Fodd on π∗ (Kmot ), together with the usual action of Gm defined by the grading, is consistent with the existence of a spectral sequence ∗ ) =⇒ K ∗ (Z) ⊗ Q E2∗,∗ = Hc∗ (Galmot (Q/Q), Kmot

of descent type, as suggested above: using the Hochschild-Serre spectral sequence for the semidirect product decomposition of the Galois group (and confusing continuous cochain with Lie algebra cohomology) we start with ∗ Hc∗ (Fodd , Kmot )∼ = Qhe2k+1 | k ≥ 1i[b] ,

where angled brackets denote a vector space spanned by the indicated elements (with e2r+1 in degree (1, 0), and the Bott (-Tate?!) element b in degree (0, −2)). The Gm -action sends b to ub, where u is a unit in whatever ring we’re over; similarly, e2k+1 7→ u−2k−1 e2k+1 . Thus e2k+1 b2k+1 ∈ E21,−4k−2 is Gm -invariant, yielding a candidate for the standard generator in K4k+1 (Z) ⊗ Q. 5. Quantization and asymptotic expansions 5.1. The motivic Galois group appears in Kontsevich’s work through a conjectured action on deformation quantizations of Poisson structures [cf. [53]]. The framework of this paper suggests a plausibly related action on an algebra of asymptotic expansions for geometrically defined functionals on manifolds, interpreted in terms of the cobordism ring of symplectic manifolds [18, 19]. This is isomorphic to the complex cobordism (abelian Hopf) algebra MU∗ (BGm (C)) of circle bundles, and the dual (rationalized) Hopf algebra MUQ∗ (BGm (C)) ∼ = MUQ∗ [[}]] , P where } = k≥1 CPk−1 ek /k, can be interpreted [46] as an algebra generated by the coefficients of a kind of universal asymptotic expansion for geometrically defined heat kernels (or Feynman measures, via the Feynman-Kac formula [20 §3.2]), as the Chern class e of the circle bundle approaches infinity. A Poisson structure on an even-dimensional manifold V is a bivector field (a section of the bundle Λ2 TV ) satisfying a Jacobi identity modelled on that satisfied by the inverse of a symplectic structure. A symplectic manifold is thus Poisson, and although I am aware of no useful notion of Poisson cobordism (but cf. [7]) one expects a natural restriction map from asymptotic invariants Poisson manifolds to the corresponding ring for symplectic manifolds. If the conjectures above are correct, then one might further hope that (some motivic version of) such a restriction map would be equivariant, with respect to some motivic Galois action.

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5.2. Working in the opposite (local to global) direction, Connes and Kreimer have recently developed a systematic program for understanding classical quantum field theoretic renormalization in terms of its symmetries. The standard methods (eg dimensional regularization) for dealing with the singular integrals which appear in classical perturbation theory replaces them with certain meromorphic functions, and through work of Broadhurst, Kreimer, and others it has become more and more clear that the polar coefficients of these meromorphic functions are frequently elements of the polyzeta algebra. [Kontsevich has suggested that this is always so, but his program of proof fails, by the arguments of [4], and at present the question seems to be open.] Connes and Kreimer[12, 21, 26, 35 §12] have developed a systematic approach to the theory of Feynman integrals through certain Hopf algebras related to automorphism groups [42] of operads defined by graphs of various sorts. There are deep connections between the Grothendieck - Teichm¨ uller group and the Lie algebras of these automorphism groups [29, 54], and it seems likely that they (and the theory of quasisymmetric functions, via free Lie algebras) will eventually be understood to be intimately related; the appearance of polyzeta values in the theory of quantum knot invariants (cf. eg [36]) is another source of recent interest in this subject. Perhaps the deepest (and most precise) approach to the relations between these topics may be the work of Goncharov, who associates to a field F a certain Hopf algebra T• (F ) of F -decorated planar trivalent trees and a closely related Hopf algebra I• (F ) of motivic iterated integrals. According to the correspondence principle of [21 §7], the renormalization Hopf algebra corresponding to certain types of Feynman integrals should be closely related to a precisely defined subgroup of the motivic Tate Galois group of F . A slight strengthening of this correspondence principle would settle the question of the role of polyzeta values in perturbative expansions of Feynman integrals. 5.3. In exemplary cases Connes and Kreimer construct a very interesting representation of the prounipotent groupscheme underlying their renormalization algebra in the group of odd formal diffeomorphisms of the line [10 §1 eq. 20, §4 eq. 2]. It also seems quite possible that the action of their groupscheme on asymptotic expansions defined by Feynman measures associated to suitable Lagrangians [30] factor through an action of the motivic Galois group on cobordism, along the lines suggested in §4.5 above. appendix: motivic models for the little disk operad 1. In 1984 Grothendieck suggested the study of the action of the Galois group Gal(Q/Q) as automorphisms of the moduli of algebraic curves, understood as a collection of stacks linked by morphisms representing various geometrically natural fusion operations. There are remarkable analogies between his ideas and contemporary work in physics on conformal field theories, and in 1990 Drinfel’d [15] unified at least some of these lines of thought by constructing a

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pronilpotent group GT of automorphisms of certain braided tensor categories, together with a faithful representation of the absolute Galois group in that group. This program has been enormously productive; the LMS notes [42, 51] are one possible introduction to this area of research, but evolution has been extremely rapid. In the late 90’s Kontsevich [33, 34] recognized connections between these ideas, Deligne’s question on Hochschild homology, deformation quantization, and other topics, while physicists [9-12] interested in the algebra of renormalization were developing sophisticated Hopf-algebraic techniques, which are now believed [5 §8,9] to be closely related to the Hopf-algebraic constructions of Drinfel’d. The point of this appendix is to draw attention to a central conjecture in this circle of ideas: that the Lie algebra of GT, which acts as automorphisms of the system of Malcev completions of the braid groups, is a free graded Lie algebra, with one generator in each odd degree greater than one. The braid groups in fact form an operad, and I want to propose the related problem of identifying the automorphisms of the operad of Lie algebras defined by the braid groups (cf. [8]), in hope that this will shed some light on this question, and the closely related conjecture that Deligne’s motivic group acts faithfully on the unipotent motivic fundamental group [13] of P)1 −{0, 1, ∞} (with nice tangential base point). 2. For the record, an operad (in some reasonable category) is a collection of objects {On , n ≥ 2} together with composition morphisms Y cI : Or(I) × Oi → O|I| i∈I

P where I = i1 , . . . , ir is an ordered partition of |I| = ik with r(I) parts. These compositions are subject to a generalized associativity axiom, which I won’t try to write out here; moreover, the operads in this note will be permutative, which entails the existence of an action of the symmetric group Σn on On , also subject to unspecified axioms. Not all of the operads below will be unital, so I haven’t assumed the existence of an object O1 ; but in that case, and under some mild assumptions, an operad can be described as a monoid in a category of objects with symmetric group action, with respect to a somewhat unintuitive product [cf. eg. [17]]. The moduli {M0,n+1 } of stable genus zero algebraic curves marked with n + 1 ordered smooth points form such an operad: if the final marked point is placed at infinity, then composition morphisms are defined by gluing the points at infinity of a set of r marked curves to the marked points away from infinity on some curve marked with r + 1 points. This is an operad in the category of algebraic stacks defined over Z, so the Galois group Gal(Q/Q) acts on many of its topological and cohomological invariants.

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By definition, a stable algebraic curve possesses at worst ordinary double point singularities. The moduli of smooth genus zero curves thus define a system M0,∗ ⊂ M0,∗ of subvarieties, but not a suboperad: the composition maps glue curves together at smooth points, creating new curves with nodes from curves without them. However, the spaces M0,n+1 (C) of complex points of these moduli objects have the homotopy types of the spaces of configurations of n distinct points on the complex line, which are homotopy-equivalent to the spaces of the little disks operad C2 . Behind the conjectures of Deligne, Drinfel’d, and Kontsevich lies an apparently unarticulated question: is the little disks operad defined over Q?; or, more precisely: is there a version of the little disks operad in which the morphisms, as well as the objects, lie in the category of algebraic varieties defined over the rationals? Thus in [33] (end of §4.4) we have The group GT maps to the group of automorphisms in the homotopy sense of the operad Chains(C2 ). Moreover, it seems to coincide with Aut(Chains(C2 )) when this operad is considered as an operad not of complexes but of differential graded cocommutative coassociative coalgebras . . . and in [34] (end of §3) There is a natural action of the Grothendieck-Teichm¨ uller groups on the rational homotopy type of the [Fulton-MacPherson version of the] little disks operad . . . although the construction in [34 §7] is not (apparently) defined by algebraic varieties. It may be that the question above is naive [cf. [53]], but a positive answer would imply the existence of a system of homotopy types with action of the Galois groups, whose algebras of chains would have the properties claimed above; moreover, the system of fundamental groups of these homotopy types would yield an action of the Galois group on the system of braid groups, suitably completed. 3. This is probably a good place to note that Gal(Q/Q) acts by automorphisms of the etal´e homotopy type of a variety defined over Q, which is not at all the same as a continuous action on the space of complex points of the variety. In fact one expects to recover classical invariants of a variety (the cohomology, or fundamental group, of its complex points, for example) only up to some kind of completion. Various kinds of invariants [etal´e, motivic, Hodge-deRham, . . . ] each have their own associated completions, some of which are still quite mysterious; the fundamental group, in particular, comes in profinite [49] and prounipotent versions. Drinfel’d works with the latter, which corresponds to the Malcev completion used in rational homotopy theory. A free group on n generators corresponds to the graded Lie algebra defined by n noncommuting polynomial generators, and the Lie algebra pn defined by the pure braid group Pn on n

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strands [the fundamental group of the space of ordered configurations of n points in the plane] is generated by elements xik , 1 ≤ i < k ≤ n subject to the relations [xik , xst ] = 0 if i, k, s, t are all distinct, and [xik , xis ] = −[xik , xks ] if i, k, s are all distinct [32]. The fundamental groups of the little disks operad define a (unital) operad {Pn } (the symmetric group action requires some care [47 §3]); in particular, cabling Y cI : Pr(I) × Pi → P|I| i∈I

of pure braids defines composition operations which extend to homomorphisms Y cI : pr(I) × pi → p|I| i∈I

of Lie algebras, defining an operad {pn } in that category as well. The natural product of Lie algebras is the direct sum of underlying vector spaces, so cI is a sum of two terms, the second defined by the juxtaposition operation pi1 ⊕ · · · ⊕ pir → pi1 +···+ir . The remaining information is contained in a less familiar homomorphism c0I : pr(I) → p|I| defined on the first component by any partition I of |I| in r parts. It is not hard to see that X c0I (xst ) ≡ xpq + . . . ; p∈is ,q∈it

it would be very useful to know more about this expansion . . .

4. Groups act on themselves by conjugation, and thus in general to have lots of (inner) automorphisms; Lie algebras act similarly on themselves, by their adjoint representations. It would also be useful to understand something about the relations between the automorphisms of an operad in groups or Lie algebras (as monoids in a category of objects {O∗ } with {Σ∗ }-action, as in §2 above), and systems of inner automorphisms of the objects On : thus the adjoint action of a system of elements φn ∈ pn defines an operad endomorphism if c0I (φ∗ ) = φ|I| for all partitions I. From some perspective, the classification of such endomorphisms is really part of the theory of symmetric functions. This may be relevant, because the action GT on the completed braid groups is relatively close to inner. Drinfel’d [15 §4] describes elements of GT as pairs (λ, f ), where λ is a scalar (ie, an element of the field of definition for the kind of Lie algebras we’re working with: in our case, Q, for simplicity), and f

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lies in the commutator subgroup of the Malcev completion of the free group on two elements. The pairs (λ, f ) are subject to certain restrictions which I’ll omit here, on the grounds that the corresponding conditions on the Lie algebra are spelled out below. It is useful to regard elements f = f (a, b) of the free group as noncommutative functions of two parameters a, b; if σi is a standard generator of the braid group, and yi = σi−1 · · · σ2 σ12 σ2 . . . σi−1 ∈ Pn for 2 ≤ i ≤ n (with y1 = 1), then the action of GT on the braid group is defined by (λ, f )(σi ) = f (σi2 , yi )σiλ f (σi2 , yi )−1 ; the omitted conditions imply that when i = 1, this reduces to an exponential automorphism σ1 7→ σ1λ defined in the Malcev completion. 5. Here are some technical details about the Lie algebra of GT, reproduced from [15 §5]. Drinfel’d observes [remark before Prop. 5.5] that the scalar term λ can be used to define a filtration on this Lie algebra, and he describes the associated graded object grt. The following formalism is useful: fr is the free formal Q-Lie algebra defined by power series in two noncommuting generators A, B: it is naturally filtered by total polynomial degree, with the free graded Lie algebra on two generators as associated graded object. The algebra grt consists of series ψ = ψ(A, B) ∈ fr which are antisymmetric [ψ(A, B) = −ψ(B, A)] and in addition satisfy the relations ψ(C, A) + ψ(B, C) + ψ(A, B) = 0 and [B, ψ(A, B)] + [C, ψ(A, C)] = 0 when A + B + C = 0, as well as a third relation asserting that ψ(x12 , x23 + x24 ) + ψ(x13 + x23 , x34 ) − ψ(x12 + x13 , x24 + x34 ) equals ψ(x23 , x34 ) + ψ(x12 , x23 ) , assuming that the xik satisfy the relations defining p. The bracket in grt is hψ1 , ψ2 i = [ψ1 , ψ2 ] + ∂ψ2 (ψ1 ) − ∂ψ1 (ψ2 ) , where ∂ψ is the derivation of fr given by ∂ψ (A) = [ψ, A], ∂ψ (B) = 0. Drinfel’d shows [15 §5.6] that this Lie algebra is in fact isomorphic to the Lie algebra of GT [omitting the subalgebra corresponding to the scalars, used in this description to define the grading], but the isomorphism is defined inductively, so describing its action on the braid groups is not immediate. Nevertheless, Ihara (cf. [15 §6.3]) has shown that for each odd n > 1 there are elements ψn ∈ grt such that X n ψn (A, B) ≡ (adA)m−1 (adB)n−m−1 [A, B] m 1≤m≤n−1

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modulo [fr0 , fr0 ] (where fr0 is the derived Lie algebra of fr]. It is conjectured that grt is free on these generators. Aside from [41], the cabling described in §3 doesn’t seem to have been considered very closely, in this context.

