Algebraic Topology, Poznan 1989: Proceedings of a Conference Held in Poznan, Poland, June 22-27, 1989 (Lecture Notes in Mathematics)

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zfirich F. Takens, Groningen

1474

S. Jackowski B. Oliver K. Pawayowski (Eds.)

Algebraic Topology Poznafi 1989 Proceedings of a Conference held in Poznafi, Poland, June 22-27, 1989

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Editors Stefan Jackowski Instytut Matematyki Uniwersytet Warszawski ul. Banacha 2 PL-00-913 Warszawa 59, Poland Bob Oliver Matematisk Institut Ny Munkegade 8000 Aarhus C, Denmark Krzystof PawaJfowski Instytut Matematyki UAM ul. Matejki 48/49 60-769 Poznafi, Poland

Mathematics Subject Classification (1980): 57-06, 55-06, 19-06

ISBN 3-540-54098-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54098-9 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper

Preface

In June, 1989, the International Conference on Algebraic Topology was held in Poznafi, Poland. The conference was part of the scientific activity in connection with the 70-th anniversary of the Adam Mickiewicz University in Poznafi. It was supported by the Adam Mickiewicz University, Warsaw University, and Polish government grant RP.I.10. There were many of our colleagues and students from both Pozna~ and Warszawa who helped to contribute to the success of the conference. We would especially like to mention Agnieszka Bojanowska, Adam Neugebauer and Bogdan Szydto, who helped with the organizational work, and the two conference secretaries Danuta Marciniak and Katarzyna Kacperska-Panek. The conference consisted of 10 plenary talks, as well as 49 talks in special sessions in various fields. These proceedings contain papers presented at the conference, as well as some other papers (mostly) submitted by conference participants. We tried--and with some success--to encourage the submission of survey papcrs. All papers in the volume have been refereed. We would like to thank the referees for their work, and Andrzej Weber for proofreading of several manuscripts which had to be retyped during the editorial process.

Stefan Jackowski Bob Oliver Krzysztof Pawatowski Warszawa//~,rhus/Poznafi, November 1990

Table o f Contents

Survey Articles: Allday, C., Puppe, V. Some Applications of Shifted Subgroups in Transformation Groups . . . . . . . . . . . . . . . . .

1

Andrzejewski, P. The Equivariant Finiteness Obstruction and its Geometric Applications: A Survey 20

Dula, G. On Conic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Faxrell, T., Jones, L. Computations of Stable Pseudoisotopy Spaces for Aspherical Manifolds . . . . . . . . . . . 59

Johnson,F. E. A., Rees, E. The Fundamental Groups of Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Murasugi, K. Invariants of Graphs and Their Applications to Knot Theory . . . . . . . . . . . . . . . . . . . .

83

Pazhitnov, A. Morse Theory of Closed 1-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Wiirgler, U. Morava K-theories. A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

G e o m e t r y of Manifolds: Connolly, F., Koiniewski, T. Examples of Lack of Rigidity in Crystallographic Groups . . . . . . . . . . . . . . . . . . . . . . . .

139

Hausmann, J. C. •

/

Sur la Topologie des Bras Arhcules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

Koschorke, U. Semicontractible Link Maps and Their Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160

Rosenberg, J. The KO Assembly Map and Positive Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . .

170

Sadowski, M. Equivariant Splittings Associated with Smooth Toral Actions . . . . . . . . . . . . . . . . . . . . .

183

Troitsky, E. V. Lefschetz Number of C*-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193

Homotopy Theory Baues, H-J. On the Homotopy Category of Moore Spaces and an Old Result of Barrat . . . . . . . . 207

Jaworowski, J. An Additive Basis for the Cohomotogy of Real Grassmannians . . . . . . . . . . . . . . . . . . .

231

Nguyen Huynh Phan On the Topology of the Space of Reachable Symmetric Linear Systems . . . . . . . . . . . 235

Schwaenzl, R., Vogt, R. Homotopy Ring Spaces and Their Matrix Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

Stomil~ska, J. Homotopy Colimits on EI-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vershinin, V.

273

V[JJ

On Bordism Rings with Principal Torsion Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295

T r a n s f o r m a t i o n Groups: Assadi, A. Localization and the Sullivan Fixed Point Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kim,S. S. Characteristic Numbers and Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Morimoto, M., Uno, K. Remarks on One Fixed Point As-actions on Homology Spheres . . . . . . . . . . . . . . . . . .

310 325 337

Other: Arlettaz, D. On the rood2 Cohomology of S L ( Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas, Ch. Characteristic Classes and 2-modular Representations for Some Sporadic Groups-H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weintraub, S. The Abelianization of the 0 Group in Low Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

382

List of talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current addresses of authors and participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

389 393

365

371

SOME

APPLICATIONS OF SHIFTED SUBGROUPS IN T R A N S F O R M A T I O N GROUPS by C. Allday and V. Puppe

If a torus G of rank r acts on a compact space X, and if all isotropy subgroups have rank at most s, then there is a subtorus K c G of rank r - s such that the action of K on X is almost-free. W h e n G is an elementary abelian p-group (i.e.,G ---(][/(p))r, where p is a prime number), then there is no immediate analogue of the very useful fa~t above, since a finitenumber of rOper subgroups can cover G. In order to overcome this difficulty,and others, shifted subgroups to be defined in detail in Section 2 below), which have been used in modular representation theory r some time (see,e.g.,[Benson, 1984]), have been introduced into the cohomological study of finite transformation groups. The use of shifted subgroups is quite natural; and indeed they seem to have appeared in transformation groups through the work of at least four differentauthors: A. A d e m introduced them explicitlyin his thesis ([Adem, 1986], and see [Adem, 1988]); they also appear explicitlyin the work of A. Assadi ([Assadi, 1988], [Assadi, 1989a], [Assadi, 1989b] and [Assadi]);and shifted subgroups of rank one appeared implicitly in our paper [Allday, Puppe, 1985]. In this paper we intend to give a survey of some of these applications of shifted subgroups. W e shall concentrate on the work of A. A d e m and ourselves and closely related results. Since it would require a substantial amount of background material, we have not included Assadi's work concerning applications of the theory of varietiesof G-modules in transformation groups: for this see Assadi's papers cited above. W e have included one of Adem's theorems (Theorem (4.14) below), the proof of which makes substantial use of varietiesof G-modules, but, for the same reas()n,we have not included Adem's proof. Otherwise, for the most part, we have included proofs, although we have referred some proofs, especiallythe proofs of some technical details,to our forthcoming book ([Allday, Puppe D. In the firsttwo sections we summarize some background material from algebra, including, in Section 2, the definition and some of the main properties of shifted subgroups. In the third section we give some of the basic topological notations and definitionswhich we shall use. W e have chosen to work with paracompact finitisticspaces. There is only a small amount of technical difficultyin extending the results given here from finite-dimensional G-CW-complexes to paracompact fiuitistic G-spaces; and yet many more applications are included amongst the latter,for example, continuous actions on topological manifolds. The last three sections give some of the applications of shifted subgroups. In Section 4 we treat equivariant Tate cohomology (as defined by R. Swan), in Section 5 we give an application in the manner of P. A. Smith's original method, and in Section 6 we give an application to equivariant cohomology (as defined by A. Borel). I.

k[G]-modules

Here we collect a few useful facts about k[G]-modules. Throughout this section G will be a finitegroup and k will be a field.

(1.1) Theorem. tl/ A k[G]-module is projectiveifandonlyifitisinjective. Any product of projective k[G]-modules is projective. Proof. (2) follows at once from (1). (1) follows from [Brown, 1982], Chap. VI, Corollaries (2.2) and (2.3). (1) also follows since k[G] is a symmetric algebra, and hence Frobenius: see, for example, [FuIler, 1989]. (1.2) Corollary. If M is a projective k[G]-module, then the dual module M* := HOmk[G](M , k[G]) is also projective. h

(1.3) D~finRion. We shall say that a k[G]-module M is Tate acyclic if H*(G; M) = 0. (This is a slight simplification of the notion of a cohomologically trivial module: see [Brown, 1982], Chap. VI, Sec. 8.) (1.4) Theorem. Suppose that G is a finite p-group, where p is any prime number, and that k is a field of characteristic p. Then the following conditions on a k[G]-module M are equivalent.

M is projective. M is Tate acyclic. (4) Hî (G; M) = 0 for at least one i E ~. Proof. This is contained explicitly in [Brown, 1982], Chap. VI, Theorem (8.5). (1.5) Corollary. Let G and k be as in Theorem (1.4). Then (1) any direct limit of free k[G]-modules is free; and (2) if k ' is an extension field of k, if M i s a k[G]-module, and if M ® k ' k free k ' [ G ] - m o d u l e , then M is a free kIG]--module.

isa

Proof. Since G has a complete resolution of finite type, P . , say, for any i E ~, and k[G]-module M, Homk[G](Pi, M ) ~ P . * =

®

M. Hence, if { M j l j E J } is a directed system of k[G]-modnles,

1 k[G]

^

h

then H*(G; llm Mj) ~ limj H*(G; Mj). ^

^

Similarly for (2), H*(G; M ® k ' ) =~H*(G; M) ® k ' . k k 2.

Shifted subgroups

In this section we recall the defnition of shifted subgroups and state some of their basic properties. Throughout this section G will be an elementary abelian p-group (also known as a p-torus), where p is a prime number; i.e. G -~ (~/(p))r for some r > 0: and k will be a field of characteristic p. The number r is called the rank of G, denoted rk-G. Suppose that G is generated by g l ' ""' gr" For 1 < i < r, let r i = 1 - gi E k[G]. Let vi = (1 - gi )p-1 for 1 _ n, a spectral sequence h

h

{~*~]. (4) E p ' q = H q ( G ; C p ) , H*G~V h

h

(3.3) Corollary. The map P, ~ P, induces a natural homomorphism 0": HG~'~* {c,*~j ~ HG~V* {~*~j. h •

And, if HJ(c *) = 0 for all j > n, then 0": H J ( c *) ~ H J ( C *) is an isomophism for all j > n. Proof. The existence of 0* is immediate. So suppose HJ(C *) = 0 for all j > n. Let C~ be the cochain complex with C ti = c i

for i < n , C nt = Zn, the cocycles of degree n, and C ti = 0 for

i > n. The inclusion C~ ~ C* is a weak equivalence, and hence the first spectral sequences show ^

h

that HG(Ct) * * ) and HG(Ct) * * ~HG(C * * ~HG(C * *). ^,

^

.