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References 1. M. Atiyah, F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962) 25 - 45. 2. A. Beilinson, Height pairing between algebraic cycles, in K-theory, arithmetic and geometry, Springer LNM 1289 (1987) 3. N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators, Springer Grundlehren 298 (1992) 4. P. Belkale, P. Brosnan, Matroids, motifs, and a conjecture of Kontsevich, available at math.AT/0012198 5. P. Cartier, A mad day’s work . . . BAMS 38 (2001) 389 - 408 6. ——–, Functions polylogarithms, nombres polyz´etas et groupes prounipotents, Sem. Bourbaki 885, in Asterisque 282 (2002) 7. A. Cattaneo, The lagrangian operad (private communication) 8. F. Cohen, T. Sato, On groups of homotopy groups, and braid-like groups (private communication) 9. A. Connes, Symetries Galoisiennes et renormalisation, available at math.QA/0211199 10. ——–, D. Kreimer, Renormalization in quantum field theory and the Riemann - Hilbert problem I : the Hopf algebra structure of graphs and the main theorem, available at hep-th/9912092 11. ——–, ——–, II: the β-function, diffeomeorphisms, and the renormalization group, available at hep-th/0003188 12. ——–, ——–, Insertion and elimination: the doubly - infinite Lie algebra of Feynman graphs, available at hep-th/0201157 13. P. Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois groups over Q, MSRI Publ. 16 (79 - 313) 1989 14. ——–, A. Goncharov, Groupes fondementaux motiviques de Tate mixte, available at math.NT/0302267 15. V. Drinfel’d, Quasitriangular quasiHopf algebras and a group closely related to Gal(Q/Q), Petersburg Math. Journal 2 (1991) 16. W. Dwyer, S. Mitchell, On the K-theory spectrum of a ring of algebraic integers, K-Theory 14 (1998) 201 - 263 17. B. Fresse, Lie theory of formal groups over an operad, J. Alg 202 (1998) 18. V.L. Ginzburg, Calculation of contact and symplectic cobordism groups, Topology 31 (1992) 767-773 19. ——–, V. Guillemin, Y.Karshon, The relation between compact and non-compact equivariant cobordisms, in the Rothenberg Festschrift (1998), 99 - 112, Contemp. Math. 231, AMS (1999) 20. J. Glimm, A. Jaffe Quantum physics. A functional integral point of view, Springer (1981) 21. A. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, available at math.AG/0208144 22. A. Grothendieck, Esquisse d’un programme, published in [49] below 23. M. Hazewinkel, The algebra of quasi-symmetric functions is free over the integers, Adv. Math. 164 (2001) 283 - 300 24. M. Hoffman, The algebra of multiple harmonic series, J. Alg. 194 (1997) 477 - 495 25. ——-, Periods of mirrors and multiple zeta values, available at math.AG/9908065 26. ——-, Combinatorics of rooted trees and Hopf algebras, available at math.CO/0201253 27. K. Iwasawa, On solvable extensions of algebraic number fields, Ann. of Maths. 58 (1953) 548 - 572

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28. U. Jannsen, Motives, numerical equivalence, and semisimplicity, Inv. Math. 107 (1992) 447 - 452 29. R. Kaufmann, On spineless cacti, Deligne’s conjecture, and Connes-Kreimer’s Hopf algebra, available at math.QA/0308005 30. D. Kazhdan, Introduction to QFT, in Quantum fields and strings 377 - 418, AMS (1999) 31. R. Kiehl, R. Weissauer, Weil conjectures, perverse sheaves, and l-adic Fourier transform, Springer Ergenbisse III no. 42 (2001) 32. T. Kohno, S´erie de Poincar´e-Koszul associ´e aux groupes de tresses pures, Invent. Math. 82 (1985) 33. M. Kontsevich, Operads and motives in deformation quantization, available at math.QA/990405 34. ——–, Y. Soibelman, Deformations of algebras over operads and Deligne’s conjecture, available at math.QA/000115 35. D. Kreimer, Knots and Feynman diagrams, Cambridge Lecture Notes in Physics 15, CUP (2000) 36. TTQ Le, J. Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions. Top. Appl. 62 (1995) 193 - 206 37. M. Levine, Tate motives and the vanishing conjectures for algebraic K-theory, in Algebraic K-theory and algebraic topology, Lake Louise 1991, Kluwer (1993) 38. ——–, F. Morel, Algebraic cobordism, available at www.math.uiuc.edu/K-theory 39. ——–, Mixed Motives, AMS Surveys 57 (1998) 40. A. Libgober, Chern classes and the periods of mirrors, Math. Res. Lett 6 (1999) 141 149 41. P. Lochak, L. Schneps, The Grothendieck-Teichm¨ uller group and automorphisms of braid groups, p. 323 - 358 in [42] below 42. ——–, ——–, Geometric Galois actions I, London Math Soc. notes 242, CUP (1997) 43. I. MacDonald, Symmetric functions and Hall algebras, 2nd ed, OUP 44. I. Moerdijk, On the Connes-Kreimer construction of Hopf algebras, available at math.phy/9907010 45. J. Morava, Smooth correspondences, in Prospects in topology (Browder Festschrift), Ann. of Math. Studies 138, Princeton (1995). 46. ———, Cobordism of symplectic manifolds and asymptotic expansions, in Proc. Steklov Inst. Math. 225 (1999) 261 - 268 47. ——–, Braids, trees, and operads, available at math.AT/0109086 48. ——–, Heisenberg groups in algebraic topology, to appear in the Segal Festschrift; available at math.AT/0305250 49. H. Nakamura, L. Schneps, On a subgroup of the Grothendieck-Teichm¨ uller group acting on the tower of profinite Teichm¨ uller modular groups, Invent. Math. 141 (2000) 503 560 50. D. Quillen, letter to Milnor, in Algebraic K-theory (Northwestern 1976), Lecture Notes in Math. 551 (Springer) 51. L. Schneps (ed.), The Grothendieck theory of dessins d’enfants, London Math Soc. notes 200, CUP (1994) 52. ——–, The Grothendieck-Teichm¨ uller group: a survey, p. 183 - 204 in [51] above 53. D. Tamarkin, Action of the Grothendieck-Teichm¨ uller group on the operad of Gerstenhaber algebras, available at math.QA/0202039 54. P. van der Laan, I. Moerdijk, The renormalization bialgebra and opeads, available at hep-th/0210226

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55. V. Voevodsky, Triangulated categories of motives over a field, in E. Friedlander, A. Suslin, V. Voevodsky, Cycles, transfers, and motivic homology theories, Annals of Maths. Studies 143, Princeton (2000) 56. ———, Open problems in the motivic stable homotopy theory I, available at www.math.uiuc.edu/K-theory 57. D. Zagier, A note on the Landweber-Stong elliptic genus, in Elliptic curves and modular forms in algebraic topology, ed. P. Landweber, Lecture Notes in Mathematics 1326, Springer (1988) Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: [email protected]

TOWARD HIGHER CHROMATIC ANALOGS OF ELLIPTIC COHOMOLOGY DOUGLAS C. RAVENEL Abstract. We show that the Jacobian of a certain Artin-Schreier curve over the field Fp has a a 1-dimensional formal summand of height (p − 1)f for any positive integer f . We give two proofs, the classical one which was known to Manin in 1963 and which requires knowledge of the zeta function of the curve, and a new simpler one using methods of Honda. This is the first s tep toward contructing the cohomology theories indicated in the title.

1. Introduction A starting point for elliptic cohomology is a homomorphism ϕ from a cobordism ring to some ring R, which is often called an R-valued genus. When the cobordism theory is MU∗ , we know by Quillen’s theorem [Qui69] that ϕ is equivalent to a 1-dimensional formal group law over R. It is also known that the functor X 7→ MU∗ (X) ⊗ϕ R

is a homology theory if ϕ satisfies certain conditions spelled out in Landweber’s Exact Functor Theorem [Lan76]. Now suppose E is an elliptic curve defined over R. It is a 1-dimensional algebraic group, and choosing a local paramater at the identity leads to a b the formal completion of E. Thus we can apply the formal group law E, machinery above and get an R-valued genus. For example, the Jacobi quartic, defined by the equation y 2 = 1 − 2δx2 + x4 , is an elliptic curve over the ring R = Z[1/2, δ, ]. The resulting formal group law is the power series expansion of p √ x 1 − 2δy 2 + y 4 + y 1 − 2δx2 + x4 ; F (x, y) = 1 − x2 y 2

Date: October 26, 2006. 2000 Mathematics Subject Classification. Primary: 55N34; Secondary: 14H40, 14H50, 14L05, 55N22.

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this calculation is originally due to Euler. The resulting genus is known to satisfy Landweber’s conditions [LRS95], and this leads to one definition of elliptic cohomology. The rich structure of elliptic curves leads to interesting calculations with the cohomology theory and to the theory of topological modular forms due to Hopkins et al, [HM] and [AHS01]. In [HM] they consider the elliptic curve defined by the Weierstrass equation y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 . Under the affine coordinate change x 7→ x + r

and

y 7→ y + sx + t

we get

(1.1)

a6 → 7 a6 + a4 r + a 3 t + a2 r 2 +a1 r t + t2 − r 3 7 a4 + a3 s + 2 a 2 r a4 → +a1 (r s + t) + 2 s t − 3 r 2 a3 7→ a3 + a1 r + 2 t 2 a2 7→ a2 + a1 s − 3 r + s a1 7→ a1 + 2 s.

This can be used to define an action of the affine group on the ring A = Z[a1 , a2 , a3 , a4 , a6 ]. Its cohomology is the E2 -term of a spectral sequence converging to the homotopy of tmf, the spectrum representing topological modular forms. However it is known that the formal group law associated with an elliptic curve over a finite field can have height at most 2; see Corollary 2.4 below. Hence elliptic cohomology cannot give us any information about vn -periodic phenomena for n > 2. The purpose of this paper is to suggest a way to construct similar cohomology theories that go deeper into the chromatic tower. Suppose we have an algebraic curve C of genus g. Then its Jacobian J(C) is an abelian variety b of dimension g. J(C) has a formal completion J(C) which is a g-dimensional b formal group. If J (C) has a 1-dimensional summand, then Quillen’s theorem gives us a genus associated with the curve C. A result in this direction is the following. Theorem 1.2. Let C(p, f ) be the Artin-Schreier curve over Fp defined by y e = xp − x

where e = pf − 1

for a positive integer f . Then the Jacobian J(p, f ) of this curve (possibly after extension of scalars) has a 1-dimensional formal summand of height (p − 1)f .

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This result was stated by Manin in [Man63]. The case f = 1 was treated by Gorbunov and Mahowald in [GM00]. Most of what is needed for the proof can be found in Katz’s 1979 Bombay Colloquium paper [Kat81]. Koblitz’ Hanoi notes [Kob80] covers much of the same material in a less formal way. In §2 we will sketch the original proof since some of the ideas behind it will be needed later. In §3 we will give a new proof using some methods developed by Honda. It has the advantage of being both simpler and more flexible than the classicial proof. In a future paper we will use it to explore deformations of the Artin-Schreier curve and changes of coordinates leading to formulae analogous to (1.1). It is a pleasure to thank the Isaac Newton Institute for their generous hospitality during the preparation of this paper, and to thank Neil Strickland, Spencer Bloch and Mike Hopkins for helpful conversations. 2. The classical proof of Theorem 1.2 An important tool in studying commutative formal groups in characteristic p is the theory of Dieudonn´e modules. Theorem 2.1. [Die55] The category of commutative formal groups over a finite field k is equivalent to the category of modules over the ring D(k) = W(k)hF, V i/(F V = V F = p)

where W(k) is the ring of Witt vectors over k, w 7→ w σ is its Frobenius automorphism, F w = w σ F and V w σ = wV for w ∈ W(k). F is the Frobenius or pth power map, and V is the Verschiebung, the dual of F . A W(k)-module equipped with an action of such an F is called an F crystal, and a similar module over Q ⊗ W(k) is called an F -isocrystal. D(k) is the endomorphism ring of a certain projective object P in the category of commutative formal groups, and the Dieudonn´e module D(G) of a formal group G is group of homomorphisms from P to G. There is also a contravariant Dieudonn´e module D ∗ (G) defined as the set of morphisms from G to a certain injective object I also having D(k) as its endomorphism ring. See Hazewinkel [Haz78] for more information. Here are some examples. • The Dieudonn´e module for the formal group associated with the nth Morava K-theory is D(Fp )/(V − F n−1 ),

so in it we have F n = p. • More generally, for m and n relatively prime, let Gm,n = D(k)/(V m − F n ).

It corresponds to an m-dimensional formal group of height m + n. Theorem 2.2. [Dieudonn´e [Die57]]

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(i) Structure theorem. Any simple Dieudonn´e module M is isogenous over W(Fp ) to some Gm,n . (This means there is a map M → Gm,n with finite kernel and cokernel.) (ii) Let the characteristic polynomial for F in M be X Q(T ) = T m + ci T m−i i>0

for ci ∈ W(k). If its Newton polygon has a line segment of horizontal length n and slope j/n, then up to isogeny over W(k), M has a summand of the form Gj,n−j .

The Newton polygon is the lower convex hull of the set of points {(i, ordp (ci )) : 0 ≤ i ≤ m} , where c0 = 1. The condition on Q(T ) above is equivalent to the existence of n roots having p-adic valuation j/n. For more information, see [Kob80, pp. 19–23]. There are severe restrictions on the formal group attached to an abelian variety, as the following result indicates. Theorem 2.3. (i) [Manin [Man63]] Riemann symmetry condition. If A is an abelian b and its Dieudonn´e module D(A) b variety with formal completion A, has a summand Gm,n up to isogeny over W(Fp ), then it also has a summand Gn,m . (ii) [Tate [Tat66]] More precisely, if A has dimension g and is defined over Fq with q = pa , then the characteristic polynomial for F a has the form X Qa (T ) = T 2g + ci T 2g−i + q g 0 2, then the dimension of A is at least n. The key to finding the Dieudonn´e module of the formal completion of the Jacobian of a curve C is the following fortunate isomorphism. Theorem 2.5. [Grothendieck, Berthelot] Let C be a smooth curve of genus g over Fq , where q = pa . Then its crystalline (or de Rham) H 1 is a free W(Fq )-module of rank 2g isomorphic to the contravariant Dieudonn´e module b of its Jacobian D ∗ (J(C)), with the induced action of the Frobenius F˜ relative to Fq coinciding with the action of F a .

This result is originally due to Grothendieck in 1966. A proof can be found in [MM74, Appendix 2] and in [Ill79]. The crystalline cohomology of C is the same as the de Rham cohomology of a lifting of C to characteristic 0, i.e., from Fq to W(Fq ). For the curve C(p, f ), this H 1 is the free Zp -module with basis xi y j dx ωi,j = e−1 : 0 ≤ i ≤ p − 2, 0 ≤ j ≤ e − 2 , y the genus of the curve being g = (p−1)(e−1)/2. This can be derived from the fact that the curve is a d-fold branched cover of the projective line with p + 1 branch points. We will have occasion to extend scalars in the ground field, in which case H 1 gets tensored with the appropriate ring of Witt vectors. The space of holomorphic 1-forms on C(p, f ) is spanned by (2.6)

{ωi,j : ei + pj < (e − 1)(p − 1) − 1} .

This space is isomorphic to the cotangent space of the Jacobian. We will need it later in an alternate proof of Theorem 1.2. At this point is useful to state the Weil conjectures, although we will only need them in the case of algebraic curves. Given a smooth d-dimensional variety X over Fq , its zeta function is defined by ! X Tn Z(X, T ) = exp |X(Fqn )| . n n>0

where |X(Fqn )| denotes the number of points of X defined over Fqn . The following statements were conjectured by Weil in 1949 [Wei49] and proved by the indicated authors. Expository accounts have been given by Katz [Kat76] and Mazur [Maz75]. Theorem 2.7. (i) (Dwork [Dwo60]) Z(X, T ) is a rational function of T .

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(ii) Z(X, T ) satisfies a functional equation 1 Z X, d = ±q dχ/2 T χ Z(X, T ), q T where χ denotes the Euler-Poincar´e characteristic of X. (iii) (Artin, Grothendieck et al, [SGA73], [Gro95], [Gro77] and Lubkin [Lub68]) More precisely, Z(X, T ) =

P1 (T )P3 (T ) · · · P2d−1 (T ) P0 (T )P2 (T ) · · · P2d (T )

where P0 (T ) = 1 − T , P2d (T ) = 1 − q d T , and for 0 < i < 2d, Pi (T ) is a polynomial whose degree is the rank of H i (X) suitably defined. (iiv) Riemann hypothesis in characteristic p. (Deligne [Del74] and [Del80]) Each reciprocal root of Pi (T ) has absolute value q i/2 . (v) If Xis the reduction of a variety X defined over a number field K, then Pi (T ) = det(1 − T F˜ |H i (X(C))) where F˜ is the Frobenius relative to Fq . In particular the degree of Pi (T ) is the rank of H i (X(C)), the ith Betti number of X. Hence (ii) follows from an analog of the Lefschetz fixed point formula. The formula (iii) follows from an analog of the Lefschtez Fixed Point Theorem once one has defined a cohomology theory for varieties in charatcteristic p with suitable properties. These statements were proved for curves by Weil in [Wei48]. If X is a smooth curve of genus g, then Z(X, T ) =

P1 (T ) , (1 − T )(1 − qT )

where the factors (1 − T )−1 and (1 − qT )−1 correspond to H 0 and H 2 . P1 (T ), which corresponds to H 1 , has degree 2g with X P1 (T ) = 1 + ci T i + q g T 2g , 00 where Cnγ is the number of points in x in X(Fp ) satisfying γ(x) = F˜ n (x).

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Observe that if the action of G on X is trivial, and ρ is irreducible, then ! X 1 X Tn L(X, ρ, T ) = exp Tr(ρ(γ)) |X(Fqn )| |G| γ∈G n n>0 ! X T n X Tr(ρ(γ)) n = exp |X(Fq )| n γ∈G |G| n>0 1 if ρ is nontrivial = Z(X, T ) if ρ is trivial.

If ρ is the regular representation, then |G| if γ = e Tr(ρ(γ)) = 0 otherwise,

so L(X, ρ, T ) is just the zeta function. We also have

L(X, ρ1 ⊕ ρ2 , T ) = L(X, ρ1 , T )L(X, ρ2 , T ).

Recall that the regular representation decomposes as a sum of irreducible representations X degree(ρ)ρ, ρ irreducible

so

Z(X, T ) =

Y

L(X, ρ, T )degree(ρ) .