Now f ~ ( C ~ ) = ~J(C~) for j> n. Hence H~(C~)~ H~(C~) for j > n. In order to work with paxacompact finitistic spaces using Alexander-Spanier or (~ech cohomology we need to review some notation and terminology concerning coverings. (3.4) Definitions. Let X be a paracompact G-space, and let A c_ X be a closed invariant subspace. Let ~ be an open covering of X. (1) Let ~ ' A = { U ~ ~ g l U n A ¢ ¢ } . v (2) The (~ech nerve of ?g, denoted ?g, is the abstract simplicial complex with vertices the V

non-empty members of ~', and {U0 .... , Un} a simplexof ~, where U i e ~' for 0 < i < n, if V

Y

n~ Ui ~ ~b. The subcomplex ?gA is defined by saying that a simplex {U 0 ..... Un} of ?g is a iv simplex of ~A if f i n U i n A ¢ ¢ . i=0 (3) The Vietoris nerve of ~, denoted ~, is the abstract simplicial complex with vertices the points of X, and {x0,...,Xn} asimplexof ~ if {x0 .... ,Xn}CU for some U e ~ . The subcomplex ~ A is defined by saying that a simplex (x o .... , Xn} of ~ is a simplex of ~ A if {x 0 .... , Xn} c_ A. ¥

(4) ?/ is said to be finite--dimensional if ?g is a finite---dimensionai abstract simplicial v complex, in which case the geometric realization I ~'1 is a finite--dimensional CW--complex. (5) X is said to be finitistic if every open covering of X has a finite-dimensional refinement: i.e. if finite-dimensionai coverings are cofinai. 6) ~g is said to be an i nvariant covering of X if for any U E ?g and g E G, gU E ~. 7) ~g is said to be a Cech---G-covering of X if ~' is invariant and if gU fl U ~ ¢ implies g U = U for any U E ~ and g E G . In this case, for a n y U E ~ ' , let G U = { g E G I g U = U } . ((~ech--G---coverings are just called G--v_overings in [Bredon, 1972]). (8) ~ is said to be faithful if ~/ is a Cech---G-covering, and if, for any U e ?g, there is a x E X such that G U C G x. (3.5) I~,~ma. If X is a paracompact G-space, then locally finite faithful (~ech--G-coverings are cofinal. If X is also finitistic, then locally finite finite--dimensional faithful Cech---G---coverings are eofinal. Proof. Most of this is contained in [Bredon, 1972], Chap. III, Theorem 6.1. Since G is finite, X has a covering by open slices, which is a fQthful Cech-G-covering. And clearly any Cech-G--covering which refines a faithful Cech---G-covering is also faithful. (3.6) Definitions. Let X be a paracompact G--space, let A c_X be a closed invariant subspace, and let h be a k-module. (1) Let g*(X, A; A) = l~m C*(~, ~A; h), where ~' ranges over the faithful (~ech--G-coverings of X, and C*(~, ~A; A) is the ordered cochain complex of the pair (~, ~A)

with coefficients in A. Then U*(X, A; A) is the Alexander-Spanier cochaln complex of (X, A) with coefficients in A as defined, for example, in [Spanier, 1966], Chap. 6, sec. 4. Clearly U*(X, A; A) is a cochain complex of k[G]-modules. If G is an elementary abelian p--group, k is a field of characteristic p, and if P C_k[G] is a shifted subgroup, then U*(X, A; A) is also a cochain complex of k[P]-modules. h

h

(2) Let H~(X, A; A) = H~(~*(X, A; A)), and let H~(X, A; A) = H~(C*(X, A; A)). h

Define H~(X, A; A) and H~(X, A; A) similarly if G is an elementary abelian p-group, k is a field of characteristic p, and F c k[G] is a shifted subgroup. (3) If X is a G--CW--complex, and A is G---CW---subcomplex, or more generally, if (x, A) a relative G-CW-complex, then let W,(X, A; k), respectively W*(X, A; A), be the cellular chain complex of (X, A) with coefficients in k, respectively the cellular cochaln complex of (X, A) with coefficients in A. The following lemma is proven in detail in [Allday, Puppe]. (3.7) Lemm~. If X is a paracompact G-space and A C X is a closed invariant subspace, then A

h

V

V

H~(X, A; A) ~=l~/m H~({ ?Z'I, {~'A{; A) A,

,

V

V

l~zm tIG(W (I ~'1, I ~'AI; A)), h

and similarly with H~ instead of H a. If G is an elementary abelian p-group, k is a field of characteristic p, and if F c_k[G] is h

a shifted subgroup, then the corresponding results also hold for H~. and H~. Furthermore H~(X, A; A) ~ H*(XG, AG; A), the Alexander-Spanier cohomology of the pair (XG, AG) with coefficients in A, where XG, for example, is the Borel construction on X; i.e. X G = (EG x X)/G. h

(3.8) Remarks. (1) H~(X, A; A), respectively H~(X, A; A), is called the equivariant, respectively the equivariant Tate, cohomology of (X, A) with coefficients in A. A

(2) H~, like Ha, has natural long exact sequences for pairs, Mayer-Vietoris sequences, and tautness properties. For example, if A and B are closed invariant subspaces of X with X = A tJ B, then there is a long exact Mayer-Vietoris sequence ^. ^. ^. ^. ~j~ ... ~ Ha(X; A) --4 Ha(A; A) ¢ HJ(B; A) --~ H~(A n B; A) --~ ~J+I(x; A) --~ ... and A,

h

HG(A; A) ~ l i r a H~(V; A), where V ranges over the closed invariant neighborhoods of A. The V A

same holds for H~ and H~. (See [Allday, Puppe], § 4.6.) We finish this section by recalling W.-Y. Hsiang's definition of the p--rank of a space. (3.9) Definitions. Let G ~ (~/(p))r, and let • : G x X --~ X be an action of G on a space X. (1) The rank of • is rk¢ : = r-max{rk Gx{ x e X}. Thus if rk~ = p, then pP is the order of the smallest orbit, rk~ = r if and only if G is acting freely; and rk~ = 0 if and only if

X G ~ ¢. Thus rk(I) measures in a certain sense the extent to which the action is free. (2) The p-rank of X is rkp(X) : = sup rk(I), where ¢ ranges over all elementary abelian p---group actions on X. (3.10) Re.mark. We could also define the free p---rank of X to be frkp(X): = sup {r I (Z/(p))r can act freely on X}. There are well known examples where rkp(X) > frkp(X). For example, rk3(£P2 ) = 1 but frk3(£P2 ) = 0: see Examples (5.5)(2) below. 4.

Equivariant Tate cohomolog~v.

Equivariant Tate cohomology and finitistic spaces were introduced by Swan in [Swan, 1960]. We shall recall Swan's main theorem immediately following the next definition. (4.1) Definition. Let G be a compact Liegroup and let X be a paracompact G-space. Then the sing'ular set of X is defined to be X 1 : = {x e XIG x ¢ {1}}. X 1 is clearly invariant, and by the Slice Theorem it is closed. (4.2) Theorem. Let G be a finite group, k a commutative ring with identity, A a k-module, X a paracompact finitistic G---space and A C X is a closed invariant subspace. Then restriction induces an isomorphism h

h

HS(X , A; A) ~, Hf(X1, A1; A). For a proof see [Swan, 1960] or [Allday, euppe], §4.6. In applying Swan's Theorem the following easy lemma to be found in [Adem, 1988] and inspired by [Heller, 1959] is useful. (4.3) Lemmm Let G be a finite group and let k be a field of characteristic p where p divides [ G I. Let C* be a cochain complex of k[G]-modules such that C i = 0 for all i < 0 and HJ(C *) = 0 for all j > N, where N is some integer. Then, for any integer m, di "A'm+l. . . . "A'm+l/"*~ EN Am " mkn (~; H°(C*)) < u~mkn G (,~ j + dimkH -J(G; HJ(c*)). j=l (4.4) Corollary ([Heller, 1959], [Adem, 1988]). Let X be a paracompact finitistic space such that H*(X; U:p) ~ H*(saxsb; ]:p) as graded If:p-Vector spaces where a and b are integers such that 0 < a < b. Then frkp(X) < 2. (See Remark (3.10)) A

Proof. Suppose that G = (~/(p))3 is acting freely on X. By Swan's Theorem, Ha(X; Fp) = 0. Now Lemma (4.3) with m = a + b and C* = U*(X; ~:p) yields a contradiction. Now we would like to prove that, under the conditions of the Corollary (4.4), rkp(X) < 2. In dAdem, 1988], Adem did this by introducing shifted subgroups. Here then, following the next efinition, is a shifted version of Swan's Theorem. (4.5) Definition. Let G ~ (Z/(p)/r and let k be a field of characteristic p. Let X be a • paracompact G--space. Then, using the notation introduced immediately above Proposition (2.4), for any shifted subgroup F C_k[G], let X(r;k) = {x e X IV r n VGx ~ 0}. Note that X(F; k) is invariant, and, by the Slice Theorem, it is closed. Also X(G; k) = {x e X I VGx~t 0} = X 1.

4.6) Theorem. Let G =~(Z/(p)) r, let k be field of characteristic p, let X be a paracompact nitistic G-space, let A c_X be a closed invariant subspace, and let F c k[G] be a shifted subgroup. Then restriction induces an isomorphism h

^

H~(X, A; k) ~ H~(X(r; k), A(F; k); k). Proof. Thanks to the long exact sequences (see Remarks (3.8)(2)) it is enough to prove the result when A = ¢. Suppose the result has been proven in case X(F; k) = ¢. If X(F; k) # ¢, let W 1 be a closed invariant neighbourhood of X(F; k) and let W 2 be the complement of the interior of W 1. So W2(F; k) = ¢ and (W 1 13W2)(F; k) = ¢. By the Mayer-Victoris sequence, therefore, ^

^

H~(X; k) ~ H~(W1; k). The result now follows by the tautness property (Remarks (3.8(2))). h

So it remains to show that H~(X; k) = 0 if X(F; k) = ¢. Let ~ be a faithful v finite-dimensional (~ech---G-covering of X. For any y E [ ~g[, there is a x E X such that v

Gy C G x. (Since ~ is a (~ech---G---covering, the maximaJ isotropy groups of ] ?z'I occur at the vertices; and these are all contained in isotropy groups of X since ?Z is faithful.) Now, by y

¥

Proposition (2.4), each k[G/Gy], for y E I ~'1, is a free k[F]-module. Thus each Wi( I ~'1; k), •

V

and hence, by Theorem (1.1)(2), also each W1(I ?z'l; k), is a free k[F]-module. h

V

By the second spectral sequence (Proposition (3.2)(4)), tt~(W*(I ~'l; k)) =: 0. So ^

H~(X; k) = 0 by aemmas (3.5) and (3.7). (4.7) Corollary. Let G =~([/(p))r, let X be a paracompact finitistic G--space, and let A c X be a closed invariant subspace. Let p be the rank of the action on X - A : i.e. p = r-max {rk Gx[X E X - A}. Then there is a finite field k of characteristic p and a shifted subgroup F C_k[G] of rank p such that ^

H~(X, A; k) = 0. Proof. Let H1, ..., H n be the isotropy groups of G on X - A. By Corollary (2.5) and its proof, there is a field k, which is of finite degree over ~:p, and a shifted subgroup F c k[G] of rank p, such that V F 13VHi = 0 for 1 < i < n. So X(F; k) = A(F; k); and the result follows. Combining Corollary (4.7) with Lemma (4.3) as in the proof of Corollary (4.4) we get immediately the first part of the following corollary. The second part of the following requires a little more work with the first spectral sequence. (4.8) Corollary. Let X be a paracompact finistic G---space such that It*(X; ~:p), ~ H*(S a x sb; U-p) as graded fp-Vector spaces where a and b are integers such that 0 < a < b. Then (1) rkp(X) rk(G). (4) Let G ~ (~/(p))r be generated by gl' "'" gr' let k be a field of characteristic p, and let F C k[G] be a rank one shifted subgroup. Let r i = 1 - gi e m and let ~i be the image of r i in

11 V = m / m 2 for l < i < r . Suppose that V F is spanned by a17 l + . . . + a r 7 r, where a i E k for l _ < i < r . Now, if p is odd, H * ( G ; k ) ~ h ( S l , . . . , S r ) ® k [ t l , . . . , t r ] , where degs i = l and deg t i = 2; and H*(F; k) ~ A(s) ® k[t], where deg s = 1 and deg t = 2. i F : k[r] ~ k[G] induces a homomorphism i~ : H*(G; k) ~ H*(r; k). i~ is multiplicative; and if s 1 .... , Sr, tl, ..., t r correspond to r 1 .... , r r and s, t correspond to a l r 1 + ... a r t r, then i~ is given by i ~ ( s i ) = ais and i ~ ( t i ) = ~ t , for l _ < i < r . If p = 2, H*(G; k) ~ k[tl, ..., tr] and H*(r; k) = k[t], where deg t i = deg t = 1, for 1 _pP. Hence dim0:pWi(X; 0:p) is a multiple of pP; and so pPlx(X). (3) A special case of (2) appears in [Gottlieb, 1986]. Let G = (Z/(p)) r and let M be a smooth closed connected oriented manifold with G acting on M in a smooth orientation preserving way with rank p. In particular, by the results of [Illman, 1978], M is a finite G---CW--complex. Let x E Hn(M; ]Z), where n = dim M, be the orientation class. Then Gottlieb shows that pPx is the least positive integer multiple of x in ira[i* : H~(M; ][) --~ Hn(M; ][)], where i : M ~ M G is the inclusion of the fibre in the bundle M G --~ BG. It is easy to see that the Euler class e(M) is in im i*. Pence pP] ~(M). (To see that e(M) E ira i*, choose N )) n and let E~ be a N-connected compact free G-manifold. Let M~ = (E~× M)/G and B N = E~/G. Let PN: MN---~ BN be the bundle map. Let T(M~) be the tangent bundle of M~ and let ~ = ker(dPN) C T(M~). Then il~[(~) = T(M), where iN: M ~ MG N is the inclusion of the fibre. Hence ii~(e(~)) = e(M). Since N is very large, e(M) E im i*.) Proof of Theorem (5.1). Suppose G ~ (][/(p))m is acting on X with rank r. By Corollary (4.7) and its proof, there is a finite field k of characteristic p and a shifted subgroup F C k[G] such h

that rkr = r, HE(X; k) = 0, and (by Proposition (2.4)) k[G/G ] is a free k[r]-module for all X

x E X. Now let ~' be a (~ech---G---covering of X (see Definitions (3.4) and Lemma (3.5)). Then each ordered chain group Ci(~; k) is a free k[F]-module; and hence each ordered cochain group ci(~; k) is a free k[F]-module by Theorem (1.1)(2). Thus each Alexander-Spanier cochain group ui(x; k) I i m ci(-~; k) is a free k[F]-module by Corollary (1.5)(1). So C* : U*(X; k) is a cochain complex of free k[.F]-modules. Let n = m a x { j I HJ(X;Fp)¢0}. Define C~ by C ti = c i for i < n , ctn = Zn thecocycles of degree n, and c i = 0 for i > n. The inclusion C~ ~ C,* is clearly a weak equivalence. So, by t

14 A

A

A

* * 8 H~(C*) = H~(X; k) = 0. Now, by the the first spectral sequence (Proposition (3.2)(2)), HF(Ct)

^

second spectral sequence (Proposition (3.2)(4)), H*(F; C~) = 0. So C nt is a free k[F]-module by Theorem (I.4). Thus C~ is also a cochain complex of free k[F]-modules. Let (C~) F denote the cochain subcomplex of C~ consisting of all cochains fixed by F. We shall show that xH(C~) = pr xH((C~)F). The theorem will then follow since tt(C~) = H(C*) = H*(X; k) ~ H*(X: ~:p) ® k. 0= P Now we can use induction on rkF. To see this, suppose that F = F 1 × F2, and write k[F] =~k[X1, ..., Xr]/(xP , ..., X p) where 1 - X1, ..., 1 - X s generate F 1 (where s = rkF1) and 1 -

x s + l . . . . . 1 - x r g e n e r a t e r 2. T h e n k [ r l r l

consists of all multiples of X p-1 ... x P - I : i.e.

k[r] rl

is the free k[F2] = k[Xs+l, ..., Xr]/(xP+I , ..., xP)-module generated by X~---1 ... X p-1. Hence, by induction, we are reduced to the following lemma, which is standard in Smith theory (see, e.g., [Bredon, 1972], Chap. III). (5.3) Lemma. Let G = ~/(p), let k be afield of characteristic p, and let C* be a cochain

complex of free k[G]-modules such that C i - - 0 for i < 0, Ci = 0 for i > n, where n is some positive integer, and dimkH(C* ) < ®. Then dimkH((C*) G) < ~; and )/H(C*) = pxH((c*)G). Proof. Let g generate G, let r = 1 - g E k[G], and let v = ( 1 - g ) p - 1 . Then ( c * ) G = vC*. By the first spectral sequence H~(C*) has finite type, since dimkH(C* ) < ~. By the second spectral sequence HG(C ) ~ H ( ~ * ) , since C* is free over k[G]. By the second spectral sequence (for equivariant Tate cohomology), HG(C * *) = 0. So, by Corollary (3.3), H (C*) = 0 for all j > n. Thus dimkH(VC* ) < ~. Now for each i, 0 _H/

and

i: . 0 ( w H ) * - - .

~0(wH)* x .o(WOY~ ,

denote inclusions. Moreover, each obstruction

~(X

j: -0(wq)~

x y) , where

--.

(5,7)

-o(WH)* x

,~o(WOya

is not o f a product form, equals zero.

The same formula was also obtained (algebraically) by Lfick [34] t h m 14.19 and cor.14.48. This product formula was used by tom Dieck and Petrie in the study of homotopy representations of finite groups ( [14] § 8 ). Finally a few words about the restriction formula. Let H denote a closed subgroup of G . Recently S. Illman ([27] §8) has defined (geometrically) the restriction homomorphism

rest: WhG(X) -* Wh.(X). If f: X

--*

Y

is a G-homotopy equivalence then, by definition,

resHG(rG(f))

=

"rH(0 and we obtain

resHG(~G(X)) = ¢H(X) . An algebraic description of the restriction homomorphism is given in [26] §1 and in [34] def. 14.36, t h m 14.37 and prop. 14.40 and it allows to determine H-obstructions

wK(x)

in terms of G-

obstructions. Illman has also announced [27] that the restriction homomorphism satisfies the natural asaociativity property

rest" res. = res 4. Some geometric applications The product formula of the preceding section gives rise immediately to the following nice geometric result.

Proposition 4._! Let

G be a finite group and X a G-complex 6-domlnated by a finite one. Let V be any unitary

complex representation of the group G and s(V) its unit sphere. Then the product X x s(V) with the diagonal Gaction has the G-homotopy type o f a finite G-complex.