ρ irreducible

Deligne proved an alternating product formula for L(X, ρ, T ) similar to Weil’s for Z(X, T ), in which Piρ (T ) is the characteristic polynomial of F˜ restricted to HomG (ρ, H i (X) ⊗W(Fq ) K). Now suppose X is a curve of genus g and A is a finite abelian group with action defined over Fq such that in H 1 (X) ⊗W(Fq ) K, each character of A occurs with multiplicity at most 1. A always acts trivially on H 0 and H 2 , so we have P1 (T ) Z(X, T ) = (1 − T )(1 − qT ) Y ρ 1 = P (T ) (1 − T )(1 − qT ) ρ 1 Y 1 L(X, ρ, T ) = (1 − T )(1 − qT ) ρ ! Y 1 1 X = 1+ Tr(ρ(a))C1a T , (1 − T )(1 − qT ) ρ |A| a∈A

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where the product is over the 2g 1-dimensional representations ρ of A that occur as summands of H 1 (X) ⊗W(Fq ) K. Since P1 (T ) has degree 2g, each of its factors P1ρ (T ) must be linear. Our curve C(p, f ) has an action of a certain finite group G on defined over the field Fpn , where n = (p − 1)f . The group is the semidirect product G = Fp o µ(p−1)e .

It fixes the point at infinity and acts on the rest of the curve via (x, y) 7→ (ζ e x + a, ζ p y),

where ζ and a are generators of µ(p−1)e (the group of (p − 1)eth roots of unity) and Fp (regarded as an additive group) respectively. (G is isomorphic to a maximal finite subgroup of the extended Morava stabilizer group Gn . It will follow from the isomorphism above that G acts on the 1-dimensional formal summand as expected.) The smallest field containing these roots is Fpn , and its Frobenius F n respects the eigenspace decomposition associated with the action of µ(p−1)e . Under this action we have X i ωi,j 7→ ai−k ζ 1+ek+pj ωk,j . k 0≤k≤i

It is easily seen to have following properties. (a) In the restriction to the subgroup µ(p−1)e , each character which is nontrivial on µe occurs with multiplicity 1. Hence the subspace spanned by each ωi,j is also an eigenspace for F n . (b) The subspace spanned by the ωi,j for a fixed j is an eigenspace (with nontrivial eigenvalue) for the subgroup µe . We will denote it by H 1,χ where χ is the corresponding nontrivial character of µe . The action of µ(p−1)e is the induction (from µe to µ(p−1)e ) of χ. (c) In the restriction to the subgroup Z/(p), each subspace H 1,χ is an irreducible representation of degree p − 1 isomorphic to the augmentation ideal in the group ring of Z/(p), i.e., to the sum of the p − 1 nontrivial characters of Z/(p). (d) In the restriction to the abelian subgroup A = Z/(p) × µe , each character which is nontrivial on both factors occurs with multiplicity 1. We will denote by H 1,ψ,χ ⊂ H 1,χ the subspace corresponding to the nontrivial characters ψ and χ on Z/(p) and µe . Note that these 1dimensional eigenspaces are not the same as those for the subgroup µ(p−1)e . (e) Each H 1,χ is an irreducible representation of the full group G.

Next we need to define some Gauss sums associated with the characters ψ and χ. Let q = pf = e + 1. Let L be a number field containing the epth roots of unity. We extend the characters ψ and χ to F× q m in the following way. We

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compose the additive character ψ on Fp with the trace map Fqm → Fp , and × we compose the multiplicative χ with with the map F× q m → Fq sending x to m x(q −1)/(q−1) . We denote these composite characters by ψqm and χqm . Then our Gauss sum is X gqm (ψ, χ) = − ψqm (x)χqm (x). x∈F× qm

The action of F f commutes with the action of the subgroup Z/(p) × µe , so it respects the corresponding eigenspace decomposition of H 1 . This means that the action of F f n on H 1,ψ,χ is multiplication by a scalar, and that scalar is known to be gqm (ψ, χ). This is originally due to Hasse-Davenport [HD34] and is explained by Katz in [Kat81, Lemma 2.1]. The proof involves counting certain points in C(p, f ). It follows that the characteristic polynomial for the action of F f n on H 1 is Y (2.7) Pqm (T ) = (T − gqm (ψ, χ)), ψ,χ

where the product is over all nontrivial ψ and all nontrivial χ. Recall that in light of Grothendieck’s isomorphism, this is also the characteristic polynomial F f n acting on the Dieudonn´e module for the Jacobian of our curve. The numerator of the zeta function of C(p, f ), regarded as a curve over Fqm is Y T 2g Pqm (T −1 ) = (1 − gqm (ψ, χ)T ). ψ,χ

It turns out that for our purposes all we need to know about the gq (ψ, χ) is their p-adic valuations. These were originally determined by Stickelberger [Sti90], but we will derive them from the Gross-Koblitz formula. To state it we need to pick a p-adic place p in L. Its residue field is isomorphic to Fq . This choice allows us to identify (via reduction mod p) the target of χ with F× q . This means that χ can be defined as the ath power map for some integer a with 0 < a < e, so we will denote χ by χa . The choice of ψ amounts to choosing a primitive pth root of unity λ, and for each such λ there is a unique solution π to the equation π p−1 = −1 such that λ ≡ 1 + π mod

(π)2 ,

so we will write such a ψ as ψπ . Finally, let α(k) denote the sum of the digits in the p-dic expansion of k. Then the Gross-Koblitz formula says (2.8)

g (ψπ , χa ) = u(a) qm

m

qm π α((qm −1)a/e)

.

295

Higher Chromatic Analogs Here u(a) is the p-adic unit u(a) = (−1)

f

Y

0≤j 0. Suppose that K has an endomorphism σ such that there is a power q of p with aσ ≡ aq modulo m for any a ∈ A. Let Aσ hhT ii be the ring of noncommutative power series in T over A subject to the rule T a = aσ T . Let Mn (Am ) denote the ring of n × n-matrices over Am , and define the ring Mn (Am )σ hhT ii in a similar way. F is characterized by its logarithm f , which is a vector of n power series f1 , . . . , fn over K in n variables x1 , . . . , xn with fi ≡ xi modulo term of degree 2. (Honda calls f the transformer of F .) F is given by the formula F (x, y) = f −1 (f (x) + f (y)), where x and y are n-dimensional vectors.

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i

Let f σ be the power series from f by applying σ i to each coeffiP obtained i cient. Given a matrix H = i Ci T in Mn (Am )σ hhT ii, define X i i (H ∗ f )(x) = Ci f σ (xq ). i

If f is the logarithm for F , we say that H is a Honda matrix for F (or for the vector f ) and that F is of type H, if H ≡ πIn modulo T (where π is a uniformizing element for m and In is the n × n identity matrix) and (H ∗ f )(x) ≡ 0 modulo m. (Honda calls such a matrix special with respect to F .) We say that two matrices H1 , H2 ∈ Mn (Am )σ hhT ii are equivalent if H1 = U H2 for an invertible matrix U ∈ Mn (Am )σ hhT ii. Theorem 3.1. (i) [Hon70, Theorem 4] Suppose that the field K as above is unramified at p, i.e., that Am is the ring of Witt vectors W (k) and m = (p). Then the strict isomorphism classes of n-dimensional formal group laws over A correspond bijectively to the equivalence classes of matrices H ∈ Mn (Ap )σ hhT ii congruent to pIn modulo degree 1. H and f are related by the formula f (x) = (H −1 ∗ p)(x). (ii) [Hon70, Corollary to Theorem 4] Moreover, given a set B ⊂ A of representatives of the residue field A/(p), for each F there is a unique Honda matrix of the form X H = pIn + Ci T i i>0

such that each Ci has coefficients in B. (iii) [Hon70, 5.5] Suppose that the coefficients of Ci are all invariant under the endomorphism σ, let ξ denote the Frobenius endomorphism of the mod p reduction F of the formal group F , and let X det H = pn + ci T i . i>0

Then this is also the characteristic polynomial (or power series) of the Frobenius endomorphism of F . Here are some examples of Honda matrices. Example 3.2. For n = 1 and A = Z, let H be the 1 × 1 matrix with entry u = p − T h for a positive integer h. Then X −1 u−1 = p−1 1 − p−1 T h = p−1 p−i T hi i≥0

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so f (x) =

X xphi i≥0

pi

and F is the formal group law for the Morava K-theory K(h)∗ . More generally the mod p reduction of a 1-dimensional formal group law over Z, Z (p) or Zp has height h iff u is congruent to a unit multiple of T h modulo (p, T h+1 ). Example 3.3. Let A = Zp [[u1 , u2 , . . . uh−1 ]] for a positive integer h, and let uσi = upi . Let H be the 1 × 1 matrix with entry X u = p − Th − ui T i . 00

Then f (x) is the logarithm for the universal p-typical formal group law, i.e., the usual formal group law over BP∗ . Example 3.5. The Honda matrix for the Dieudonn´e formal group Gn,m is

where

H = pIn − C1 T − Cm+1 T m+1

C1 =

0 .. . 0 0 0

1 ··· .. . 0 ··· 0 ··· 0 ···

0 0 .. . 1 0 0 1 0 0

and

Cm+1

0 ... = 0

0 ··· 0 .. .. . . . 0 ··· 0

1 0 ··· 0

Details can be found in [Hon70, 5.2]. Next we recall some results from [Hon73] about the formal completion of the Jacobian J of an algebraic curve C of genus g over K. Let {ω1 , . . . ωg } be a basis of the holomorphic 1-forms on C. (For our curve C(p, f ), such a basis is given in (2.6).) Choose a local parameter z at some point P ∈ K(C) and denote by ωi (z) the expansion of ωi at P . There are power series ψi (z) over K with ψi (0) = 0 and dψi (z) = ωi (z). Let y = (y1 , . . . , yg ) be a system of local parameters at the origin of the Jacobian J(C), and let ν1 , . . . , νg be the invariant differentials of J such that νi ◦ Λ = ωi , where Λ : C → J is the canonical map corresponding to the point P . Let νi (y) denote the local expansion of νi at the origin. There are

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power series φi (y) over K with φi (0) = 0 and dφi (y) = νi (y). Then the vector b φ(y) = (φi (y)) is the logarithm for the formal completion J(C) of J(C) with respect to the local parameters y, whichl we denote by F (C).

Theorem 3.6 (Honda [Hon73]). With notation as above there is a finite set S of prime ideals in K such that if m ∈ / S and H ∈ Mg (Am )σ hhT ii is a Honda matrix for the vector ψ(z), then it is also a Honda matrix for the logarithm φ(y), so F (C) is of type H as a formal group law over Am . This means there is a close connection between power series expansions of holomorphic 1-forms on the curve C and the formal group law for its Jacobian J(C). For the curve C(p, f ) we will use y as a local parameter about the origin. A basis for the space of holomorphic 1-forms is given in (2.6). Since y e = xp − x, we can solve for x and get a power series expansion of the form x = y e g0 (y m )

where m = (p − 1)e,

for some power series g0 . It follows that dx = y e−1 g1 (y m )dy, xi y j dx ωi,j = y e−1 = y ei+j g˜i,j (y m )dy, and

ψi,j = y ei+j+1gi,j (y m )

for some power series g1 , g˜i,j ∈ Zp [[y m ]] and gi,j ∈ Qp [[y m ]]. These conditions on the ψi,j put some restrictions on the Honda matrix. Let (3.7)

E = {ei + j + 1 : i, j ≥ 0, ei + pj < (p − 1)(e − 1) − 1} ⊂ Z/(m).

Note that these conditions on i and j are the same as those in (2.6). In particular this set has g elements, where the genus g is (p − 1)(e − 1)/2. The following description of it is useful. Lemma 3.8. The set E of (3.7) is the image of p−1−i −1 ei + j + 1 : 0 ≤ i ≤ p − 2, 0 ≤ j < e p Proof. By definition E is the image of [ {ei + j + 1 : ei ≤ ei + pj < (p − 1)(e − 1) − 1} 0≤i≤p−2

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We have ei + pj < pe − p − e + 1 − 1 = m − p pj < (p − 1 − i)e − p p−1−i j < e − 1, p and the result follows.

Thus for k ∈ E, the values of i and j are uniquely determined. We will denote ψi,j and gi,j by ψei+j+1 and gei+j+1 respectively. Let hs,t ∈ ZhhT ii denote the coefficient in the tth column of the sth row of H(p, f ), where it is understood that some rows and columns are empty. Thus we have Lemma 3.9. With notation as above, hs,t (for appropriate s and t) is nonzero only if s ≡ pn t mod (m) for some integer n, and in that case X hs,t = hs,t,n T n , n≥0

where the sum is over all such nonnegative integers n. This implies that H(p, f ) has a block decomposition as follows. The pth power map acts on the group Z/(m) ∼ = µm of eigenvalues. The set E ⊂ µm is the union of its intersections with the orbits of the action on µm . The cardinality of each orbit is a divisor of (p − 1)f . Each nonempty intersection of cardinality g 0 corresponds to a g 0 × g 0 factor of H(p, f ), and hence to a g 0 -dimensional formal summand of the Jacobian J(p, f ). Example 3.10. Consider the case (p, f ) = (3, 2), for which m = 16. The values of ei + j + 1 for the integrals of the seven holomorphic 1-forms ω i,j listed in (2.6) are indicated in the following table. j 0 1 2 3 4 i=0 1 2 3 4 5 i = 1 9 10 Meanwhile, the orbits in Z/(m) under the multiplication by p include {1, 3, 9, 11} , {15, 13, 7, 5} , {2, 6} , {14, 10} , and {4, 12} . Each value listed in the table is in one of these. The first orbit contains three such elements, and the other orbits contain one each.

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It follows that the Honda matrix H (with empty rows and columns omitted) has the form b h1,1 (T 4 ) 0 T 3b h1,3 (T 4 ) 0 0 T 2b h1,9 (T 4 ) 0 2 b 0 h2,2 (T ) 0 0 0 0 0 Tb 4 4 3b 4 b 0 h3,3 (T ) 0 0 T h3,9 (T ) 0 h3,1 (T ) 2 b 0 0 0 h4,4 (T ) 0 0 0 4 b 0 0 0 0 h5,5 (T ) 0 0 4 4 4 b b T 2b h9,1 (T ) 0 T h9,3 (T ) 0 0 h9,9 (T ) 0 b 0 0 0 0 0 0 h10,10 (T 2 )

where each power series b hi,i has constant term p, so

det(H) = b h2,2 (T 2 )b h4,4 (T 2 )b h5,5 (T 4 )b h10,10 (T 2 )D, where D = b h1,1 (T 4 )b h3,3 (T 4 )b h9,9 (T 4 ) + T 8b h1,3 (T 4 )b h3,9 (T 4 )b h9,1 (T 4 )

+T 4b h1,9 (T 4 )b h3,1 (T 4 )b h9,3 (T 4 ) − T 4b h9,1 (T 4 )b h3,3 (T 4 )b h1,9 (T 4 ) −T 4b h9,3 (T 4 )b h3,9 (T 4 )b h1,1 (T 4 ) − T 4b h9,9 (T 4 )b h3,1 (T 4 )b h1,3 (T 4 ).

We will study this by computing the various factors modulo (3, T n ) for appropriate n. Recall that H is congruent to 3I7 modulo T , so b h2,2 (T 2 ) ≡ h2,2,2 T 2 b h4,4 (T 2 ) ≡ h4,4,2 T 2 b h5,5 (T 2 ) ≡ h5,5,4 T 4 b h10,10 (T 2 ) ≡ h10,10,2 T 2 and D ≡ h1,3,0 h3,9,0 h9,1,0 T 4 so det(H) ≡ h2,2,2 h4,4,2 h5,5,4 h10,10,2 h1,3,0 h3,9,0 h9,1,0 T 14

mod(3, mod(3, mod(3, mod(3, mod(3, mod(3,

T 4 ), T 4 ), T 8 ), T 4 ), T 8) T 16 ),

where the 3-adic integer hi,j,k is the coefficient of T k in the matrix entry hi,j . On the other hand by Theorem 3.1(iii), det(H) is the characteristic polynomial of the Frobenius, so by Theorem 2.3(ii) it must have the form T 14 + · · · + 37 . It follows that h5,5,4 is a unit, which gives us the desired 1-dimensional formal summand of height 4. In order to generalize the calculation above we need the following three easy lemmas. Lemma 3.11. Each orbit of Z/(m) other than {ei} for 0 ≤ i ≤ p − 2 has a nonempty intersection with E. If a is in such an orbit, then k ∈ E iff e−k ∈ / E. Lemma 3.11 is needed for the following.