Since the assertion of the proposition 4.1 depends on the fact that

s(V) has well-behaved equivariant

Euler characteristic it is also true for other G-spaces with vanishing G-Euler characteristic (cf [33] cor. 6.4 ). In many situations when studying group actions one encounters the problem whether the G-space (or Gcomplex) obtained as a result of various constructions has the equivariant homotopy type of a finite G-complex. What is often guaranteed is a weaker assumption - equivariant finite domination - and we are resulting in a question when does the equivariant finiteness obstruction of a given G-complex vanish ? Such situations appear most frequently when one studies smooth, locally smooth or topological actions on manifolds. On the other hand there is an old problem posed by K. Borsuk in early fifties (and solved finally by J. West in 1977 [62]) which says whether a compact ANR-space has the homotopy type of a finite complex and it is natural to ask on the equivariant version of this question. It is well-known that any compact G-ANR is finitely G-dominated ( see [37] prop. 10.1 or [30] ). In particular, it applies to G-manifolds. If M is a compact manifold with smooth G-action then the triangulation results of S. Illman [22], [23] show that M carries the structure of a finite G-complex. If we assume that the action of the group G is locally smooth then the situation changes drastically. Namely for a finite group G an n-dimensional, locally smooth G-manifold M n has the G-homotopy type of an n-dimensional G-complex [32] but on the other hand examples given by Quinn [44], Dovermann and B.othenberg [18] §12 and Weinberger [60] p. 532 or [61] thm 16(a) show that the equivariant finiteness obstruction of M can be non-zero for certain M and a suitable finite

29 group G . Even more, Steinberger and West [49] §8 using their equivariant topological s-cobordism theorem have constructed compact, locally smooth G-manifolds realizing elements of a subgroup of the group I~06(X) for any compact G-ANR

X . It is worth to note here that even in the ease of a finite G-complex X not all

elements of the group I(0G(X) can be realized as the equivariant finiteness obstruction of compact G-manifolds [50] ( cf also §5 ). These examples show that equivariant version of the Borsuk conjecture fails in the ease, when G-manifold M (or G-ANR) has at least two orbit types. However~ if G acts freely on the manifold M then M/G is a compact topological manifold and in this case the equivariant finiteness obstruction of M vanishes. The following results yield an affir-mative answer to the equivariant Borsuk conjecture in some special cases.

Prooosition 4.._.22[3] Let G be a compact Lie group and X a compact G-ANR-space with one orbit type. Then X has the G-homotopy type of a finite G-complex.

Proposition 4.3 [31] Let -in denote a torus group and let X be a compact Tn-ANR. If for every isotropy subgroup H c Tn occurring on X the fixed point set X H is simply connected then

X

has the equlvarlant homotopy type o f

a finite Tn-complex.

One should point out that in the case of finite G there is a very close relation between the equivariant finiteness obstruction and G-surgery on equivariant complexes [42]. The idea that originates from Milnor and Swan was successfully developed by Oliver and Petrie [38], [43], [42] and shortly looks as follows. Let f: X -~ Y be a G-map of finite G-complexes. First one constructs an appropriate if-resolution fl: Xl "* Y extending f. The resolution fl has "correct" homology except one dimension Hn(M(fl) ) which is projective lG-module and defines an element (-1)n[ Hn(M(fz) ) ] of /~0(ZG) (so called Swan invariant). One can then attach infinitely many free G-n-celis to M(fl) to obtain a G-complex W which has the "right" homology and turns out to be finitely G-dominated. Its equivariant finiteness obstruction wG(W ) corresponds to the projective class group invariant (-1)n[ Hn(M(fl) ) ] above. Results similar to those of Oliver and Petrie was also obtained (in less generality) by Ku and Ku [29]. In particular they have proved the following generalization of Swan's theorem.

Theorem 4.4 ( [29] thm 1.3 ) Let G be a finite group of order q with periodic cohomology o f period n and let d = (q, ¢(q)) where ~b is the Euler C-function. Suppose F is a simply connected r-dimensional integral homology rsphere. Then there exists a finite G-complex X homotopy equivalent to Sr÷dn. Moreover G acts semifreely on X with X G = F.

The swan invariant detects the equivariant finiteness obstruction also in homology propagation of group actions (see e.g. [9], [59] ~2). It turns out that - at least under some restrictions - the equivariant finiteness obstruction is closely connected with the problem of existence of regular neighbourhoods. This observation was made by F. Quinn [44] prop. 2.1.2 and cot. 2.1.3 and K. H. Dovermann [16]. For instance, if a finite group G acts locally smoothly and semifrecly on a compact manifold neighbourhood in M

M

then the fixed point set

M G has an equivariant closed regular

iff all equivariant finiteness obstructions wH(M)'s

vanish.

In 1965 L. C. Siebenmann in his thesis [46] applied Wall finiteness obstruction to geometric question giving necessary and sufficient conditions for attaching the boundary to an open manifold. It is natural to ask whether the equivariant finiteness obstruction can be applied to produce an equivariant boundary in a smooth open G-manifold. The answer is yes although the author did not yet ver!fy all details. Up to the author's knowledge nobody has presented the equivariant version of Siebenmann's theorem in final written form and we only outline some constructions in the ease of a smooth G-manifold.

30 The existence of the G-collar neighbourhood of the boundary reduces the question to the following : when does a G-end g of a smooth open G-manifold M admit an equivariant collar neighbourhood ? Recall that a G-end $ is called (equivariantly) tame if (1) there exists a decreasing sequence of G-neighbourhoods N 1 ~ N 2 3 ... of $

with

NN i = 0 satisfying

certain ~rt-isomorphism conditions and (2) each N i is finitely G-dominated. The inclusions N i + 1 C N i induce isomorphisms of fundamental groups rl((NH_I)H~)

----- rl((Ni)Ha)

and the sequence of obstructions {w~(Ni)}i defines an element wHa(M,g) of the group

~0(z[.o(w~(g))*]) = i ~ ~0(Z[%(WH(N~))*]). 1

Both the group I~0(Z[%(Wtt($))*]) and the invariant w~(M,g) do not depend on the choices involved [3]. Moreover, if the G-end g has a G-collar then it is tame and wH(M,g) ----0 . Now one can formulate the following version of Siebenmann's boundary theorem for smooth G-manifolds. Theorem. 4.5 Let G be a finite group and M a smooth G-manlfold with one G-end ~. Suppose M satisfies a certain variant o f the "gap hypothesis" i.e. if ~ w~(M,~)

=

~

~

then dim ~

- dim M~

0 for each element [ M H ] of the set CI(M) . If dim MH~ >_ 6

>_ 3 . Let ~ be tame and

then there exists a G-collar around

the end ~.

The proof of this theorem goes by induction like in theorems 2.3 and 2.7. A weaker version of theorem 4.5 can be found in [3]. Another particular case of the equivariant boundary theorem for actions on Euclidean spaces was established by A. Assadi in [5] § 6. Proposition 5.1 in [38] and above theorem yield the following characterization. Provosition 4.6 Let M n be a smooth G-manlfold with compact non-empty boundary and one G-end ~ . Suppose M satisfies the above gap hypothesis and additionally 1) the inclusion

cgM c M

2) the natural map

is G-(n-?)-connected.

Ro(Z[.o(WH(g))*]) ~

I~o(Z[,o(WH(M))*~])

is an isomorphism.

3) dimMHa > 6. Then M is G-dlffeomorphic to OM x 0 and for any

nelgh-bourhood

V0 of the end ~ there isasequence

Vi of neighbourhoods of tbe end ~ such that V i c

Fr(Vi_l)

Vi - ~-1

FrVi

obstruction

and

is a G-ei-h-cobordlsm from

r(g(p)) E lira 1 WhG($(p) )

to

aVi4.1. Moreover, i f ¢r(g(p))

Vi_ l -

---- 0 there is an

which vanishes iff the end ~ admits an equivariant collar over B .

The next use of the controlled equivariant finiteness obstruction (in a relative form) is a controlled splitting obstruction for elements of WhG(X,p)e. It strictly follows Chapman's treatment in [10] §§ 9-11 and it is summarized in [49] p. 197. The controlled equivariant boundary theorem is closely related to:the equivariant product structure theorem. Namely, let a G-manifold M be provided with a locally smooth PL structure ~0 consider locally smooth PL structures ~

near aM . We

on M x R which agree with ~0 x R near aM x R (so-called

structures rel 8). We say that a structure 5] on M x R tel a admits a product structure if 5] is G-isotopic rel a to a structure of the form e x R , where e

is a locally smooth PL structure on M which agrees with

~'0 near a M . If now 5] is a locally smooth PL structure tel a on M x R one can consider the projection p: M x R ~

M

and define the positive end obstruction e(~+(p)) of 5] over M. Steinberger and West have observed that the product structure theorem of Kirby and Siebenmann [28], Essay III, fails equivariantly and then they have successfully identified the controlled equivariant end obstruction a(g(p)) to be the obstruction to equivariant product structure [51]. Using the equivariant end theorem 5.5 they showed in [51] the following. Theorem 5.6

5] admits a product structure re/ (3 iff the positive end obstruction of

~ ,

~(g+(p))

over M

vanishes.

Theorem 5.6 has very interesting consequences. We mention here three corollaries. Let representation of the group

G . Let

TopG(V), PLG(V )

homeomorphisms, the group of equivariant automorphisms of V , respectively.

and

OG(V )

V be a linear

be the group of equivariant self-

PL automorphisms and the group of equivariant linear

34

Corollary 5.7 Let V be a G-representation satisfying "gap hypothesis" and with dim VG >_ 5 . Then stab#]zatlon maps

~° \ Pt~(v) )

--* ~o \ Pt~(v • R) ]

and

{

"° \ oa(v) / are

(To dL R_)l

-~

~° \ oG(v • R) /

onto.

Corollarv 5.8

Let

V and W be as above. Then V ~ R is G-homeomorphlc

to

W ~ R

iff V is G-

homeomorphic to W .

Corollary 5.9 Let V be a G-representation with dim ~3 > 5 which satisfies a certain "vanishing condition" ( [4~, p. 70 ) in degrees >_ k , where 1 < k < dim VG . Then the space TOPG(V ~ R)/TopG(V )

is (dim V 3 - k)-

connected. Similar results were obtained independently by Madsen and Rothenberg [36] thm B and cor. C. More interestingly yet, Steinberger and West have made an analysis of Kirby-Siebenmann program [28], Essay III to prove the existence of the handlebody decomposition on a manifold and they remarked t h a t only the product structure theorem fails equivariantly.