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Lemma 3.12. The determinant of a block of the Honda matrix H(p, f ) associated with an orbit of cardinality n is congruent to an integer multiple of T n modulo (p, T n+1 ), and the sum of all such cardinalities is 2g. Lemma 3.13. The orbit of (m−1) has (p−1)f elements, but its intersection with E has only one element, namely (m − 1)/p. Lemma 3.12 impies that det(H(p, f )) is congruent to an integer multiple of T 2g modulo (p, T 2g+1 ). By Theorems 3.1(iii) and 2.3(ii), the coefficient of T 2g is 1, so the coefficients in 3.12 are all units. In particular the one for the orbit of (m − 1)/p is a unit, which gives us the desired 1-dimensional formal summand of height (p − 1)f . This completes our alternate proof of Theorem 1.2. Proof of Lemma 3.11. The only singleton orbits are the ones cited since pk ≡ x modulo (m) implies that k is a multiple of d. These orbits do not intersect E. Given k not divisible by d with 0 < k < e, we write k = ei + j + 1 and e − k = ei0 + j 0 + 1. Then i0 = p − 2 − i and j 0 = e − 1 − j. Now k ∈ E iff p−1−i 0≤ j <e p p−1−i p − 1 − i0 0 e−1≥ j >e −1=e −1 p p which holds iff k − e ∈ / E.

Proof of Lemma 3.12. Suppose e1 ∈ E is in such an orbit. Define elements ei inductively by letting ei+1 ∈ E be congruent modulo (m) to pk ei for the smallest possible positive value of k. Hence the intersection of the orbit with E has the form {e1 , e2 , . . . es } ki ks with P ei+1 ≡ p ei for 1 ≤ i < s, and e1 ≡ p es . The cardinality n of the orbit is ki . The entries in the corresponding block of H(p, f ) are ( T ki +···+kj−1 b hei ,ej (T n ) if i ≤ j hei ,ej = n−kj −···−ki−1 b n T hei ,ej (T ) if j < i Since hei ,ei ,0 = 0, it follows that the determinant of this block is congruent to he1 ,e2 ,k1 he2 ,e3 ,k2 · · · hes−1 ,es ,ks−1 hes ,e1 ,ks T n

modulo (p, T 2n ) as desired. To find the sum of the cardinalities, we need Lemma 3.11. If an orbit intersecting E is self-conjugate, that is it contains both a and −a, then 3.11 implies that exactly half the elements in the orbit are in E. Otherwise if s out of n elements of an orbit are in E, then E also contains n − s elements

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in the conjugate orbit. In either case half of the elements in the union of an intersecting orbit and its conjugate are in E. Since E has g elements, the sum of the cardinalities is 2g. Proof of Lemma 3.13. The orbit of 1 is k p : 0 ≤ k < (p − 1)f = pk + ei : 0 ≤ k < f, 0 ≤ i ≤ p − 2 since modulo (m)

pf +k (d + 1)pk ≡ e + pk . It follows that the orbit of m − 1 is e − pk − ei : 0 ≤ k < f, 0 ≤ i ≤ p − 2 = ei + e − pk : 0 ≤ k < f, 0 ≤ i ≤ p − 2 ,

which has (p − 1)f elements. (m − 1)/p is among them since m−1 = pf − pf −1 − 1 = e − pf −1 . p We need to show that no other elements of this orbit are in E. Using 3.8 we see that ei + e − pk is in E only if p−1−i k e−p < e p −1 − i k −p < e p i+1 k , p > e p which means i = 0 and k = f − 1.

References [AHS01] M. Ando, M. J. Hopkins, and N. P. Strickland. Elliptic spectra, the Witten genus and the theorem of the cube. Invent. Math., 146(3):595–687, 2001. ´ [Del74] Pierre Deligne. La conjecture de Weil. I. Inst. Hautes Etudes Sci. Publ. Math., (43):273–307, 1974. ´ [Del80] Pierre Deligne. La conjecture de Weil. II. Inst. Hautes Etudes Sci. Publ. Math., (52):137–252, 1980. [Die55] Jean Dieudonn´e. Lie groups and Lie hyperalgebras over a field of characteristic p > 0. IV. Amer. J. Math., 77:429–452, 1955. [Die57] Jean Dieudonn´e. Groupes de Lie et hyperalg`ebres de Lie sur un corps de caract´eristique p > 0. VII. Math. Ann., 134:114–133, 1957. [Dwo60] Bernard Dwork. On the rationality of the zeta function of an algebraic variety. Amer. J. Math., 82:631–648, 1960. [GM00] V. Gorbounov and M. Mahowald. Formal completion of the Jacobians of plane curves and higher real K-theories. J. Pure Appl. Algebra, 145(3):293–308, 2000. [Gro77] Cohomologie l-adique et fonctions L. Springer-Verlag, Berlin, 1977. S´eminaire de G´eometrie Alg´ebrique du Bois-Marie 1965–1966 (SGA 5), Edit´e par Luc Illusie, Lecture Notes in Mathematics, Vol. 589.

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[Gro95] Alexander Grothendieck. Formule de Lefschetz et rationalit´e des fonctions L. In S´eminaire Bourbaki, Vol. 9, pages Exp. No. 279, 41–55. Soc. Math. France, Paris, 1995. [Haz78] M. Hazewinkel. Formal Groups and Applications. Academic Press, New York, 1978. [HD34] H. Hasse and H. Davenport. Die Nullstellensatz der Kongruenz zeta-funktionen in gewissn zyklischen F¨ allen. J. Reine Angew. Math., 172:151–182, 1934. [HM] M. J. Hopkins and M. A. Mahowald. From elliptic curves to homotopy theory. Preprint in Hopf archive at http://hopf.math.purdue.edu/Hopkins-Mahowald/eo2homotopy. [Hon68] Taira Honda. Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan, 20:83–95, 1968. [Hon70] Taira Honda. On the theory of commutative formal groups. J. Math. Soc. Japan, 22:213–246, 1970. [Hon73] Taira Honda. On the formal structure of the Jacobian variety of the Fermat curve over a p-adic integer ring. In Symposia Mathematica, Vol. XI (Convegno di Geometria, INDAM, Rome, 1972), pages 271–284. Academic Press, London, 1973. ´ [Ill79] Luc Illusie. Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. (4), 12(4):501–661, 1979. [Kat76] Nicholas M. Katz. An overview of Deligne’s work on Hilbert’s twenty-first problem. In Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), pages 537–557. Amer. Math. Soc., Providence, R. I., 1976. [Kat81] Nicholas M. Katz. Crystalline cohomology, Dieudonn´e modules, and Jacobi sums. In Automorphic forms, representation theory and arithmetic (Bombay, 1979), volume 10 of Tata Inst. Fund. Res. Studies in Math., pages 165–246. Tata Inst. Fundamental Res., Bombay, 1981. [Kob80] Neal Koblitz. p-adic analysis: a short course on recent work, volume 46 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1980. [Lan76] P. S. Landweber. Homological properties of comodules over M U∗ (M U ) and BP∗ (BP ). American Journal of Mathematics, 98:591–610, 1976. [LRS95] P. S. Landweber, D. C. Ravenel, and R. E. Stong. Periodic cohomology theories ˇ defined by elliptic curves. In Mila Cenkl and Haynes Miller, editors, The Cech Centennial, volume 181 of Contemporary Mathematics, pages 317–338, Providence, Rhode Island, 1995. American Mathematical Society. [Lub68] Saul Lubkin. A p-adic proof of Weil’s conjectures. Ann. of Math. (2) 87 (1968), 105-194; ibid. (2), 87:195–255, 1968. [Man63] Ju. I. Manin. Theory of commutative formal groups over fields of finite characteristic. Uspehi Mat. Nauk, 18(6 (114)):3–90, 1963. [Maz75] B. Mazur. Eigenvalues of Frobenius acting on algebraic varieties over finite fields. In Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pages 231–261. Amer. Math. Soc., Providence, R.I., 1975. [MM74] B. Mazur and William Messing. Universal extensions and one dimensional crystalline cohomology. Springer-Verlag, Berlin, 1974. Lecture Notes in Mathematics, Vol. 370. [Qui69] D. G. Quillen. On the formal group laws of oriented and unoriented cobordism theory. Bulletin of the American Mathematical Society, 75:1293–1298, 1969. [SGA73] Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 3. Springer-Verlag, Berlin, 1973. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1963–1964 (SGA

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WHAT IS AN ELLIPTIC OBJECT? GRAEME SEGAL ALL SOULS COLLEGE, OXFORD

1. Elliptic cohomology A generalized cohomology theory is a sequence of contravariant functors {hi }i∈Z from spaces to abelian groups which are linked together in a wellknown way. The theories that arise in nature are of two types: K-theories, and cobordism theories. (Classical cohomology can be approached in so many different ways that I shall leave it aside for the moment.) On a compact space X the isomorphism classes of complex vector bundles form an abelian semigroup Vect(X) under the operation of direct sum, and K 0 (X) is the abelian group got by formally adjoining inverses to the semigroup Vect(X). Then K 0 is a homotopy functor, and the functors K −i , for i > 0, defined — roughly — by composing K 0 with the i-fold suspension functor, have the properties of “half” a cohomology theory. That much is true for any representable homotopy functor, but the functors K i are special because of the Bott periodicity theorem, which gives a canonical equivalence between K i and K i−2 for i ≤ 0, and enables us to define K i for all i ∈ Z by periodicity. There is a completely different reason, however, unrelated to Bott periodicity, why the functor K 0 forms part of a cohomology theory, and it applies in a much more general context. For any (discrete) ring A we have a contravariant functor X 7→ ModA (X), where ModA (X) is the semigroup of isomorphism classes of bundles of finitely generated projective A-modules on X. It is a representable homotopy functor, though not a very interesting one, as it sees only the fundamental group of X. But if, instead of making the semigroup ModA (X) into a group separately for each space X, we perform the group-completion on the representing space, i.e. we look for the universal abelian-group-valued representable homotopy functor F with a transformation ModA → F , then we obtain a much more interesting functor KA0 . In fact KA0 (X) = [X; ΩB|PA |]. Here |PA |, the “space” of the category PA of finitely generated projective A-modules, which is the representing space for ModA (X), is a topological semigroup under ⊕, and ΩB|PA | is the loop-space of the classifying space of |PA |. The remarkable thing is that for any category C with a composition law 306

What is an elliptic object?

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which, like ⊕, is commutative up to coherent canonical isomorphisms we can iterate the classifying space functor |C| 7→ B|C|, and can define a cohomology theory by KCi (A) = [X; ΩB i+1 |C|]. When C = PA this is Quillen’s algebraic K-theory for the ring A, and when C is the category of finite sets under disjoint union then we get stable cohomotopy theory [S1]. The theories KC∗ are the first basic class of cohomology theories. Classical cohomology with coefficients in an abelian group A is the case coming from the category whose objects are the elements of A, and which has no morphisms except identity morphisms. The second basic class of cohomology theories are cobordism theories. They arise as homology rather than cohomology theories.1 The basic example is oriented bordism, where hn (X) is defined as the cobordism classes of maps φ : M → X, where M is a compact oriented smooth n-manifold. (A cobordism between (M0 , φ0 ) and (M1 , φ1 ) is a cobordism W between M0 and M1 with a map φ : W → X restricting to φ0 and φ1 at the ends.) By considering manifolds M with various kinds of additional structure, or allowing manifolds with singularities of various specified kinds, we get a variety of different homology theories. For example, framed manifolds lead to stable homotopy theory, while classical integral homology corresponds to allowing singular manifolds with arbitrary singularities of codimension two. The example we shall be interested in is complex cobordism MU∗ , corresponding to manifolds with weakly almost-complex structure. A noticeable difference between cobordism theories and K-theories is that, with the former, classes of any degree n are equally well represented geometrically, not just those of degree 0. For any cobordism theory h∗ there is an appropriate space Mn of n-manifolds such that h−n (X) = [X; Mn ], and Mn has an interpretation even for n < 0. As far as I know, the two classes of theories just mentioned exhaust the known “natural” cohomology theories. But there are other classes of theories which can be constructed from them algebraically. The most important of these are complex-orientable theories. Definition 1.1. A multiplicative2 theory h∗ is complex-oriented if there is given an element c1 ∈ h2 (P∞ C ) which restricts to the canonical generator of 2 2 ∼ 0 ˜ h (S ) = h (point) . (Here S 2 = P1C ⊂ P∞ C .) For such a theory we have (see [Ad]) 1We

can pass at will between homology and cohomology theories, but it is significant that it is the homology classes which have a straightforward geometric interpretation. 2A theory h∗ is multiplicative if h∗ (X) is an anticommutative graded ring.

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Proposition 1.2. ∼ h∗ (P∞ C ) = A[[c1 ]], where A = h∗ (point). Because P∞ C is an H-space (its composition-law representing the tensor product of line bundles) the ring A[[c1 ]] is a Hopf algebra, and the diagonal map c1 7→ m(c1 ⊗ 1, 1 ⊗ c1 ) is a formal group-law associated to h∗ . In a complex-oriented theory h∗ there is a canonical Thom class for any complex vector bundle, and, in particular, a sequence of elements in h2n (MUn ) corresponding to a transformation MU ∗ → h∗ . This means that complex cobordism is universal among complex-oriented theories. Quillen proved that its formal group law is also universal, in the sense that a law over any graded ring R comes from that of MU ∗ by a ring-homomorphism AM U = MU ∗ (point) → R. Alternatively expressed, a formal group-law over R is the same thing as a genus for weakly almost-complex manifolds, for a genus is exactly such a homomorphism. Elliptic cohomology was conceived because of the discovery of the elliptic genus — actually, of the remarkable rigidity properties (see [L],[S2]) of a particular family of genera Φτ : AM U → C parametrized by elliptic curves Στ = C/(Z + τ Z). The Φτ can be assembled into Φ : AM U → R, where R is a ring of modular forms. Landweber observed that if we take R = Z[ 21 , δ, ε, ∆−1 ], where ∆ = ε(δ 2 − ε)2 , and δ and ε are the functions of the curve Στ which arise when its equation is written in the form y 2 = 1 − 2δx2 + εx4 , then Ell∗ (X) = MU ∗ (X) ⊗AM U R satisfies conditions that he had previously found which ensure that the functor MU ∗ ( ) ⊗AM U R is a cohomology theory. This was the original definition of elliptic cohomology. Since its proposal a great deal of work — especially by Hopkins [H] and his collaborators — has been devoted to finding an improved version, which ought not to require inverting the prime 2. It is now believed that the “correct” theory, which Hopkins calls tmf∗ , is not, in fact, quite complex-orientable, but that a tmf∗ -orientation of a manifold M should be a string structure on M in the sense described below. The coefficient ring tmf∗ (point) maps to the ring MZ of integral modular forms (i.e. modular forms whose expansion in terms of q = e2πiτ lies in Z[[q]]). If we tensor with the rational numbers Q then tmf ∗ (point) → MZ becomes an isomorphism, but tmf∗ (point) is a much more subtle and complicated ring than MZ , with a great deal of torsion. An n-manifold with a string structure has a genus in tmf−n (point) — the image of the fundamental class [M] under the map

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induced by M → (point) — and the image of this genus in MZ is the Witten genus. 2. Elliptic objects It is a very natural question whether the elliptic cohomology classes of a space have geometric representatives in the way that K-theory classes are represented by vector bundles. The main evidence that this may be true is Witten’s heuristic argument for the rigidity of the elliptic genus. The essential idea is that the elliptic genus of a compact manifold M is the Hilbert series Σq k dim(Vk ) of a (virtual) graded vector space V = ⊕Vk which is the index ker(DLM ) − coker(DLM ) of a Dirac-like differential operator DLM defined not on M but on its smooth loop-space LM. The grading on the index comes from the action of the circle T on LM by rotation of loops. This loop space perspective, however, does not by itself shed light on the modularity which is the basic property of the elliptic genus. The place where one “naturally” encounters graded vector spaces whose Hilbert series are modular is two-dimensional conformal field theory. Definition 2.1. 3 A conformal field theory is a Hilbert space H together with a trace-class operator UΣ,ξ : H⊗p → H⊗q associated to each pair (Σ, ξ), where Σ is a Riemann surface which is a cobordism from an “incoming” manifold Sp consisting of p parametrized circles to a similar “outgoing” manifold Sq , and ξ is a point of the Quillen determinant ¯ line DetΣ of the ∂-operator of Σ. The essential properties the operators UΣ,ξ are required to satisfy are (i) UΣ,0 ,ξ 0 ◦ UΣ,ξ = UΣ0 ∪Σ,ξ 0 ⊗ξ when the cobordisms Σ and Σ0 are concatenated, and (ii) UΣ,ξ ⊗ UΣ0 ,ξ 0 = UΣtΣ0 ,ξ⊗ξ 0 when the cobordisms are simply put side-by-side. If UΣ,ξ depends holomorphically on (Σ, ξ) then the theory is called chiral. If, furthermore, we have UΣ,λξ = λm UΣ,ξ for λ ∈ C× then we say the theory is of level m. If, finally, we have UΣ,ξ = UΣ∗ ∗ ,ξ for all (Σ, ξ), where Σ∗ is Σ with the conjugate complex structure, then the theory is called unitary. If the surface Σ is closed then UΣ,ξ is simply a complex number. For a chiral theory of level m, the restriction of UΣ,ξ to closed surfaces of genus 1 is a modular form of level m. 3For

more details, see [S3].

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Among the cobordisms from S1 to S1 there is a sub-semigroup formed by the annuli Aq = {z ∈ C : |q| ≤ |z| ≤ 1} for 0 ≤ |q| ≤ 1, with the boundary circles parametrized by θ 7→ eiθ , θ 7→ qeiθ . Its action on the Hilbert space H of a chiral theory gives H a grading H = ⊕k≥0 Hk by finite dimensional subspaces. Proposition 2.2. If Σq is the torus C× /q Z , then UΣq ,ξq = Σq k dim(Hk ) for any chiral theory of level m, where ξq is the canonical element coming from the annulus Aq .

4

of DetΣq

This is easily proved by regarding the cobordism Σq as the composite of two annuli Aq1 , Aq2 with q1 q2 = q, but I shall not give the details here . As chiral conformal field theories give us modular forms so naturally, we might first guess that elliptic cohomology is a K-theory made from bundles of field theories. The crudest approximation to an elliptic class is simply a graded complex vector bundle, and there is indeed a forgetful transformation tmf ∗ (X) → K ∗ (X)[[q]] corresponding to the q-expansion of a modular form. Nevertheless, to get further we must remember that the elliptic genus is the index of an operator not on X but on LX. We need the notion of a conformal field theory over X. There is no loss in assuming that X is a smooth manifold. Definition 2.3. A conformal field theory over X is a rule which assigns a vector space Hγ to each smooth loop γ : S 1 → X, and an operator UΓ,ξ : Hγ1 ⊗ . . . Hγp → Hγp+1 ⊗ . . . Hγp+q to each Riemann surface Σ which is a cobordism from the “incoming” loops γ1 , . . . , γp to the “outgoing” loops γp+1 , . . . , γp+q , and is equipped with a map Γ : Σ → X. As before, ξ ∈ DetΣ , and UΓ,ξ must have the properties (i) and (ii) of Definition 2.1. The basic example which motivates Definition 2.2 is the bundle of spinors on the loop space of an oriented Riemannian n-manifold M. The tangent bundle of LM has structure group LSOn , and this group has a projective unitary representation (see [PS] Chap.12) which is naturally regarded as its 4See

[S3] §6. The element ξq differs from the “more canonical” element of DetΣq which is unique only up to a 12th root of unity by multiplication by the square of the Dedekind η-function.

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“spin” representation.5 The condition that we can make a Hilbert space bundle on LM associated to its tangent bundle by this projective representation is the vanishing of the characteristic classes w2 and 12 p1 of M. The choice of such a spinor bundle on LM is called a string structure on M. A string structure automatically extends — using the Riemannian metric of M — to a conformal field theory of level n over M. The propagation operators UΓ,ξ form a kind of connection in the spinor bundle, though it should be remembered that even when Σ is a cylinder, i.e. Γ : Σ → M is a path in LM, the operator UΓ,ξ is a contraction operator, not a unitary isomorphism. In my talk [S2] I speculated whether a conformal field theory over X of level m defines a class in Ell−m (X). As far as I know, the question is still open. If something of the kind really is true then it seems to me quite remarkable, for, apart from cobordism theories, the only situations I know where we have geometric representatives for cohomology classes of all dimensions are real and complex K-theory (in virtue of Bott periodicity), and classical de Rham theory for smooth manifolds. The main evidence in support of the idea is that a level m theory over a compact 2n-manifold X with a string structure can be “integrated” to give a virtual conformal field theory of level m+2n, and hence a modular form in Ell−m−2n (point): the integration process is tensoring the theory with the Dirac operator on LX and forming the index of the resulting coupled operator. In the language of quantum field theory the Dirac operator in the spinor bundle on LM is a supersymmetry operator. To explain what this means we must first recall two more aspects of the formalism of conformal field theory. First, the group Diff(S 1 ) of diffeomorphisms of S 1 acts on the Hilbert space H of a conformal theory, and the action of annuli extends the action of its Lie algebra Vect(S 1 ) to the complexification, giving us a map L : VectC (S 1 ) → End(H). In the case of a chiral theory the map L is complex-linear, but in general we write L = L+ + L− , where L+ is C-linear and L− -antilinear. The maps L+ and L− define commuting (projective) actions of VectC (S 1 ) on H. The second point is that VectC (S 1 ) is the even part of a Lie superalgebra 1 V(S 1 ) whose odd part is the space Ω− 2 (S 1 ) of (− 21 )-forms on S 1 (two of which can be multiplied pointwise to give a vector field). The class of conformal field theories which are “half-supersymmetric” in the sense that there is given a Cantilinear action on V(S 1 ) extending the L− -action of VectC (S 1 ) is important 1 for elliptic cohomology, for the action of the odd element (dθ)− 2 of V(S 1 ) on 5More

precisely, there is a positive energy and a negative energy spin representation, differing by changing the orientation of the circle. We actually want the negative energy choice, which makes the theory antichiral, in the sense that the operators depend antiholomorphically on the complex structure of the surface.

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H has an index which is a virtual chiral conformal field theory. Indeed the space of these half-supersymmetric theories seems to be the correct model of the space of virtual chiral conformal theories, just as the space of Fredholm operators is the best model of the space of virtual finite-dimensional vector spaces. When we have a string structure {Hγ } on a manifold M the Dirac operator acts — in principle — on the Hilbert space H of sections of the spinor bundle {Hγ } over LM which are square-summable for a measure on LM which forces them to be concentrated in an extremely small neighbourhood of the point loops. The group Diff(S 1 ) acts on LM, and the bundle {Hγ } is equivariant with respect to it, so we expect Diff(S 1 ) to act on H. One would like to say that this action is part of a conformal field theory structure on H which is halfsupersymmetric in the above sense: the Dirac operator should be the action 1 on H on the element (dθ)− 2 of the superalgebra. In fact that is too much to hope for; but as far as homotopy theory is concerned one can proceed much more formally, replacing the space of sections of the bundle {Hγ } on LM by the space H of jets of sections along the subspace M of point loops.6 On this Hilbert space H one can much more plausibly define the half-supersymmetric conformal field theory structure, as I have attempted to sketch in [S2]. We can do this even after tensoring the spinor bundle with an arbitrary chiral conformal field theory over M as defined in 2.3. This is the “integration” operation referred to above. The Dirac operator itself is mapped to the Witten genus, while the original elliptic genus is the image of the Dirac operator tensored with the chiral — rather than antichiral — spinor bundle. If we could do this for a family of manifolds M rather than just a single one then we should have related the space of level m + n conformal field theories to the n-fold loop space of the space of level m theories. Unfortunately, one could not expect to use conformal field theories over X by themselves to define Ell∗ (X). The essential reason is that, like the loop space LX, they are not defined locally on X, and so do not have the basic Mayer-Vietoris property of a cohomology theory. Another disconcerting fact is that a chiral conformal field theory is a rigid object which is not determined up to isomorphism by its modular form (e.g. an even unimodular lattice of rank k gives rise to a conformal field theory of level k, and non-isomorphic lattices with the same modular form give non-isomorphic theories). It is plausible, however, that chiral theories with the same modular form are connected in the space of half-supersymmetric theories. Before describing the recent progress in understanding elliptic objects it seems worth mentioning one other hint — first pointed out by Grojnowski 6I

should mention at this point a large body of work by Gorbounov, Malikov, Schechtman, and Vaintrob, e.g. [MSV],[GMS], who have developed a notion of ”chiral de Rham complex” defined on the formal neighbourhood of M in LM .

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[G] — that the field theory approach may be on the right track. This is the question of G-equivariant elliptic cohomology for a compact group G: we know that the definition of equivariant K-theory in terms of G-vector-bundles is one of the things that makes K-theory such a useful tool. There is a general notion of a G-equivariant quantum field theory — in physicists’ language, a “theory with gauged G-symmetry” — which in the present situation reduces to the following. Definition 2.4. A two-dimensional G-equivariant chiral conformal field theory assigns a Hilbert space HS,P to each oriented 1-manifold S equipped with a principal G-bundle P , and an operator UΣ,Q,ξ : HS0 ,P0 → HS1 ,P1 to each conformal cobordism Σ from S0 to S1 equipped with a holomorphic GC bundle reducing to P0 and P1 at the ends. The operator should have properties analogous to those of definition 2.1, and should depend holomorphically on the pair (Σ, Q). As usual, ξ ∈ DetΣ,Q . When G is a connected group this means that HS,P is a positive-energy projective representation of the loop group LG. The attractive idea that Ell∗G (point) should be some kind of representation ring of LG has been pursued further by Devoto [Dv1] and Ando [An], but for lack of a satisfactory equivariant version of Landweber’s theorem there is still no real candidate for the equivariant elliptic theory Ell∗G . For finite groups G the field theory point of view seems to fit with what is known ([HKR],[Dv2]) about Ell∗ (BG), but again I know no definite theorem. The work of Stolz and Teichner [ST] has moved the idea of an elliptic object forward in several important ways. One is their focus on the space of the halfsupersymmetric theories already mentioned. But their main contribution concerns the Mayer-Vietoris property. The problem with the definition 2.1 of a conformal field theory is that it does not incorporate any sense in which the Hilbert space H associated to the circle S 1 is local with respect to S 1 . If H could be reconstructed from objects HI associated to small subintervals I of S 1 then we might be able to think of a conformal field theory over X as a local object on X, and could hope to construct a cohomology theory. The simplest sense in which H could be local would be if one could associate a Hilbert space HI to each closed subinterval I of the circle so that H ∼ = ⊗HIi when the circle is the union of intervals Ii meeting only at their ends. Locality of this simple kind — which would hold, for example, if H were the symmetric or exterior algebra on L2 (S 1 ) — is easily seen to be impossible in conformal field theory. In the simplest conformal field theories, the space H is a “renormalized” symmetric or exterior algebra on a space of functions such as L2 (S 1 ), where the renormalization depends on a polarization L2 (S 1 ) = L2 (S 1 )+ ⊕ L2 (S 1 )−

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of L2 (S 1 ) into positive and negative frequency parts. The projection operators defining the polarization need to be given only up to Hilbert-Schmidt perturbations: they are singular integral operators on S 1 with kernels whose supports can be chosen in an arbitrarily small neighbourhood of the diagonal in S 1 × S 1 , but which cannot be supported exactly on the diagonal. The effect for the locality of H is that if S 1 = I ∪ J, where I and J are open intervals, then H can be reconstructed from Hilbert spaces HI , HJ and a von Neumann algebra AI∩J associated to I ∩ J which acts on both HI and HJ . The reconstruction is by means of Connes’s notion of the tensor product of bimodules over von Neumann algebras. If A, B, C are von Neumann algebras, M is an (A, B)-bimodule, and N is a (B, C)-bimodule, then Connes defines a (A, C)-bimodule M ∗B N . Two important features of his theory are the existence of a neutral B-bimodule B0 with the property that M ∗B B0 ∼ =M ∼ and B0 ∗B N = N , and the fact that a (B, B)-bimodule gives us a Hilbert space M∗B = M ∗(B op ⊗B) B0 .7 The relevance of the Connes tensor product to the locality of loop group representations, and hence to two-dimensional conformal field theory, was first realized by Wassermann [W]. In the light of his work the following definition — essentially that of Stolz and Teichner — seems appropriate. Definition 2.5. A three-tier conformal field theory over X consists of the data of Definition 2.2 together with (i) a bundle of von Neumann algebras {Ax }x∈X on X, and (ii) an (Ax , Ay )-bimodule Hγ for each path γ from x to y. The properties the bimodules must have are that Hγ ∼ = Hγ ∗A Hγ 2

z

1

if the path γ from x to y is the concatenation of γ1 from x to z and γ2 from z to y, and that Hγ0 ∼ = Hγ ∗Ax if the path γ from x to x is regarded as a closed path γ0 . One can presumably construct a cohomology theory based on 3-tier conformal field theories of any chosen level, but, as far as I know, little has yet been proved, especially about why the theories at different levels should be related by suspension. Apart from the Stolz-Teichner programme there is another quantum field theory approach to elliptic cohomology which has been proposed by Baas, Dundas, and Rognes [BDR]. In one important sense it is much less ambitious: it aims only to construct elliptic objects of degree zero, relying on the 7I

have taken the notation M ∗B from Quillen, who uses it in an algebraic setting for the quotient of M which equalizes the left and right B-actions.

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machinery of algebraic K-theory to produce the theory in other dimensions. I shall try to say briefly how it fits in to the general philosophy of this talk. We can give a definition of a general d-dimensional quantum field theory along the lines of 2.1, but using manifolds equipped with a Riemannian rather than just a conformal structure. (Of course we shall have a Hilbert space HS assigned to each compact oriented Riemannian (d − 1)-manifold S, subject to HS1 ⊗ HS2 ∼ = HS1 tS2 .) Among these theories are the conformal ones, and — much more specially still — the so-called topological field theories, for which the vector spaces and the operators depend only on the smooth structure of the manifolds, without any metric at all. On the space of all quantum field theories we have the renormalization group flow: a theory is a functor from a cobordism category to vector spaces, and for any t > 0 we can compose the theory with the functor from the cobordism category to itself which multiplies the metric of every manifold by t. When d = 1 a quantum field theory is precisely a semigroup of trace-class operators in a Hilbert space — self-adjoint operators if the theory is unitary. The renormalization group flow retracts the space of 1-dimensional theories (with its natural topology) to the subspace of topological theories, which is simply the space of finite-dimensional complex vector spaces. We can also define supersymmetric unitary 1-dimensional theories. Such a theory is a mod 2 graded Hilbert space with a trace-class semigroup whose generator is given as the square of a self-adjoint operator of degree 1. Up to homotopy, this is the space of Fredholm operators Z × BU , i.e. the representing space for K-theory. When d = 2 we can not assume that the space of quantum field theories is homotopy equivalent to the space of topological theories. It nevertheless seems interesting to consider the space of 2-dimensional topological theories, and better, in the light of the discussion above, the space of “3-tier” unitary topological theories. The general definition of a 3-tier d-dimensional quantum field theory — of which Definition 2.5 is a specialization — is as a structure that assigns (i) (ii)

a linear category CZ to each closed (d − 2)-manifold Z, a functor FY : CZ0 → CZ1 to each (d − 1)-dimensional cobordism from Z0 to Z1 , and (iii) a transformation of functors UX : FY0 → FY1 to each d-dimensional cobordism X between cobordisms Y0 and Y1 from Z0 to Z1 . These data must satisfy natural conditions which I shall not spell out. (There is a discussion of the 3-dimensional case in Lecture 3 of [S4].) In Definition 2.5 the category associated to a point x is the category of modules for the von Neumann algebra Ax , and the functors are defined in the usual way by bimodules. Now in the topological 2-dimensional case we expect the whole structure to be determined by the category assigned to a point, so that a theory reduces to a semisimple C-linear category, i.e. a “module” over the

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category of finite dimensional vector spaces, or, in the language of [BDR] a “two vector space”. These objects define module spectra over the complex K-theory spectrum, and the K-theory of that ring-spectrum is the elliptic theory proposed in [BDR].

References [Ad ] Adams, J.F., Stable homotopy and generalised cohomology. Univ. of Chicago Press, 1974. [An ] Ando, M., The sigma orientation for analytic circle-equivariant elliptic cohomology. Geom. Topology 7(2003), 91-153. [BDR ] Baas, N., B. Dundas, and J. Rognes, Two vector spaces and forms of elliptic cohomology. Topology, Geometry, and Quantum Field Theory. Proc. Oxford 2002. Oxford Univ. Press 2004. [Dv1 ] Devoto, J.A., An algebraic description of the elliptic cohomology of classifying spaces. J. Pure Appl. Alg. 130(1998), 237-264. [Dv2 ] Devoto, J.A., Equivariant elliptic cohomology and finite groups. Michigan Math. J. 43(1996), 3-32. [GMS ] Gorbounov, V., F. Malikov, and V. Schechtman, Gerbes of chiral differential operators. Math. Res. Letters 7(2000), 55-66. AG/9906117. [G ] Grojnowski, I., A delocalized form of equivariant elliptic cohomology. Preprint available fron the author’s home page at www.maths.cam.ac.uk. [H ] Hopkins, M.J., Topological modular forms, the Witten genus, and the theorem of the cube. Proc. Internat. Cong. Mathematicians, Zurich 1994, 554-565. Birkh¨auser 1995. [HKR ] Hopkins, M., N. Kuhn, and D. Ravenel, Group characters and complex oriented cohomology theories. J. Amer. Math. Soc. 13(2000), 553-594. [L ] Landweber, P.S. (Ed.), Elliptic curves and modular forms in algebraic topology. Springer Lecture Notes in Mathematics 1326, 1988. [MSV ] Malikov, F., V. Schechtman, and A. Vaintrob, Chiral de Rham complex. Comm. Math. Phys. 204(1999), 439-473. AG/9803041. [PS ] Pressley, A., and G. Segal, Loop groups. Oxford Univ. Press, 1986. [S1 ] Segal, G., Categories and cohomology theories. Topology 13 (1974), 293-312. [S2 ] Segal, G., Elliptic cohomology. S´eminaire Bourbaki no.695, 1988. Ast´erisque 161-162 (1989),187-201. [S3 ] Segal, G., The definition of conformal field theory. Topology, Geometry, and Quantum Field Theory. Proc. Oxford 2002. Oxford Univ. Press 2004. [S4 ] Segal, G., Notes of lectures at Stanford, available from www.cgtp.duke.edu/ITP99/segal

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[ST ] Stolz, S., and P. Teichner, What is an elliptic object? Topology, Geometry, and Quantum Field Theory. Proc. Oxford 2002. Oxford Univ. Press 2004. [W ] Wassermann, A., Operator algebras and conformal field theory. Proc. Internat. Cong. Mathematicians, Zurich 1994, 966-979. Birkh¨auser 1995.