It leads directly to the obstruction to the existence of an

equivariant handlebody decomposition. In order to exibit Steinberger-West results in this direction we need some definitions. Let H be a subgroup of G . A G-handle of type H is G XH(Dk x Dp) sentation disk of an orthogonal representation p : H

---* O(r) of H and

action. The index of the handle is k . Thus a 0-handle of type

tt

is just

decomposition of a G-manifold M is a sequence M 0 c M 1 ¢ ... c Mm = submanifolds with each H c G ; a handle

where Dp is the unit repreD k is a k-disk with trivial It-

G XHD p . An equivariant handle M of codimension zero equivariant

M i - Mi_ 1

a disjoint union of equivariant k-handles of type tt for some k < n and

G XH(D k x Dp)

is of course attached by an (equivariant) embedding of G XH(S k-1 x Dp)

into the previous stage. For smooth actions of compact Lie groups equivariant Morse theory provides equivariant handle decompositions ([57] thin 4.6 ). Locally linear PL actions of finite groups on manifolds m a y be given equivariant handle decompositions by triangulating and

taking barycenter stars with respect to the second derived ( equi-

variant ) subdivision as in the inequivariant case. If we assume t h a t the action of the group G is locally smooth then examples of compact manifolds not equivariantly homotopy equivalent to finite G complexes do not a d m i t equivariant handle decompositions. In other words , a compact locally smooth G-manifold have the equivariant finiteness obstruction

M a d m i t t i n g an equivariant handle decomposition must

aG(M) = 0 . By sufficiently

fine

subdivision of

a

handle

decomposition it certainly has the e-homotopy type of a finite G-complex for every ¢ > 0 and we conclude t h a t ~rc(M ) = 0 . The fundamental result of Steinberger and West asserts t h a t vanishing of a c ( M ) is sufficient for M to have an equivariant handle decomposition. In order to formulate it assume t h a t each component M H has dimension at least 6. Let boundary.

N c 0M

be a codimension zero G-submanifold with equivariantly bicollared

35 Theorem 5.10 [51], [52] §7. There is an equivariant handle decomposition of M on a closed collar of N iff the cur,trolled equivariant finiteness obstruction Crc(M,N) is zero in

KGo(M)c .

The idea of the proof is to adapt the Kirby-Siebenmann argument [28], Essay III to the equivariant, locally smooth context by means of the equivariant product structure theorem above. There is also an isovariant velsion without "gap hypothesis" [47] thm 8. We have already mentioned that (generally) the control-relaxation map

is not injective. This fact was proved by D. Webb [58] who has precisely calculated that for X = RP 2 with trivial action of the g r o u p 721 the control-relaxation homomorphism ¢

has non-trivial kernel. This calcu-

lation along with Steinberger-West realization result (theorem 5.4) gives the following corollary. Proposition 5.11 For G cyclic of order 21 there is a compact , locally smooth G-manlfold M ( with fundamental group 72 ) which has the G-homotopy type of a finite G-complex but admits no equivarlant handle structure.

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37 Steinberger, J.E. West: Equivariant controlled simple homotopy theory (in preparation) Steinberger, J.E. West: Controlled finiteness is the obstruction to equivarlant handle decomposition, (preprint). 52. M. Steinberger, J.E. West: Equlvariant handles in finite group actions, (preprint). 53. R.G. Swan: Periodic resolutions for finite groups, Ann. Math. 72 (1960), 267-291. 54. C.B. Thomas, C.T.C. Wall: The topological spherical space form problem-I, Compositio 50. M. 51. M.

Math. 23 (1971), 101-114. 55. C.T.C.WMh 56. C.T.C.Walh

Finiteness conditions for CW-complexes, Ann. Math. 81 (1965), 55-69. Finiteness conditions for CW-complexes, II, Proc. Royal Soc. London,

Set.

A,

295 (1966), 129-139. 57. A.G. Wasserman: Equivariant differentia/topology, Topology 8 (1969), 127-150. 58. D. Wehb: Equivariantly finite manifolds wltb no handle structure, (preprint). 59. S. Weinberger: Constructions of group actions: a survey of some recent developments, Group actions on manifolds, pp. 269-298~ Contemporary Math. 36 (1985). 60. S. Weinberger: an example in: Problems submitted to the AMS Summer Research Conference on Group Actions, Group actions on manifolds, ed. R.E. Schultz, pp. 513-568, Contemporary Math. 36 (1985). 61. S. Weinberger: Class numbers, the Novikov conjecture and transformation groups,

Topology 27 (1988), 353-365. 62. J.E. West: Mapping Hi/bert cube manifolds to ANRs, Ann. Math. 106 (1977), 1-18.

On Conic Spaces GIORA DULA*§** Abstract. In this paper, the notions of conic spaces, Thorn spaces, Hopf Invariants mid construction Xk are surveyed.

Introduction The main purpose of this survey paper is to advocate the usage of conic ,paces• Section 1. called Conic Spaces or CS for short, gives their definition (CS1), and points out that every CH; complex (CS2), and every simplieial complex (CS3) are particular cases of conic spaces• It follows by examples of conic (non C W and non simplicial) presentations of tlu'ee important spaces, and by some theorems concerning conic spaces. The second section (TS) takes on one of the theorems mentioned in CS, nmnely, the fact that Thorn spaces of bundles over conic bases are conic• Since this is a review paper, the section revmws some older results in the same direction, The third section (HI) reviews the subject of Hopf Invariants. Those Hopf invariants come into the description of the relative attaching maps in Thorn spaces so that some of the constructions in HI are mentioned in TS too, but HI discusses some older constructions. The fourth section Xk reviews the construction Xk. Some of this work have not been published. Xk could be viewed as Thorn spaces of bundles over the conic space Jk(Sn), and hence the forth seotion is a continuation of CS and of TS. •

t

C S 1 - C o n i c spaces A space B is a conic *pace if B is the limit li,__2~° B , , where B , is defined inductively as the mapping cone of .fn : A , _ 1 - - * B , - 1 , and Bu is a discrete set. B , is called the n th conic ,keleton (and is Called an n cone in [FHLT]. See also [May2]), and fn is called the n th attaching map. It is assmned throughout that each Ai has the homotopy type of countable C W complex and that B is the limit of B , as topological spaces. C A , - I is called the ,~th conic cell, Of particular importmlce is the ease when B0 is a singlton {*}. T h e n the first attaching map .fi i A0 ~ {*} is the trivial map and BI is suspension of A0. Such conic space B is called connected, and most considerations will involve connected conic spaces. T h e composition m a p A , - 1 - - ~ B , - I ~ B n - 1 / B n . - 2 of the n th attaching m a p and the quotien}, m a p p is called the n th relative attaching map. It measures the way the 11.t h conic cell attaches to the n. - 1't. CS2-CW complexes It is easy to observe that every C W complex is a particular case of conic space where A,,-1 is a disjoint union of spheres of dimension n - 1. In this case the relative attaching map gives the *Departtnent of Mathematics, Purdue University, West Lafayette, Indiana 47907. §This is a survey paper of material related to the author's Ph.D. thesis. The thesis was carried out. under supervision of Prof. M.td.Barratt. I wish to thank Prof. Barratt for his help during many ylmrs. I wish to thank Prof, Mahowald far nlany discussions concerning this study, and for presenting me with [CDGM}. I wish to thank Prof. B. Gray for presenting me with his unpublished work [Grl]'and discussing it with me, and Prof. H. Marcum for" disciJ~¢ing with me his work [Marl] and [MarS]. I wish to thank Bob Oliver for suggestions concerning the i~rese|it~tlori of the paper. I wish to thank Greg Henderson for proof reading the paper. I wish to thank Alex Nofech for mentioning [May2]. **current address: Department of Mathematics, Bar-Ilan University, 52900 Ranmt-Gan, Israel.

39 boundary homomorphism in the chain complex calculating the homology of B. In the connected cttse the disjoint union becomes a bouquet,

~SS-s|mplicial complexes i~ is easy to observe that every simplicial complex is a particular case of a C W complex in which the attaching map carries every sphere in the domain homeomorphically onto the boundary of an n simplex in B , - I . C S 4 - E x a m p l e s o f conic ( n o n C W ) spaces The fact that every C W and simplieial presentations are also cc~lic presentations, supplies ma1~y examples of conic presentations. The following are three examples of conic (non CI~") presentations. E x a m p l e J,,(EP) Given a space X, the James model for f/EX, denoted J(X), was first considered in [J1], (mid was denoted X¢¢ ). Filtration one is given on X, while the basepoint * of X is given filtration zero. There is an induced filtration on XX,, the n -th cartesiml power of X, and on lira X × ' , denoted X ~ , such that under the induced equivalence relation X ~ / ~ denoted J ( X ) becomes the free monoid on X, mad X × " / ~ denoted J , ( X ) gives a filtration on J ( X ) . In the particular case that X is a suspension EP, the points of exact filtration one in X form a cone CP. Then points of filtration k in J n ( X ) are of the form ( C P ) ×k, the k-~h cartesian power of CP, and using the homotopy equivalence of pairs ( C P × CQ, P * Q ) ~- ( C ( P * Q ) , P * Q ) (where P ~-Q denotes the join of the spaces P and Q), it turns out that J , ( E P ) has the following conic presentation: J n ( ~ P ) = (.-.((* U C P ) U C ( P * P ) U ... U C ( P * P * ... * P)), where the last join is of n copies of P. The attaching maps in tlds complex, wk : P * P * . . . * P ~

Jk-l(X),

were first studied by Porter in [P], and were called generalized Whitehead Product maps. The conic structure presented in this example specializes to a C W structure if P is a sphere, but o~herwise is a conic non C W structure of J ~ P , having less cells than may C W presentation of this space. Example B(G) Given a topological group G, whose underlying space is a countable C W complex, the following model for the classifying space of G, B( G ), was given in a paper of Milnor [Mi2]. E , ( G ) denotes the itet'a.ted join of G with itself, where G is taken n + l times. This is n G space, with diagonal G action on the G coordinates of the join, and orbit space Bn(G). In the case that the underlying space G has the homotopy type of a countable CW complex, the same is true for E , ( G ) and B,,(G). The projection map ha(G) : E , ( G ) ------*B , ( G ) is a fibration with fiber G. The pal"ticular cases h1(S ~) : S 2i+1 ~ S i+1, for i=0,1,3 and 7, are the classical Hopf fibrations. There is an indlusion map in(E) : E , ( G ) ~ E , + I ( G ) , obtained by using the base point of the last G in the join. It induces a map i , ( B ) : B,{G) ~ B , + I ( G ) so that the following diagraan commutes:

40 i.(E)

E.(G)

* En+I(G)

i.(B)

B.(G) The limit of the maps h. is a fibration

, B,,+,(G)

hop = h : E(G) --~ B(G) with fiber G, from E(G) which is a contractible free G space to B(G), the classifying space of G. The sequence Bo(G) C BI(G) C ... C B . ( G ) C ...B(G) is an ascending filtration of B(G), and it turns out that B . + I ( G ) is homotopy equivalent to the mapping cone of h . : En(G) ~ B.(G), B . ( G ) Uh. CE.(G). This gives B . ( G ) the conic presentation B . ( G ) = (,..(({*} U CG) Uh~ C(G* G) U.,. Uh._~ C ( G * G * . . . * G ) ) , where the last join has n G's mad n - 1 stars. The conic structure presented in this example specializes to a C W stntcture if G is a sphere, but otherwise is a conic non C W structure of B(G), having less cells than any C]~V presentation on this space. Example ~S a In [CDGM] a conic presentation is given to ~ S a. Actually, the filtration F n f ~ S 3 was given in [May1]. In [ C D G M , lemma 2.1], the authors show that F . is actually a conic space with the presentation F . = F . - 1 Or CM.,(O[ m ), where M,,,(X) is defined by: M.,(X) =

F ( R 2 , m ) ×~., X

F(RL,.) x~,. {*}'

where F( R 2, m) is the c[)|lection of m distinct points in the plane, and X needs to be a ~ . set. Both F ( R 'z, m) and 0I'" axe ~., sets, and thus M . d O l " ) is well defined. The composition map F.(I%'2Sa ) ~ ~2S~ ---. S 1 when localized at p, is denoted .T'. - - * S.1. A . denotes the fiber of 5v.p+l - - * S~, and the authors show in theorem 1.3, that as a corollary to the conic structure of lemma 2,1, the ilbration }-.p+l ---* S1 is a product ilbration. Tlfis gives properties of Brown-Gitler spectra.

CS5-A t h e o r e m about finite conic spaces in [FHLT] the n ~h conic skeleton is called an n cone. The following theorem is proved in §5. THEOREM D. Given a simply connected space X which is a finite conic .space. having a finite Postnikov tower of fi~fite type localized at some prime p, then it follows that all the homolog3" groups of A- ~'ith coett~cients hi Zp in positive dimensions vanish. In particular, if X is a p local C W complex, it is contractible. This theorem D is derived from theorem A. THEOREM A. A simply connected s.pace X with all the rood p homology groups finite di: menslonM has the property that the depth of the algebra H . ( ~ X , lp) is not bigger than the Lusternik-Schnilvhnaml category of X . Every finite conic space composed of n cones has Lusternik-Schlfirelmann category lesser than n~ and thus theorem A applies to finite simply connected conic spaces with finite dimensional ~tod p homology groups.

CS6-A t h e o r e m about conic spaces

41

THEOREM. A Thorn space of a bundle over a conic space is conic, In this last theorem the %undle' can be a fiber bundle, The Thorn space of the fiber bundle is defined [HS] as the ma,pping cone of the projection map, In the usual case of vector bundles, the associated sphere bundle has a Thorn space in the new defiifition Milch is the same as the classical Thorn complex of the original vector bundle. This theorem follows from the work of Held and Sjerve [HS], which followed that of Wall tWill(mentioned in TS6 below). In §3 they prove that given a cofiber sequence A ~ f X ~ C f, and a. fiber bundle ( over CI, the Thorn space over CI is a mapping cone T(X) Og C ( A * Y ) , where Y is ~he fiber of the bundle, , denotes the join, T(X) is the Thorn complex of the bundle over X, i*(5), induced fl'om ( by i, and ( A , Y ) ~ T ( X ), the attaching ma.p, is given a description in [HS, theorem 3.4]. The description is a composition of four maps, one of which uses a trivialization of the induced bundle (.f o i)*(~) over A. [HS, theorem 3.5], states that in the particular case that X is a point and C I is homotopic to ZA, (A * Y) g-~ ~I" is homotopic to the Hopf construction of the clutching nmp A x ]" ---* A after both maps are suspended once. The following obvious extension of [HS, 3.4] appears in [D I, §1.17]. B the base of ( has a conic presentation B0 C B1 C ... C B , C ...B, where B,, = (...((* U CA1) U C(A2)) tO ... U C(An-1)), then it follows that the Thorn space over B is conic with a conic presentation To C T1 C ... C T,, C .,.T, where T, = (...((T(,) U C(A1 A T(,))) U C(A2 A T(,))) U ... U C(A,,_I A T(,))), where T(*) is the Thorn space of the fibration induced from ( by the inclusion of the basepoint 1.} in B. C S T - W o r k of H . M a r c m n In [Marl] (see also [Mar2]), the main theorem 1.3 generalizes the theorem of [HS] about the s t r u c t u r e of Thorn spaces over conic spaces. Given a diagram:

*)

E1

~ E1

Eu

J" ~ E

v'l

~1

B

, M,

m which p is a fibration with fiber F, EB is the pull back of p and fl and given a homotopy pushout

6'

f

~ A

(**) B

, M,

42 then it follows that there exist a map W and a space E ( f *PA} such that the following diagram ts a homotopy pushout: W

E ( f *pA)

t A

~, l(pB,j)

"l .., l ( p , pa ),

where l(p, Pl ) is the double mapping cylinder of p and Pl, mad p is the obvious map between the double mapping cylinders. The theorem of Held and Sjerve is obtained as the particular case when both A and E1 equal a point. Then ** is a presentation of M as a nmpping cone M = B Ug CC, and * says that there exists a ill)ration p : E ~ B which restricts to a fibration PB : EB ----* B. Then l(p, pa) is the Thorn space over M, T(M), / ( P B , j ) is T(B) and E(.f "*:PA)specializes to be C * F. Thus the conclusion of theorem 1.3 presents T(kI) as a mapping cone T(B) Uw C(C * F). Iterated usage of ** gives the result stated in CS6. !i2. A t t a c h i n g n m p s in T h o r n spaces over Conic Spaces

TSl-Introductlon Finding the cells of the Thorn space over the conic space B is not enough information for building up the space. The attaching maps are needed too. [HS, theorem 3.4] gives a description of the attaching maps in the mapping cone, but this description is not homotopy invariant, as it rises a choice of trivialization of the bundle over the cone. This section reviews the literatm'e describing the homotopy type of the a.ttazhing maps. It turns out that the two key concepts at~peaxing are '3-homomorphism' and 'Hopf invariants'. T S 2 - T h e structure of T1-The J h m n o n m r p h i s m In the particular case that the base space B equals B1, which is by definition EA0, the Thom space of any bundle over B has a conic presentation as T(*) UT~/~) CAo A T(*), and is totally determined by the attaching map T(.fl ). As T(*) equals the suspension of the fiber, T(fl ) is a map A0 * F ----* EF. In the case that both A0 = S i and F = S j are spheres, and ~ is a spherical fibration with O(j ) action, classified by a homotopy class c~ : S i+1 ----* BO(j), T(fl) equals J(a), the image under the J homomorphism of a, discussed first in the paper [WG1] by G.W.Whitehead. In the case that ~ has a nlore general structure group, still T(fl) equals J(a), where tlfis J was defined in the paper [At] by Atiyah. Held and Sjerve in [HS, theorem 3.5] discuss this map for fiber bundles over suspensions without assunfing that the base or the fiber are spheres. They prove that the map T ( f l ) and the Hopf construct.ion of the clutching function are honmtopic, after both are suspended once (as mentioned in CS6), They do not call this map a 'J homomorphism'. The same construction appears in the thesis of C.H.Hanks [ H a l , Ha2], to which he gives the name J Homomorphism. As an application of his main theorem, Hanks deduces that this ,] Homomorphism is not a homomorplfism in general. In [ H a l , theorem 4.8] he cMeulates the obstruction for the J 'Homomorphism' from indeed being a homomorplfism. He gives an example of a case where the obstruction is non zero. This example is l~epeated in [D, I, §3.13b]. This obstruction vanishes in the case that the base of ~ is a double suspension. In all the classical cases, ~he base is a highly connected sphere and the obstruction vanishes: T S 3 - T h e structure of T2

43 Given a presentation B = B2 = ((* U C A o ) U CA1 ),

the following presentation is obtained: T = T2 = ((T(*) U C(Ao A T(*))) U C(A1 AT(*))), with attaching maps T ( f l ) mid T(f2). Thus, as T ( f l ) is known, only T(f~) is needed in order to give a full description of T2. The main theorem of [Hal], was shown in [D, I, §6], to describe the homotopy class of the relative attaching map Po o T(f2), (in the case that A1 is a suspension) which is an element in the set [A1 AT(*}, EA0 AT(*}]. Tlfis is a group as A1 is a suspension, and in that group the following statement can be derived from [Hall: oo

Po o T ( f 2 ) = ,EiJ,,(c~) o ~, o H I , ( . h ) , where H I , ( f . z ) : A1 A T(*) ~ ~Aon" A T(*) is the identity map on T(*) smashed with a higher Hopf Invariant of the map ,fl, s , is a. combinatorial type map permuting the order of the smash factors in ~ a^(") A T(*), and for a : EAI ----* B ( G ) classifying the fibration ~. "~'~0 J(c~) : A0 A T(*) ~ T(,) is the J map defined in [Hal] while J,(c~) : EAo ("~ AT(*) ~ T(*) is a nmp defined in [Hall by iterating J(o,). T S 4 - T h e m a p T(.fi) It is desirable to describe the attaching map T ( f i ) : Ai-1 A T(*) ---* Ti-1. There is the description of [HS, theorem 3.4]. However in general, this description is not a homotopy invariant, for example one of the composition factors uses a trivialization map for the bundle { restl~cted to the cone C A i _ I . The main theorem of [D],[D, I, §1.32], gives a primary homotopy invariant of T(.fi ), namely the relative attaching map Pi-2 o T ( f i ) : Ai-1 A T(*) ~ ~ ( A i - 2 A T(*)), in the case that A i - i is A suspension. This is aal element in the group [Ai-1 A T(*), EAI-2 A T(*)], and in it the following statement holds

Pi-'2 o T(f~) = ((eA) A T(*)) o J(eB) o (GHI(.fi) A T(*)) + (Pi-2 o A) A T(*), where the second summand (which is simpler than the first ) is the smash of the relative attaching map p~_~ o .fi : Ai-1 ----+ Bi-1 -----* B i - 1 / B i _ 2 = EA,_2, and the identity map of T ( * ) , T ( * ) . It is the same for all bundles ~ over B, that have the same T(*). In partictflar, it comes from the trivial bundle, while the first summand vanishes on the trivial bundle. The first summand is nmre complicated than the second and is a composition of three maps. The first and the last of those three, are the same for All bundles as they are smaslaes of a map fl'om the base with T(*). Only the middle factor does depend substantiaUy on (. Tile first factor is the smash of T(*) with a version of the Hopf Invariant of .fi, discovered first by T.Ganea in [Ga], which is called the Ganea Hopf Invariant of fi. Tile second map is essentially a J homomorphism as defined by Hanks [Hall, applied to the nlap o'oeB : EFtB, ----* B(G), which is the composition tff ~he evaluation map eB : E ~ B , ~ B,,, with the nm.p o' : Bn ~ B ( G ) classifying ~. The last fa.c~or is T(*) smashed with the evaluation map eA : EFtEA~ -----4EA,.