SPIN BORDISM, CONTACT STRUCTURE AND THE COHOMOLOGY OF p-GROUPS C.B. THOMAS

Among the interesting mathematical developments of the last twenty years has been the use of the topological technique of surgery in differential geometry. This has led, for example, to the establishment of necessary and sufficient conditions for the existence of a metric of positive scalar curvature on a simply-connected manifold, and to the construction of contact forms on certain classes of manifold satisfying the necessary tangential condition. There are also a few scattered results for metrics of positive Ricci curvature, together with an outline argument (due to S. Stolz) that a necessary condition for the existence of such a metric on a simply-connected spin manifold is the vanishing of the Witten genus. In part Stolz’ argument consists in transferring a problem about Ricci curvature on M to one of scalar curvature on the loop space LM, whose ‘K-Theory’ is elliptic cohomology. For a survey of what is known, and additional references, see [6]. The aim of the present paper is to formulate a programme for extending these results for simply-connected manifolds to manifolds having finite fundamental group. Leaving aside a few special cases this has already achieved for groups with periodic cohomology (p-rank = 1), and there is evidence that similar results hold when the p-rank of G = π1 M equals 2. For such groups the even-dimensional cohomology subring H even (G, Z) is generated by ‘transfered Euler classes’, and many of them satisfy the stronger condition Ch(G) = H even (G, Z). Given the results and conjectures already mentioned for positive scalar and positive Ricci curvature (using the Aˆ and the Witten genus respectively) our use of group cohomology suggests that for nonsimply-connected manifolds the proper setting may involve a G-equivariant genus taking values in KO ∗ (BG) or Ell∗ (BG). Note that for groups of low p-rank the Morava building blocks K(n)∗ (BG) for several cohomology theories can be described in terms of Chern or transfered Euler classes of complex representations, see [10] and [8]. The links between contact geometry and ‘bundle-like’ cohomology theories are more tenuous. As J. Devoto explains elsewhere in these proceedings 3dimensional contact manifolds appear to play a role in the construction of

Date: 7 May 2003.

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K3-objects in the v4 -periodic theories generalising elliptic cohomology. Another clue may be provided by the relationship between contact automorphism groups of regular contact manifolds fibering over a K3-surface and the Mathieu simple group M23 . We explain a little of this below. The results of this paper are stated for 5-dimensional contact manifolds. There are easily formulated analogues for 5-manifolds with positive scalar curvature, and in contrast to the contact case there is no geometric obstruction to extending these results to higher dimensions. However here more delicate calculations in cohomology may be needed, especially if part of H even does not come from transfered Euler classes. This work is part of an on-going collaboration with Hansj¨org Geiges. 1. Basic Definitions As usual we write Ωspin n (X) for the bordism group of n-dimensional closed spin manifolds together with maps into the space X. For small values of n the coefficients are as follows: 1 2 Z/2 Z/2 S 1 (S 1 )2

3 0 —

4 Z K3

5,6,7 8 L 0 Z Z — HP2 Bott

More can be said: there is a natural transformation Ωspin n (X) → kon (X) which is a 2-local isomorphism in dimensions 6 7. We will be concerned with the case X = BG, the classifying space for the finite group G (usually of order pt ). We also recall that a contact form α on M 2n−1 is a 1-form such that α∧ (dx)n 6= 0. Such forms arise naturally in theoretical physics, being defined for example on (suitable) constant energy levels in Hamiltonian systems. The form α, or more generally the scalar multiple λα, is associated with a codimension one distribution in TM of symplectic type, called a contact structure. Contact structures are examples of surgery-compatible geometric structures, others being metrics of positive scalar curvature and perhaps positive Ricci curvature. We restrict attention to dimension 5 because: (a) Ωspin 5 (BG) is open to calculation, partly because of the relation with connective K-theory, and (b) we can sidestep framing problems in contact surgery. Technically this amounts to requiring the first Chern class c1 to evaluate as zero on 2-spheres. Contact surgery along any 1-sphere is then possible for any topological framing, and for 2-surgeries we have no choice since π2 (SO3 ) vanishes. There is some hope that the argument may extend to dimension 7, but in higher dimensions more restrictions may be needed.

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Note that the spin assumption (w2 = 0) implies that the tangential structural group of an oriented 5-manifold reduces from SO5 to U2 ⊕ 1 (almost contact condition). Given the possibility of doing contact surgery the problem of integrating some almost contact structure on M 5 reduces to finding a spin-bordant N 5 ∈ Cont5 (G) ⊆ Ωspin 5 (BG). Here Cont5 (G) = set of classes with representatives of the form (f : N → BG, σ), where N admits a contact structure ξ defining the orientation given by the spin structure σ and c1 (ξ) = 0. Definition . G is said to be a contact group if Cont 5 (G) = Ωspin 5 (BG). If M ∼ (N, α) with c1 (ξ) a torsion class, G is a weak contact group. One justification for restricting attention to the class of p-groups is that Cont5 (·) inherits what one can call homological Frobenius reciprocity from equivariant spin bordism. It is this reciprocity which reduces the determination of Cont5 (G) from the group itself to a representative family of Sylow subgroups. Since this use of subgroups to detect the existence of geometric structure also applies to certain classes of p-group it is worthwhile recalling the definitions needed here. Let h : G → G0 be a homomorphism. There is an induced homomorphism spin 0 (Bh)∗ : Ωspin 5 (BG) → Ω5 (BG ), (f : V → BG, σ) 7→ ((Bh)f : V → BG0 , σ),

and, if h is an inclusion, a transfer homomorphism in the reverse direction spin 0 (Bh)! : Ωspin 5 (BG → Ω5 (BG).

Here σ stands for spin structure. To define transfer note that, given (f 0 : 0 V → BG0 , σ 0 ) in Ωspin 5 (BG ), we have a pull-back diagram f

V −−−→ BG y yBh f0

V 0 −−−→ BG0 .

Lift the spin structure σ 0 on V 0 to σ on V , via the covering V → V 0 , and then define (Bh)! (f 0 : V 0 → BG0 , σ 0 ) = (f : V → BG, σ). It is easy to see that both Bh∗ and Bh! map Cont5 to Cont5 , although care must be taken with both their compositions. For G0 the map (Bh)∗ (Bh)! can be studied by means of its analogue in ordinary cohomology; for G the map (Bh)! (Bh)∗ is a little more transparent, at least when G is normal in G0 . Reciprocity properties of this kind are discussed at some length, but with reference to the calculation of bordism groups, rather than to contact or curvature questions by P. Conner and E. Floyd in [2].

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2. Known Results t Proposition 1. The spin bordism group Ωspin 5 (BZ/p )(p = odd) is gener5 t 5 t ated by L (p ; 2, 1, 1) and L (p ; 2, 2, 2) with their canonical spin and Z/pt structures. In particular (for p > 5) Ωspin (BZ/q) ∼ = Z/q ⊕ Z/q. 5

The prime p = 3 is anomalous in that Ω5 ∼ = Z/9. Hence Z/pt is certainly a weak contact group. Calculation with Pontrjagin numbers shows that Proposition 2. If p > 5 or G ∼ = Z/3 (i.e. t = 1), Z/pt is a contact group. As one might expect the prime 2 is more complicated/interesting. Proposition 3. Ωspin (Z/2) = 0. For k > 2, Ωspin (Z/2k ) ∼ = Z/2k ⊕ Z/2k−2 , 5

5

generated by fibre spaces of the form L3 −→ N 5 −→ S 2 . π

Corollary . Z/4 is a weak contact group. (Ω5 is generated by L3 (4, 1, 1) → N 5 → S 2 , which can be shown to be contact with c1 a torsion class.) More tantalising is the result for the quaternion group Q2k : ∼ Proposition 4. Ωspin 5 (BQ2k ) = Z/2 ⊕ Z/2 (k > 3). One of the Z/2-factors is generated by the map f : (S 3 /Q2k ) × T 2 → BQ2k , where on the first factor f classifies the standard linear action and on the second factor f is trivial. (The product admits a contact form, by a branched cover construction.) For these results see [3]. What is suggestive about the proof of Proposition 4 (using the AtiyahHirzebrach spectral sequence for bordism) is that the factors correspond to 2 2 E4,1 and E3,2 . Looking at the low-dimensional spin bordism generators listed on page 3 this suggests that we consider a K3-surface with its natural symplectic form and the contact manifold V 5 obtained by the ‘regular contact construction’ over it. If X is a K3-surface then X admits a nowhere vanishing holomorphic 2form ω. The group G of automorphisms of X which preserve ω is the group of (complex) symplectic automorphisms of X. This group is finite and contained in the Mathieu group M23 , see [5, table on page 184]. Here are some explicit examples: • For the standard quartic x4 + y 4 + z 4 + t4 = 0 in CP 3 G is isomorphic to the extension (42 ) o S4 of order 384. • For the surface in CP 4 defined by the pair of equations x31 + · · · + x34 = 0 x1 x2 + x3 x4 + x25 = 0

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G is a semi-direct product of (32 ) and D8 . • There is a double cover of CP 2 with branch defined by x6 + y 6 + z 6 − 10(x3 y 3 + y 3 z 3 + z 3 x3 ) = 0 and with group isomorphic to the small solvable Mathieu group M9 ∼ = (32 ) · Q8 . The form ω can be written as the sum of the real symplectic forms ω1 +iω2 , and hence using the regular hypercontact/contact construction we obtain a hypercontact form on an S 3 -bundle over X 4 , and (in two ways) a contact form on the total space N 5 of an S 1 -fibration over X 4 . Furthermore, using a construction of A. Banyaga [1] symplectic automorphisms of X 4 lift to strict contact automorphisms of N 5 . The explicit examples given above strongly suggest that, at least for small values if k, a contact generator for the second 2 factor Z/2 ∼ in Ωspin = E4,1 5 (BQ2k ) can be constructed by this method. Full details of the groups and manifolds arising from Mukai’s examples will appear elsewhere [9]. 3. Cohomology of p-Groups, p > 5. At least in principle Ωspin 5 (BG) can be calculated from the terms of total degree 5 on the E 2 -page of the Atiyah-Hirzebruch spectral sequence Hr· (G, Ωspin s (pt)) ⇒ spin Ωr+s (BG). Since |G| is odd, universal cycles are carried by subquotients of H5 (G, Z) and H1 (G, Z) = Gab . [In the case of cyclic groups both these summands are cyclic of order pt , partly explaining Proposition 1.] For finite groups G it is easier to look at cohomology rather than homology, and we have H 6 (G, Z) ∼ = Hom(H6 , Z) ⊕ Ext(H5 , Z), with the first summand equal to zero. The subring H even (G, Z) is finitely generated as a module over the possibly smaller subring Ch(G) generated by Chern classes of irreducible complex representations. For the sake of clarity we will consider this ring rather than the larger and in general more useful one generated by transfered Euler classes of representations of proper subgroups. From [7] we recall the following: Let H(1) (G, Z) be the subring of H even (G, Z) generated by elements of the form {h∗ xj for x ∈ H 2 and some inclusion G1 → G}. Then Proposition 5. If G is a p-group and k < p − 1, H(1) (G, Z) ∩ H 2k (G, Z) = Chk (G). The proof uses Blichfeldt’s theorem on the representations of supersolvable groups together with the ‘Riemann-Roch’ formula for the Newton polynomials in the Chern classes of an induced representation. Dualising from cohomology to spin bordism this allows us to construct contact generators in Ωspin 5 (BG) spin by a transfer process applied to contact generators in Ω5 (BG1 ). The programme now is to construct contact generators for subquotients of the two

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2 2 homology groups H5 (G, Z) ∼ and H1 (G, Z) ∼ . For the first, at = E5,0 = E1,4 6 3 least when H (G, Z) = Ch (G), we use transfer from proper subgroups; for the second a variant of the construction of a regular contact manifold over a K3-surface. If G does not belong to the restricted class of subgroups of M23 considered there, we will need to look for a symplectic manifold bordant to K3 admitting G as a group of automorphisms. Even for groups such that Ch(G) is properly contained in H even , the first part of the argument may apply, since we are only looking at one low dimension. One hint that Chern classes may suffice is provided by the relation between spin bordism and connective K-theory quoted in section one.

An example will bring us down from speculation to rigorous calculation. 2

Using the ‘Atlas’ notation: p1+2 = {a, b : bp = ap = 1, aba−1 = b1+p }, i.e. − 1+2 p− is metacyclic with normal subgroup of index p generated by b. H ∗ (p1+2 − , Z) = Z[α, χ, ζ, β1 . . . βp−1 ], deg α = 2, deg βi = 2i, deg ζ = 2p, deg χ = 2p + 1 with relations p2 ζ = pα = pχ = pβi = 0, χ2 = 0, βi α = βi χ = βi βj = 0 (all i, j). If β ∈ H 2 (< b >, Z) generates the cohomology of the normal subgroup , < b >, then we may take βi = corestriction(β i ). For the details see [4]. 1+2 It follows from the last sentence that H even (p1+2 − , Z) = Ch(p− ).

Let us interpret these calculations for spin bordism, noting first that because of our assumption that p > 5, odd dimensional cohomology arises at 3 worst in dimension 11. H 2 ∼ = Z/αp ⊕ Z/βp 3 . Here = Z/αp ⊕ Z/βp 1 and H 6 ∼ β1 = cor(β), β3 = cor(β 3 ). The relations imply that the monomials α2 β1 , αβ12 , β13 , αβ2 and β1 β2 all vanish. Next recall that for the subgroups < a > and < b > we have already proved that Ωspin is the sum of two copies of a cyclic group. 5 For the subgroup < b > Ω5 ∼ = 2(Z/p2 ), both factors being generated by lens spaces having c1 (ξ) = 0. The p1+2 − action on the transfered 5-sphere can be described in a similar way to an induced representation module. The generator b acts on the disjoint union of p copies of S 5 by (some conjugate of) the original action, while the generator a permutes the factors. This non-connected manifold admits the standard contact form on each sphere. By naturality and another application of the Riemann-Roch formula, cor(c1 (ξ)) = c1 (i∗ ξ) = 0. For the non-normal subgroup generated by a two arguments are available. Transfer can again be used, although the non-normality of the subgroup makes it less transparent. On the other hand we can note that < a > is a retract of p1+2 − , so that the naturality of Cont 5 implies that Cont5 (B < a >),

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as with ordinary cohomology, splits off as a direct summand. Motivation for the transfer argument was provided to the author by [11]. Proposition 6. If p > 5 the metacyclic group p1+2 is a contact group. − A similar argument applies to other split metacyclic p-groups. More interesting is the group p1+2 of exponent p, the finite Heisenberg group. Its even + cohomology, again see [4], is generated by Chern classes. However there are • odd dimensional generators of degree 3 (in contrast to 2p + 1), and • elements which restrict to zero on every proper subgroup. Along with other minimal non-abelian groups the Heisenberg group provides the first test for the validity of the general argument sketched above. References [1] A. Banyaga, Sur le groupe de diff´eomorphismes qui pr´eservent une forme de contact r´eguli`ere, CR Acad. Sci. Pari.; 281 (1975) A707–A709. [2] P.E. Conner, E.E. Floyd, Differentiable Periodic Maps, Ergebruisse der Mathematik 33 (1964) (Springer, Heidelberg). [3] H. Geiges, C.B. Thomas, Contact structures, equivariant spin bordism, and periodic fundamental groups, Math. Annalen 320 (2001) 685–670. [4] G. Lewis, The integral cohomology rings of groups of order p 3 , Trans. Amer. Math. Soc. 132 (1968) 501–529. [5] S. Mukai, Finite groups of automorphisms of K3-surfaces and the Mathieu group, Invent. Math. 94 (1988) 183–221. [6] J. Rosenberg, Surgery Theory Today - what it is and where it’s going, Annals of Math. Studies 149 (Princeton University Press 2001) 3–47. [7] C.B. Thomas, An integral Riemann - Roch formula for flat line bundles, Proc. London Math. Soc. (3) (1977) 87–101. [8] C.B. Thomas, Morava K-theory for groups of small p-rank, (in preparation). [9] C.B. Thomas, Automorphisms of hypercontact manifolds fibering over K3-surfaces, (in preparation). [10] N. Yagita, Cohomology for Groups of rank G = 2 and Brown-Peterson Cohomology, J. Math. Soc. Japan 45 (1993) 627–644. [11] N. Yagita, Complex cobordism of the dihedral group, J. Math. Soc. Japan 48 (1996) 195–203.

University of Cambridge May 2003 [email protected]

BRAVE NEW ALGEBRAIC GEOMETRY AND GLOBAL DERIVED MODULI SPACES OF RING SPECTRA ¨ AND GABRIELLE VEZZOSI BERTRAND TOEN Abstract. We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, brave new algebraic geometry. We discuss various model topologies on the model category of commutative algebras in S, and their associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry). Two examples of geometric S-stacks are given: a global moduli space of associative ring spectrum structures, and the stack of elliptic curves endowed with the sheaf of topological modular forms.

Key words: Sheaves, stacks, ring spectra, elliptic cohomology. MSC-class: 55P43; 14A20; 18G55; 55U40; 18F10. Contents 1. Introduction 2. Brave new sites 2.1. The brave new Zariski topology 2.2. The brave new ´etale topology 2.3. Standard topologies 3. S-stacks and geometric S-stacks 3.1. Some descent theory 3.2. The S-stack of perfect modules 3.3. Geometric S-stacks 4. Some examples of geometric S-stacks 4.1. The brave new group scheme RAut(M) 4.2. Moduli of algebra structures 4.3. Topological modular forms and geometric S-stacks References

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Homotopical Algebraic Geometry is a kind of algebraic geometry where the affine objects are given by commutative ring-like objects in some homotopy theory (technically speaking, in a symmetric monoidal model category); these affine objects are then glued together according to an appropriate homotopical modification of a Grothendieck topology (a model topology, see [To-Ve 1, 4.3]). More generally, we allow ourselves to consider more flexible objects like stacks, in order to deal with appropriate moduli problems. This theory is developed in full generality in [To-Ve 1, To-Ve 2] (see also [To-Ve 3]). Our motivations for such a theory came from a variety of sources: first of all, on the algebro-geometric side, we wanted to produce a sufficiently functorial language in which the so called Derived Moduli Spaces foreseen by Deligne, Drinfel’d and Kontsevich could really be constructed; secondly, on the topological side, we thought that maybe the many recent results in Brave New Algebra, i.e. in (commutative) algebra over structured ring spectra (in any one of their brave new symmetric monoidal model categories, see e.g. [Ho-Sh-Sm, EKMM]), could be pushed to a kind of Brave New Algebraic Geometry in which one could take advantage of the possibility of gluing these brave new rings together into an actual geometric object, much in the same way as commutative algebra is helped (and generalized) by the existence of algebraic geometry. Thirdly, on the motivic side, following a suggestion of Y. Manin, we wished to have a sufficiently general theory in order to study algebraic geometry over the recent model categories of motives for smooth schemes over a field ([Hu, Ja, Sp]). The purpose of this paper is to present the first steps in the second type of applications mentioned above, i.e. a specialization of the general framework of homotopical algebraic geometry to the context of stable homotopy theory. Our category S − Aff of brave new affine objects will therefore be defined as the the opposite model category of the category of commutative rings in the category S of symmetric spectra ([Ho-Sh-Sm]). We first define and study various model topologies defined on S − Aff. They are all extensions, to different extents, of the usual Grothendieck topologies defined on the category of (affine) schemes, like the Zariski and ´etale ones. With any of these model topologies τ at our disposal, we define and give the basic properties of the corresponding model category of S-stacks, understood in the broadest sense as not necessarily truncated presheaves of simplicial sets on S − Aff satisfying a homotopical descent (i.e. sheaf-like) condition with respect to τ -(hyper)covers. A model topology on S − Aff is said to be subcanonical if the representable simplicial presheaves, i.e. those of the form Map(A, −), for some commutative ring A in S, Map being the mapping space in S − Aff op , are S-stacks.

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As in algebraic geometry one finds it useful to study those stacks defined by smooth groupoids (these are called Artin algebraic stacks), we also define a brave new analog of these and call them geometric S-stacks, to emphasize that such S-stacks host a rich geometry very close to the geometric intuition learned in algebraic geometry. In particular, given a geometric S-stack F , it makes sense to speak about quasi-coherent and perfect modules over F , about the K-theory of F , etc.; various properties of morphisms (e.g. smooth, ´etale, proper, etc.) between geometric S-stacks can likewise be defined. Stacks were introduced in algebraic geometry mainly to study moduli problems of various sorts; they provide actual geometric objects (rather than sets of isomorphisms classes or coarse moduli schemes) which store all the fine details of the classification problem and on which a geometry very similar to that of algebraic varieties or schemes can be developed, the two aspects having a fruitful interplay. In a similar vein, in our brave new context, we give one example of a moduli problem arising in algebraic topology (the classification of A∞ -ring spectrum structures on a given spectrum M) that can be studied geometrically through the geometric S-stack RAssM it represents. We wish to emphasize that instead of a discrete homotopy type (like the ones studied, for different moduli problems, in [Re, B-D-G, G-H]), we get a full geometric object on which a lot of interesting geometry can be performed. The geometricity of the S-stack RAssM , with respect to any fixed subcanonical model topology, is actually the main theorem of this paper (see Theorem 4.2.1). We also wish to remark that the approach presented in this paper can be extended to other, more interested and involved, moduli problems algebraic topologists are interested in, and perhaps this richer geometry could be of some help in answering, or at least in formulating in a clearer way, some of the deep questions raised by the recent progress in stable homotopy theory (see [G]). In this direction, we will explain in §4.3 how topological modular forms give rise to a natural geometric S-stack which is an extension in the brave new direction of the moduli stack of elliptic curves (see Theorem 4.3.1). This fact seems to us a very important remark (probably much more interesting than our Theorem 4.2.1), and we think it could be the starting point of a very interesting research program. We also present a brave new analog of the stack of vector bundles on a scheme, called the S-(pre)stack Perf of perfect modules (Section 3.2), and we expect it to be a key tool in brave new algebraic geometry. The prestack Perf is a stack if and only if the model topology we are working with is subcanonical (Thm. 3.2.1 whose proof is postponed to [To-Ve 2]). This is another instance of the relevance of the descent problem, i.e. the question whether a given model topology is subcanonical or not (see Section 3.1). Though we prove that some of the model topologies we introduce (namely the standard and the semi-standard ones, Section 2.3) are subcanonical, at present we are not

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able to settle (nor in the positive nor in the negative) the descent problem for the three most promising model topologies we define, namely the Zariski, ´etale and thh-´etale ones. Though this is at the moment quite unsatisfactory, we believe that the descent problem for these topologies is a very interesting question in itself even leaving outside its crucial role in brave new algebraic geometry. For the Zariski model topology, we have a partial positive result in this direction. By definition, for a model topology τ the property of being subcanonical depends on the notion of stacks we consider; if instead of defining a stack as a prestack (i.e. a simplicial presheaf) satisfying homotopical descent with respect to all homotopy τ -hypercovers, we simply require descent with ˇ respect to all Cech τ -hypercovers (i.e. those arising as nerves of τ -covers), ˇ we obtain a notion of Cech stacks, recently considered by J. Lurie ([Lu]) and Dugger-Hollander-Isaksen ([DHI]). We prove (Corollary 3.1.4) that the Zariski model topology is in fact subcanonical with respect to the notion of ˇ Cech stacks. Moreover, by replacing in Theorem 4.2.1 the word “stack” with ˇ the weaker “Cech stack”, the statement remains true for any model topology ˇ which is subcanonical with respect to the notion of Cech stacks. It is therefore natural to ask why we did not choose to formulate everyˇ thing only in terms of Cech stacks. We believe that at this early stage of development of homotopical algebraic geometry and, in particular, of brave new algebraic geometry, it is not advisable to make choices that could prevent some applications or obscure some of the properties of the objects involved, while it is more useful to keep in mind various options, some of which can be more useful in one context than in others. For example, it is clear that knowing that a given, geometrically meaningful, simplicial presheaf is a stack ˇ and not only a Cech stack adds a lot more informations, in fact exactly the descent property with respect to unbounded hypercovers ([DHI, Thm. A.6]). ˇ Moreover, Cech stacks fail in general to satisfy an analog of Whitehead theˇ orem: a pointed Cech stack may have vanishing πi sheaves for any i ≥ 0 without being necessarily contractible. This last fact is a very inconvenient ˇ property of Cech stacks, that makes Postnikov decompositions and spectral ˇ sequences arguments uncertain. On the other hand, the Cech descent condition is usually much easier to establish than the full descent condition, and as we have already remarked, some natural model topologies are easily seen ˇ to be subcanonical with respect to the notion of Cech stacks while it might be tricky to show that they are actually subcanonical. Finally, we would like to mention that in our experience we have never met serious troubles by using one or the other of the two notions, and in many interesting contexts it does not really matter which notion one uses, as the rather subtle differences actually tend not to appear in practice.

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Acknowledgments. We wish to thank the organizers of the INI Workshop on Elliptic Cohomology and Higher Chromatic Phenomena (Cambridge UK, December 2003) for the invitation to speak about our work and Bill Dwyer for his encouraging comments. We would also like to thank Michael Mandell, Peter May, Haynes Miller, John Rognes, Stefan Schwede and Neil Strickland for helpful discussions and suggestions. Notations. To fix ideas, we will work in the category S := Sp Σ of symmetric spectra (see [Ho-Sh-Sm]), but all the constructions of this paper will also work, possibly with minor variations (see [Sch]), for other equivalent theories (e.g for the category of S-modules of [EKMM]). We will consider S as a symmetric monoidal simplicial model category (for the smash product − ∧ −) with the Shipley-Smith positive S-model structure (see [Shi, Prop. 3.1]). We define S − Alg as the category of (associative and unital) commutative monoids objects in S, endowed with the S-model structure of [Shi, Thm. 3.2]; we will simply call them commutative S-algebras instead of the more correct but longer, commutative symmetric ring spectra. For any commutative Salgebra A, we will denote by A − Alg the under-category A/S − Alg, whose objects will be called commutative A-algebras. Finally, if A is a commutative S-algebra, A − Mod will be the category of A-modules with the A-model structure ([Shi, Prop. 3.1]). This model category is also a symmetric monoidal model category for the smash product − ∧A − over A. For a morphism of commutative S-algebras, f : A −→ B one has a Quillen adjunction f ∗ : A − Mod −→ B − Mod

A − Mod ←− B − Mod : f∗ ,

where f ∗ (−) := − ∧A B is the base change functor. We will denote by Lf ∗ : Ho(A−Mod) −→ Ho(B−Mod)

Ho(A−Mod) ←− Ho(B−Mod) : Rf∗

the induced derived adjunction on the homotopy categories. Our references for model category theory are [Hi, Ho]. For a model category M with equivalences W , the set of morphisms in the homotopy category Ho(M) := W −1 M will be denoted by [−, −]M , or simply by [−, −] if the context is clear. The (homotopy) mapping spaces in M will be denotedby MapM (−, −). When M is a simplicial model category, the simplicial Hom’s (resp. derived simplicial Hom’s) will be denoted by Hom M (resp. RHomM ), or simply by Hom (resp. RHom) if the context is clear. Recall that in this case one can compute MapM (−, −) as RHomM (−, −). Finally, for a model category M and an object x ∈ M we will often use the coma model categories x/M and M/x. When the model category M is not left proper (resp. is not right proper) we will always assume that x has been replaced by a cofibrant (resp. fibrant) model before considering x/M (resp. M/x). More generally, we will not always mention fibrant and cofibrant replacements and suppose implicitly that all our objects are fibrant

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and/or cofibrant when required. Since we wish to concentrate on applications to stable homotopy theory, some general constructions and details about homotopical algebraic geometry will be omitted by referring to [To-Ve 1]. For a few of the results presented we will only give here sketchy proofs; full proofs will appear in [To-Ve 2].

2. Brave new sites In this section we present two model topologies on the (opposite) category of commutative S-algebras. They are brave new analogs of the Zariski and ´etale topologies defined on the category of usual commutative rings and will allow us to define the brave new Zariski and ´etale sites. We denote by S − Aff the opposite model category of S − Alg. If M is a model category we say that an object x in M is finitely presented if, for any filtered direct system of objects {zi }i∈J in M, the natural map colimi MapM (x, zi ) −→ MapM (x, colimi zi ) is an isomorphism in the homotopy category of simplicial sets. Definition 2.0.1. A morphism A → B of commutative S-algebras is finitely presented if it is a finitely presented object in the model under-category A/(S − Alg) = A − Alg; in this case, we also say that B is a finitely presented commutative A-algebra. An A-module E is finitely presented or perfect if it is a finitely presented object in the model category A − Mod. Perfect A-modules can also be characterized as retracts of finite cell Amodules (see [EKMM, Thm. III-7.9]); in particular, there are plenty of them. If A is a commutative S-algebra, then the free commutative A-algebra on a finite number of generators (or, more generally, on any perfect A-module) is a finitely presented A-algebra. The reader will find other examples of finitely presented morphisms of commutative S-algebras in Lemma 2.1.6. 2.1. The brave new Zariski topology. Definition 2.1.1. • A morphism f : A −→ B in S − Alg is called a formal Zariski open immersion if the induced functor Rf∗ : Ho(B − Mod) −→ Ho(A − Mod) is fully faithful. • A morphism f : A −→ B is a Zariski open immersion if S − Alg is it is a formal Zariski open immersion and of finite presentation (as a morphism of commutative S-algebras).

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• A family {fi : A −→ Ai }i∈I of morphisms in S − Alg is called a (formal) Zariski open covering if it satisfies the following two conditions. – Each morphism A −→ Ai is a (formal) Zariski open immersion. – There exist a finite subset J ⊂ I such that the family of inverse image functors {Lfj∗ : Ho(A − Mod) −→ Ho(Aj − Mod)}j∈J is conservative (i.e. a morphism in Ho(A − Mod) is an isomorphism if and only if its images by all the Lfj∗ ’s are isomorphisms). Example 2.1.2. If A ∈ S − Alg and E is an A-module such that the associated Bousfield localization LE is smashing (i.e. the natural transformation LE (−) → LE A ∧LA (−) is an isomorphism), then A → LE A (which is a morphism of commutative S-algebras by e.g. [EKMM, §VIII.2]) is a formal Zariski open immersion. This follows immdiately from the fact that Ho(LE A − Mod) is equivalent to the subcategory of Ho(A − Mod) consisting of LE -local objects, by [Wo]. It is easy to check that (formal) Zariski open covering families define a model topology in the sense of [To-Ve 1, §4.3] on the model category S − Aff. For the reader’s convenience we recall what this means in the following lemma. Lemma 2.1.3. • If A −→ B is an equivalence of commutative S-algebras then the one element family {A −→ B} is a (formal) Zariski open covering. • Let {A −→ Ai }i∈I be a (formal) Zariski open covering of S-algebras and A −→ B a morphism. Then, the family of homotopy push-outs {B −→ B ∧LA Ai }i∈I is also a (formal) Zariski open covering. • Let {A −→ Ai }i∈I be a (formal) Zariski open covering of S-algebras, and for any i ∈ I let {Ai −→ Aij }j∈Ji be a (formal) Zariski open covering of S-algebras. Then, the total family {A −→ Aij }i∈I,j∈Ji is again a (formal) Zariski open covering. Proof: Left as an exercise.

2

By definition, Lemma 2.1.3 shows that (formal) Zariski open coverings define a model topology on the model category S − Aff and so, as proved in [To-Ve 1, Prop. 4.3.5], induce a Grothendieck topology on the homotopy category Ho(S − Alg). This model topology is called the brave new (formal) Zariski topology, and endows S − Aff with the structure of a model site in the sense of [To-Ve 1, §4]. This model site, denoted by (S − Aff, Zar) for the brave new Zariski topology, and (S − Aff, fZar) for the brave new formal Zariski topology. They will be called the brave new Zariski site and the brave new formal Zariski site.