TS5-Exalnples The following are examples for the application of the theorem mentioned in TS4. Example J, (~P)

44 In the case that B = B , = J , ( E P ) , the n th stage in the James model for QE2P, as defined in CS4, it is shown in [D, II, §4] that there is an element ~1 in the image of J (as defined by Hanks) sudh that all pi-2 o T(wi), for 1 < i < n, can be presented as a sum of elements, each ~f :which is a composition A o #, where p is obtained by smashing ~l with tm identity map of ~he space P 4- P , . . . . p , and ), is a combinatorial type map permuting the smash factors. This construction generalizes some aspects of the Barratt-Malmwald construction X~. ([B4],[Mah], [Gr] and [BJM]). Example B,(G) tn the case that B = B , = B n ( G ) , the n -th stage in the Milnor model for B(G), as defined in CS4, it is shown in [D, II, §3] that that there exist three elements hi : G * G ---* IEG the attaching map of B2( G), po o h~ : G * G * G ~ E G * G the relative map of the attaching map h~ ~ G * G * G ---.* B2(G) in B s ( G ) , mad y which is a certain element in the image of J (as defined by Hanks) such that all Pi-2 o T(hi), for 1 < i < n, can be built from those three elements . The way of btfilding Pi-2 o T ( h i ) is slightly more complicated thin1 in the J , ( E P ) case. In some particular cases the general result simplifies. One such case is when the fibration ( is the 'line bundle' with projection map h , discussed in CS4. Then T ( h i ) is hi+l and the element ~l equals bl, Thus the result expresses pi-2 o hi+l in terms of hi mad P0 o h2. In the more special case that G is a sphere, there is an easy way to express P0 o h2 in terms of hi, and the classical result about relative attaching maps in projective spaces is obtained, expressing pi-2 o hi41 in terms of hi.

TS6.Work of C.T.C.Wall In his paper [Will, C.T.C. Wall discussed the structure of Poincar6 complexes. In chapter 3 he, discussed Thorn complexes of vector bundles over those spaces. He called a Thorn complex reducible, if the top cell homotopy splits. This is equivalent to the fact that the attaching map of the top cell is null homotopic. He proved that every Poincard complex M has a stable spherical bundle v over it with reducible Thorn space M v. v is unique, a~d M v is Spanier Whitehead dual to ~trivlM. In the next part of his work (which predated the work of Held and Sjerve mentioned in CS6 above), he discussed the structure of the Thom complex over a suspension E K (proposition 3.7). He gave the cell structure of (9.K) a for every spherical ill)ration c~ a~d showed that the attaching map is stably equal to the characteristic map (also called clutching map) K ------*G where G is the limit of G , , the monoid of self homotopy equivalences on S "-1.

§3~ Hopf Invariants HIl-Introduction It tut'ns out that relative attaching maps in the Thom spaces of bundles over conic spaces. are expressed in terms of Hopf Im~ariaa~ts. This section reviews some of the works concerning with Hopf Invariants. The very classical works of H.Hopf [Hol], [Ho2], Steenrod's definition of functional cup products IS], the solution of the Hopf invariant one problem l a d e ] and [Ada] and the works of G.Whitehead [WG2,3], are skipped. This survey is by no mean clainmd to be complete, Another source is the book by Baues [Bau]. This is the only book devoted entirely to material related to Hopf inv~riants. We can only describe it very briefly. The Hilton-Milnor process takes place in a free group, and in the collection process commutators ard produced. The book discusses commutator calculus in general. Then it derives results for distfibutivity laws and homotopy operations on spheres, including the higher Hopf invariaats. Then it uses the previous theory with different coefficient rings to deduce results about homotopy groups.

HI2-The work of I.James

45

In his work,[J1,2], !.M.James had a contribution to the subject of Hopf Invarimrts. In [J1] he established his model J X for flEX, discussed before in CS10. Using this model he defined in [J2] for each ~' the maps Ck : J X ----* J(X^k), where A ^k is the kD smash power of A, and called those 1naps combinatorial extensions. In particular C2 is a map J X ~ J ( X 2 ). Then the ~,h Hopf invariealt of a class f : E A - - ~ E X , H k ( f ) , is obtained by first taking the adjoint (ff .f, a(,f) : A ----* f t ~ X , composing it with the k th combinatorial extension nmp, giving the composition A ~ f~EX ~ flEX ^" and then adjointing back again giving H k ( f ) : E A E X ^~. Thus not only that James' H2 extends Whiteheads' definition (up to sign) of the operator H h)r any dimension q mxd fox' every space X (giving Whiteheads ~ case as a sphere), also the other H~, called nowadays the higher James Hapf Invarianta were introduced by Janms. Thus there are many Hopf Invmlants indexed by N, and using them it is clear that tim E H P sequence stops being exact exactly when H~ becomes non zero. Thus the exactness of the E H P sequence is related ~o the va~f.shing of the higher James Hopf Invariants. HI3-The work of J.P.Hilton The work of Whitehead was generalized in another form by J.P.Hilton [Hil]. He maintained the original line of Whitehead by starting with a map f : S q ~ S p, and applying the coproduct /~ : S p ----+ S I' V S p. Using an analysis of the space f l ( S i V SJ), he was able to show that there exists an infinite sum Ew~ o H~ which equals p o f, where r varies over a Hall basis of the free lie

Mgebra generated by two variables xl, x2, of dimensions i - 1 and j - 1 respectively, X~ is a space of the form Skt A S k2 A ..-/x. S/¢I where the generator r in the free lie algebra has coordinates {kl, k2 . . . . . kt}, the nmp Hr : S q -----*~Xr is a generalization of the Hopf Invariant, and the map w~ : ~ X r ~ S p V S t' is a. successive composition of whitehead product maps. H I 4 - T h e work o f J . M i l n o r The works of James and.Hilton were generalized by that of Milnor [Mil]. He was able to produce all the results obtained by Hilton (and mentioned in HI3) to the case when S q is replaced by a space E P and S p is replaced by a space EA. His result (theorem 4) is that p o .f is an infinite sum as before, indexed by r in the Hall basis of a free Lie algebra on two variables. In order to obtain his result he defined a model for the loop space over the suspension of a space, using free group functor 5t-, rather than the free monoid functor used by James. His theorenl 3, which is a lemma for theorem 4, specializes to give an alternative definition of James' invm-iants. Theorem 3 states that 5t'(B A ~'(A)) is homotopy equivalent to the space 5t'(i=lV,ooB A A î ). Applying the classifying space operator B on the last equality gives a homotopy equivalence between E(B A ~-(A)) and the space E( V B A A^~). The particular choice of B being S °, a i=l,oo

sphere of dimension zero, gives that EF(A) is homotopy equivalent to the space ~(

V Aî).

Thus given a class f : ~ P ~ EA, it has an adjoint a(.f) : P ~ f/EA, whose suspension is a. map E P ----- E(Y(A)), using the fact that F(A) is Milnor's model for flEA. Then using the above corollary of theorem 3 in [Mill, there is a composition map E P ~ ~(~-(A)) ~ EA ^~ whose adjoint is a map E P ----* ~A î, which equals James' i th invafiant up to a sign. The next two paragraphs discuss other relations between the different generalization of the Hopf invariants. H I 5 - T h e w o r k of B a r c u s a n d B a r r a t t In the paper [BB] Ba.reus and Barratt considered the following problem: Given a map u : K ----* .¥ and an inclusion K ~ L, they wanted to enumerate the homotopy classes of the extensions of u, u ~ : L - - * X. Given a class a in ~rq(X,*), they define a map a , : 7ri(~,*) ~ Irq+l(X,*) where ~" is the function space of basepoint preserving maps K ----4 X, In the case that L is the space _Tt't3c~cq+l , obtained as a mapping cone of any representative of a, then they prove (theorem

46

3.3) that the homotopy classes of extensions are in one to one correspondence with the cokernel of ~ t ,