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Let Alg be the category of (associative and unital) commutative rings. Let us recall the existence of the Eilenberg-Mac Lane functor H : Alg −→ S − Alg, sending a commutative ring R to the commutative S-algebra HR such that π0 (HR) = R and πi (HR) = 0 for any i 6= 0. This functor is homotopically fully faithful and the following lemma shows that our brave new Zariski topology does generalize the usual Zariski topology. Lemma 2.1.4. (1) Let R −→ R0 be a morphism of commutative rings. The induced morphism HR −→ HR0 is a Zariski open immersion of commutative S-algebras (in the sense of Definition 2.1.1) if and only if the morphism Spec R0 −→ Spec R is an open immersion of schemes. (2) A family of morphisms of commutative rings, {R −→ Ri0 }i∈I , induces a Zariski covering family of commutative S-algebras {HR −→ HRi0 }i∈I (in the sense of Definition 2.1.1) if and only if the family {Spec Ri −→ Spec R}i∈I is a Zariski open covering of schemes. Proof: Let us start with the general situation of a morphism f : A −→ B of commutative S-algebras such that the induced functor Rf∗ : Ho(B − Mod) −→ Ho(A − Mod) is fully faithful. Let L = Rf∗ ◦ Lf ∗ , which comes with a natural transformation Id −→ L. Then, the essential image of Rf∗ consist of objects M in Ho(A − Mod) such that the localization morphism M −→ LM is an isomorphism. The Quillen adjunction (f ∗ , f∗ ) extends to a Quillen adjunction on the category of commutative algebras f ∗ : A − Alg −→ B − Alg

A − Alg ←− B − Alg : f∗ ,

also with the property that Rf∗ : Ho(B − Alg) −→ Ho(A − Alg) is fully faithful. Furthermore, the essential image of this last functor consist of all objects C ∈ Ho(A − Alg) such that the underlying A-module of C satisfies C ' LC (i.e. the underlying A-module of C lives in the image of Ho(B − Mod)). ¿From these observations, we deduce that for any commutative A-algebra C, the mapping space RHomA−Alg (B, C) is either empty or contractible; it is non-empty if and only if the underlying A-module of C belongs to the essential image of Rf∗ . To prove (1), let us first suppose that f : Spec R0 −→ Spec R is an open immersion of schemes. The induced functor on the derived categories f∗ : D(R0 ) −→ D(R) is then fully faithful. As there are natural equivalences ([EKMM, IV Thm. 2.4]) Ho(HR − Mod) ' D(R)

Ho(HR0 − Mod) ' D(R0 )

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this implies that the functor Rf∗ : Ho(HR0 − Mod) −→ Ho(HR − Mod) is also fully faithful. It only remains to show that HR −→ HR0 is finitely presented in the sense of Definition 2.0.1. We will first assume that R0 = Rf for some element f ∈ R. The essential image of Rf∗ : Ho(HR0 − Mod) −→ Ho(HR − Mod) then consists of all objects E ∈ Ho(HR − Mod) ' D(R) such that f acts by isomorphisms on the cohomology R-module H ∗ (E). By what we have seen at the beginning of the proof, this implies that for any commutative HR-algebra C the mapping space RHomHR−Alg (HR0 , C) is contractible if f becomes invertible in π0 (C), and empty otherwise. ¿From this one easily deduces that RHomHR−Alg (HR0 , −) commutes with filtered colimits, or in other words that HR0 is a finitely presented HR-algebra in the sense of Definition 2.0.1. In the general case, one can write Spec R0 as a finite union of schemes of the form Spec Rf for some elements f ∈ R. A bit of descent theory (see §3.1) then allows us to reduce to the case where R0 = Rf and conclude. Let us now assume that HR −→ HR0 is a Zariski open immersion of commutative S-algebras. By adjunction (between H and π0 restricted on connective S-algebras) one sees easily that R −→ R 0 is a finitely presented morphism of commutative rings. The induced functor on (unbounded) derived categories f∗ : D(R0 ) ' Ho(HR0 − Mod) −→ D(R) ' Ho(HR − Mod) is fully faithful. Through the Dold-Kan correspondence, this implies that the Quillen adjunction on the model category of simplicial modules (see [G-J]) f ∗ : sR − Mod −→ sR0 − Mod

sR − Mod ←− sR0 − Mod : f∗

is such that Lf ∗ ◦ f∗ ' Id. Let sR − Alg and sR0 − Alg be the categories of simplicial commutative R-algebras and simplicial commutative R0 -algebras, endowed with their natural model structures (equivalences are and fibration are detected in the category of simplicial modules). Then, the Quillen adjunction f∗ : sR0 − Alg −→ sR − Alg

sR0 − Alg ←− sR − Alg : f ∗

also satisfies Lf ∗ ◦ f∗ ' Id, as this is true on the level on simplicial modules. In particular, for any simplicial R0 -module M, the space of derived derivations LDerR (R0 , M) := RHomsR−Alg/R0 (R0 , R0 ⊕M) ' RHomsR0 −Alg/R0 (R0 , R0 ⊕M) ' ∗ is acyclic (here R0 ⊕ M is the simplicial R0 -algebra which is the trivial extension of R0 by M). As a consequence one sees that Quillen’s cotangent complex LR0 /R is acyclic, which implies that the morphism R −→ R0 is an ´etale morphism of rings. Finally, using the fact that the functor on the category of modules R 0 − Mod −→ R − Mod is fully faithful, one sees that Spec R0 −→ Spec R is a monomorphism of schemes. Therefore, the morphism of schemes Spec R 0 −→

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Spec R is an ´etale monomorphism, and so is an open immersion by [EGA-IV, Thm. 17.9.1]. Finally, point (2) is clear if one knows (1) and that Ho(HR − Mod) ' D(R). 2 Remark 2.1.5. The argument at the beginning of the proof of Lemma 2.1.4 shows that if f : A → B is a Zariski open immersion, the functor L(f ) := Rf∗ Lf ∗ is a localization functor on the homotopy category of A-modules in the sense of [HPS, Def. 3.1.1]. And it is also clear by definition that L(f ) is also smashing ([HPS, Def. 3.3.2]). Let us call a localization functor L on Ho(A − Mod) a formal Zariski localization functor over A if L ' L(f ) for some formal Zariski open immersion f . Let us also say that a localization functor L on Ho(A−Mod) is a smashing algebra Bousfield localization over A if L ' LB for some A-algebra B such that LB is smashing (over A). Then it is easy to verify that in the set of equivalence classes of localization functors on Ho(A − Mod), the subset consisting of formal Zariski localization functors over A coincides with the subset consisting of smashing algebra Bousfield localizations over A. In fact, if f : A → B is a Zariski open immersion, LB denotes the Bousfield localization with respect to the A-module B, and `B/A : A → LB A the corresponding morphism of commutative A-algebras, we have L(f ) ' LB ' L(`B/A ) because all three localizations have the same category of acyclics. Conversely, if LC is a smashing algebra Bousfield localization over A, and `C/A : A → LC A is the corresponding morphism of commutative A-algebras, one has LB ' L(`C/A ). Let Aff be the opposite category of commutative rings, and (Aff, Zar) the big Zariski site. The site (Aff, Zar) can also be considered as a model site (for the trivial model structure on Aff ). Lemma 2.1.4 implies in particular that the functor H : Aff −→ S − Aff induces a continuous morphism of model sites ([To-Ve 1, Def. 4.8.4]). In this way, the site (Aff, Zar) becomes a sub-model site of (S − Aff, Zar). To finish with the Zariski topology we will now describe a general procedure in order to construct interesting open Zariski immersions of commutative S-algebras using the techniques of Bousfield localization for model categories. Let A be a commutative S-algebra and M be a A-module. We will assume that M is a perfect A-module (in the sense of Definition 2.0.1), or equivalently that it is a strongly dualizable object in the monoidal category Ho(A−Mod). As already noticed, perfect A-modules are exactly the retracts of finite cell A-modules, see [EKMM, Thm. III-7.9]). Let M[n] = S n ⊗L M be the n-th suspension A-module of M, for n ∈ Z.

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We denote by D(M[n]) the derived dual of M[n], defined as the derived internal Hom’s of A-modules D(M[n]) := RHOMA−Mod (M[n], A). Consider now the (derived) free commutative A-algebra over D(M[n]), LFA (D(M[n])), characterized by the usual adjunction [LFA (D(M[n])), −]A−Alg ' [D(M[n]), −]A−Mod . The model category A − Alg is a combinatorial and cellular model category, and therefore one can apply the localization techniques (see e.g. [Hi, Sm]) in order to invert the natural augmentations FA (D(M[n])) −→ A for all n ∈ Z. One checks easily that, since M is strongly dualizable, the local objects for this localization are the commutative A-algebras B such that M ∧LA B ' 0 in Ho(B − Mod). The local model of A for this localization will be denoted by AM . By definition, it is characterized by the following universal property: for any commutative A-algebra B, the mapping space RHomA−Alg (AM , B) is contractible if B ∧LA M ' 0 and empty otherwise. In other words, for any commutative S-algebra B the natural morphism RHomS−Alg (AM , B) −→ RHomS−Alg (A, B) is equivalent to an inclusion of connected components and its image consists of morphisms A −→ B in Ho(S − Alg) such that B ∧LA M ' 0. Lemma 2.1.6. With the above notations, the morphism A −→ AM is a Zariski open immersion. Proof: Let us start by showing that AM is a finitely presented commutative A-algebra. Let {Bi }i∈I be a filtered system of commutative A-algebras and B = colimi Bi . We assume that B ∧LA M ' 0, and we need to prove that there exists an i ∈ I such that Bi ∧LA M ' 0. By assumption, the two points Id and 0 are the same in π0 (REndB−Mod (M ∧LA B)). But, as M is a perfect A-module one has π0 (REndB−Mod (M ∧LA B)) ' colimi∈I π0 (REndBi −Mod (M ∧LA Bi )). This implies that there is some index i ∈ I such that Id and 0 are homotopic in REndBi −Mod (M ∧LA Bi ), and therefore that M ∧LA Bi is contractible. It remains to prove that the induced functor Ho(AM − Mod) −→ Ho(A − Mod) is fully faithful. To see this, one first notice that by definition the functor Ho(AM − Alg) −→ Ho(A − Alg) is fully faithful, and therefore so is Ho(AM − Alg/AM ) −→ Ho(A − Alg/AM ). For any two AM -modules N and P , one consider the trivial extensions AM ∨ N and AM ∨ P of AM by N and P (these are AM -augmented commutative AM -algebras). Then, one has

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a natural equivalence RHomAM −Alg/AM (AM ∨ N, AM ∨ P ) ' RHomAM −Mod (N, P ). Furthermore one has a natural fiber sequence RHomA−Mod (N, P ) → RHomA−Alg/AM (AM ∨ N, AM ∨ P ) → RHomA−Alg/AM (AM , AM ∨ P ). But, as RHomA−Alg/AM (AM , AM ∨ P ) ' RHomAM −Alg/AM (AM , AM ∨ P ) ' ∗ this shows that the natural morphism RHomAM −Mod (N, P ) → RHomA−Mod (N, P ), is an equivalence and therefore that Ho(AM − Alg) −→ Ho(A − Alg) is fully faithful. 2 An important property of the localization A −→ AM is the following fact. Lemma 2.1.7. Let A be a commutative S-algebra, and M be a perfect Amodule. Then the essential image of the fully faithful functor Ho(AM − Mod) −→ Ho(A − Mod) consists of all A-modules N such that M ∧LA N ' D(M) ∧LA N ' 0. Note that since M is perfect, then for any A-module N , M ∧LA N ' 0 iff D(M) ∧LA N ' 0, so the two conditions in the lemma are actually one. Moreover, AM ' AD(M ) in Ho(A − Alg). Proof: As every AM -module N can be constructed by homotopy colimits of free AM -modules and − ∧LA M commutes with homotopy colimits, it is clear that AM ∧LA M ' 0 implies N ∧LA M ' 0. Since AM ∧LA D(M) ' D(AM ∧LA M) ' 0 (here the second derived dual is in the category of AM -modules), the same argument shows that N ∧LA D(M) ' 0. Conversely, let N be an A-module such that N ∧LA M ' N ∧LA D(M) ' 0. By definition, the commutative A-algebra A −→ AM is obtained as a local model of A → A when one inverts the set of morphisms LFA (D(M[n])) −→ A, for any n ∈ Z. It is well known (see e.g. [Hi, §4]) that such a local model can be obtained by a transfinite composition of homotopy push-outs of the form AO α ∂∆p ⊗L LFA (D(M[n])) /

/

Aα+1 O

∆p ⊗L LFA (D(M[n]))

in the category of A-algebras. ¿From this description, and the fact that −∧LA M commutes with homotopy colimits, one sees that the adjunction morphism N −→ N ∧LA AM is an equivalence because by assumption on N , the natural morphism N ' N ∧LA A −→ N ∧LA LFA (D(M[n])) is an equivalence. 2 Lemma 2.1.7 allows us to interpret geometrically AM as the open complement of the support of the A-module M. Lemma 2.1.7 also has a converse whose proof is left as an exercise.

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Lemma 2.1.8. Let f : A −→ B be a morphism of commutative S-algebras and M be a perfect A-module. We suppose that the functor Rf∗ : Ho(B − Mod) −→ Ho(A−Mod) is fully faithful and that its essential image consists of all A-modules N such that N ∧LA M ' 0. Then, the two commutative Aalgebras B and AM are equivalent (i.e. isomorphic in Ho(A − Alg)). Remark 2.1.9. One should note carefully that even though if the EilenbergMac Lane functor H embeds (Aff, Zar) in (S − Aff, Zar) as model sites, there exist commutative rings R and Zariski open coverings HR −→ B in S − Aff such that B is not of the form HR0 for some commutative R-algebra R0 . One example is given by taking R to be C[X, Y ], and considering the localized commutative HR-algebra (HR)M (in the sense above), where M is the perfect R-module R/(X, Y ) ' C. If (HR)M were of the form HR0 for a Zariski open immersion Spec R0 −→ Spec R, then for any other commutative ring R00 , the set of scheme morphisms Hom(Spec R00 , Spec R0 ) would be the subset of Hom(Spec R00 , A2 ) consisting of morphisms factoring through A2 − {0}. This would mean that Spec R0 ' A2 − {0}, which is not possible as A2 − {0} is a not an affine scheme. This example is of course the same as the example given in [To, §2.2] of a 0-truncated affine stack which is not an affine scheme. These kind of example shows that there are many more affine objects in homotopical algebraic geometry than in usual algebraic geometry. Remark 2.1.10. (1) Note that Lemma 2.1.7 shows that the localization process (A, M) /o /o /o / AM is in some sense “orthogonal” to the usual Bousfield localization process (A, M) /o /o /o / LM A in that the local objects for the former are exactly the acyclic objects for the latter. To state everything in terms of Bousfield localizations, this says that LAM -local objects are exactly LM -acyclic objects (compare with Remark 2.1.5). Note that however, while the Bousfield localization is always defined for any A-module M, the commutative A-algebra AM probably does not exist unless M is perfect. (2) Let Sp be the p-local sphere. If f : Sp → B is any formal Zariski open immersion then L := Rf∗ Lf ∗ is clearly a smashing localization functor in the sense of [HPS, §3]. Its category C of perfect1 acyclics (i.e. perfect objects X in Ho(Sp − Mod) such that LX is null) is then a localizing thick subcategory of the homotopy category Ho(Sp −Modperf ) of the category of perfect Sp -modules, and therefore by [H-S] it is equivalent to the category Cn of perfect E(n)-acyclics, for some 0 ≤ n < ∞, where E(n) is the n-th Johnson-Wilson Sp -module (see e.g. [Rav]); in other words L and Ln := LE(n) are both smashing localization functors on Ho(Sp − Mod) having the same subcategory of finite acyclics. Therefore, if we assume (one of the form of) the 1The

word finite instead of perfect would be more customary in this setting.

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Bertrand To¨en and Gabrielle Vezzosi Telescope conjecture (see [Mil]), we get that Ln and L have equivalent categories of acyclics and so have equivalent categories of local objects. But the category of local objects for L is equivalent to the category Ho(B − Mod) (since Rf∗ is fully faithful by hypothesis) and the category of local objects for Ln is equivalent to the category Ho((Ln Sp ) − Mod), by [Wo] since Ln is smashing. This easily implies that the two commutative Sp -algebras B and Ln Sp are equivalent (i.e. isomorphic in Ho(Sp − Alg)). In conclusion, one sees that if the Telescope conjecture is true, then, up to equivalence of Sp -algebras, the only (non-trivial) formal Zariski open immersions for Sp are given by the family U := {Sp → Ln Sp }0≤n

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