Then they go on to present c~, explicitly, For the particular case that a representative of (~ has a presentation as a composition Sq ~ Sn ~ K, they present (corollary 4.8) ~u(~) as an ilffinlte sum of elements using o,, its Hopf invariants, the homotopy groups of X, and the operations of compositions suspensions and formations of Whitehead products. The higher Hopf invariants they use, are part of the Hilton invarimlts. They choose a subfamily of the Hall basis of the free Lie Mgebra, in which one of the variables has degree one. In [B2] it is claimed that this subfamily determines the James' invariants and all the Hilton invariants. The proof of the first assertion is sketched in [B3]. The completion of the proof is sketched in [BT], as well as the procedures for finding the explicit relations. It uses twisted Lie algebras ( [B6] ) and generalized signs. H I 6 - T h e w o r k of B o a r d m a n a n d S t e e r The fact that there are so many generalizations of the Hopf invaxiants, could mean that this notion is important, but is certainly discouraging because it seems to be that the different generalizations are not related ([BT] was not yet published). Maybe this is the reason that instigated Boardman and Steer to make the study appearing in ([BS]). They were ready to suspend the Hopf invariants certain number of times in order to obtain simplifications. Thus they define the ~,t, Hot)f invariant as a.n operator A, : [~A, ~B] ~ [ ~ " A , ( ~ B ) " ] . It should be noted that the .]ames" 7~th invariant is an operator [~A,~B] ~ [~2A,~B"], so that A , has the stone domain and range as that of the n - 1~t suspension of the James invariants. They defined a 'cup 1)roduct' o. A/~ of a class a in [~"A,B] mad of a class/3 in [~'nA, C] as the composition class ~ " + ' " A ~ ' ~ z x ~ " + " A A A - - ~ ~ ' A A ZmA - ~ B A C, where A is the reduced diagonal map, and _~ is a generalized sign homeonmrphism, induced by switching the order of some of the smash f~ctors. A ladder of Hopf invariants is a family of maps A, : [~A, ~B] -----*[~"A, (ZB)"] with the following properties: (i) A1 equals the identity operator. (ii) for every n, the composition operator An o E is the zero operator [A, B] ~

tEA, EB]

[~"A, (~B)"]. (iii) for every e and/~ in [~A, ~B], there is ~n equality of An(a +/3) mad of An(a) + A n - l ( a ) A

~(/3) + . . . + ~h(-) A ~x._~(/3) + A.(/3). They first proved that a ladder, if it exists, must be unique. Then they went on to show that the ~th Jmnes' invariants, when suspended n - 1 times, form a ladder. Then they showed that the Barcus Barra.tt [BB] family of the Hilton-Milnor invariants, suspended n - 1 times form a ladder too, establishing the fact that the two invariants are the same, when sufficiently suspended. Then they went on to prove that any other Hilton-Milnor invariant, when suspended, can be expressed in terms of the ladder and of generalized signs. Finally they related the ladders to constructiou in framed cobordism. H I 7 - T w o s t e p s in t h e M i l n o r process The Hilton-Milnor process, a s described in HI4 above, had actually two conceptual steps. (i) presenting the free group on a wedge as the product of the free group on one the wedge summands, times a fiber. (ii) presenting the above fiber as a free group on some wedge. The Hilton-Milnor process is obtained by applying iteratively the two steps above. Theorem 3 mentioned in HI4 is part of the second step, of presenting the fiber as a free group on some wedge, which turns out to be the longest part in the proof. Sometimes the fiber resulting from a projection of the first step, has a wedge summand in it before the second step is applied, and

47 ~h~.ts one eo~tld apply the first step twice, rind project into two wedge summands, before the need to apply ~he second step, This point of view leads to the next work.

H I 8 - T h e work of Berstein and Hilton In [BH], I.Berstein and P.Hilton defined the flat product as follows: Given a wedge of spaces X V 1-, there is the inclusion of the wedge into the product j : X V Y ~ X x Y. The homotopy fiber of j was denoted by Xb]" and called the flat product of X and 1". There is the map w : Xbl" ---, X v 1", and [BH] showed that the fibration f~(Xbl') ~ f~(X V l ' ) ~ f~(X × 1") homotopy splits and gave a pea-ticular choice of a splitting map r : f~(X V Y) ~ f/(XbY). Thus there is a homotopy presentation of f~(X V 1") as the product f/X x f/Y x f~(X~t'). The particular case of X being EA and < L+

L_

L0

Fig. 5.1 Then PL(v, z) satisfies the following formula

(5.1)

1 -PL+(v, 7.)

z)

-

vP _

z)

=

zP o(V, z).

93 If L is a triviM knot, then (5.2)

PL(v, z) = 1,

The integer polynomial uniquely defined by (5.1) and (5.2) will be called the skein polynomial of a link L. The skein polynomial is a generalization of the Jones polynomial. E x a m p l e 5.1 P ~ O(v, z) = ( - 1 _ v)z-1 P~

(v, z)

= v-~z -1 - vz + z -1

Since P L ( v , z) involves two variables v and z, we can define the v - s p a n PI.(v, z) and the z - s p a n P L ( v , z). As is suggested by Theorem 5.1 below, however, v - span PL(v, z) is more interesting and important. T h e o r e m 5.1 [Mo, FW] For any link L, (5.3)

v - s p a n PL(v, z) < 2(b(L) - 1).

Surprisingly, equality holds in (5.3) for many knots, although inequality is sharp for some knots up to 10 crossing knots. One of the earliest conjecture on b(L) can be found in [FW]. Conjecture [FW] I f L is the closure of a positive braid, then (5.4)

v - span PL(v, z) = 2(b(L) - 1),

where a positive braid is a braid in which every Artin's generator crl appears with a non-negative exponent.

Although this conjecture was recently disproved in [MS], equality (5.4) holds for many alternating links, including 2-bridge links, alternating fibred links and alternating pretzel links IMP]. Unfortunately, there is an alternating link for which the equality is false [MP]. The simplest example is the alternating link depicted in Fig. 5.2.

-%./ Fig. 5.2 We have seen before that a Seifert graph of a closed n-braid is the block sum of n - 1 multiple edge graphs and hence the natural diagram of a closed n-braid has exactly n Seifert circles. (See Fig. 4.3.) Therefore, any link has at least one diagram Do for which s(D0) = b(L), and hence, we have rain s ( D ) < b(n), where the minimum is taken over all diagrams of L. In 1987, Yamada proved the reverse inequality. In fact, he proved

94 T h e o r e m 5.2 [Y] For any diagram D of L, s(D) >_ b(L). This theorem suggests that for any link diagram D of L, the study of the ~urplu~ s(D) - b ( L ) would eventually lead to the determination of b(L). We may ask, for example, for what diagram D, is s(D) - b(L) equal to 0? If D is a reduced alternating diagram, s(D) - 1 equals the degree of the reduced Alexander polynomial of L and hence s(D) - b(L) is a link type invariant. Our study of the surplus s(D) - h ( L ) leads to a new invaxiant of ~ graph G, called the index of G, which is a topic of the next section. §6 I n d e x o f g r a p h s D e f i n i t i o n 6.1 An edge e of a graph G is called singular if e is not a loop and no other edges ( 5 e) have the same ends as e. Let 3r = { e l , . . . , en} be a set of singular edges of G. ~ is said to be independent if there exists an edge el in ~" and a vertex v that is one end of el, such that ~ - {ei} is independent in G/star v, where star v is the smallest subgraph of G containing all edges incident to v, and G/star v is the graph obtained from G by contracting star v to a point. We define the empty set to be independent. The index of G, ind G, is the maximal number of independent edges in G. E x a m p l e 6.2 For the graph G depicted in Fig. 6.1, ind G = 2.

Fig. 6.1 We note that ind G = 0 iff G has no singular edges. We can define a slightly different, but almost equivalent invariant of G, called the cycle index of G. D e f i n i t i o n 6.3 A set ~" = { e l , . . - , e,} of singular edges in G is called cyclically independent if no k edges (1 _< k ~ n) occur on a simple cycle of length at most 2k. The cycle index, a(G), of G is the maximal number of cyclically independent edges in G. Generally ind(G) < o~(G). Very recently, however, P. Traczyk proved the reverse inequality for bipartite graphs. T h e o r e m 6.4 [Tr] Let G be a bipartite graph. Then ind(G) >_ ~(G) and hence, ind G = Theorem 6.4 is false for a non-bipartite graph. Using Theorem 6.4, it is easy to show the following T h e o r e m 6.5 (1) For a bipartite graph, ind(G) is an invariant under 2-isomorphism. (2) If G1 and G2 are both bipartite, then ind( G1V G2 ) = ind G1 + ind G2 where Ga V G2 denotes the one point union of Ga and G2. For a link diagram D of a link L, the index of D, ind D, is defined as the index of the Seifert graph FD associated with D. Recall that F o is bipartite. Now, given a link diagram D of L, it is possible to find another diagram Do of L with s(Do) = s(D) - ind D, and therefore, Theorem 5.2 implies

95 T h e o r e m 6.6 [MP] For any link diagram D of a link L,

(6.1)

b(L) _< s(D) - ind D.

For many alternating links, the equality holds in (6.1). We conjecture that this is always the case. C o n j e c t u r e D. For an alternating link diagram D of an alternating link L, b(L) = s ( D ) - i n d

D.

This conjecture is true if ind D = 0 and we have T h e o r e m 6.7 [MP] Let D be an alternating link diagram of an alternating link L. If ind D = O, then b(L) = s( D ) and, furthermore, v - span PL(V, z) = 2(b(L) - 1).

Since an alternating fibred link L admits a reduced alternating diagram D with ind D = O, the braid index of L is completely determined by v - span PL(v, z). (cf [Mu

6].) §7 A m p h i c h e i r a l i t y A Seifert graph FD of an alternating diagram D of an alternating link L is the block sum of blocks F1,.-., Fk. Let Di be the alternating diagram of a (special) alternating link Li recovered from Fi. We write then L = L I * L 2 * . . . * L k and L is called a *-product (or Muraaugi sum or product) of Li. Many important numerical invariants of L are obtained from those of Li. For example, it is known that (1) the degree of the (reduced) Alexander polynomial AL(t) of L is the sum of those of Li [Mu 1] (2) The leading coefficient co(L) of &L(t) is the product of those of Li [Mu 1] (3) the span of the Jones polynomial VL(t) is the sum of those of Li [Mu 2]. These observations would suggest a certain kind of uniqueness in the decomposition of L into its *-product for an alternating link. For example, the mirror image L of L is the .-product of the mirror images Li of Li, i.e. L = L1 * . . . * Lk. Since each Li is special alternating, it is either a positive or a negative link. An (oriented) link is called positive (or negative) if all of its crossings are positive (or negative). (See §3). If L~ is a special alternating positive link, then Li is a special alternating negative link. Therefore the uniqueness of *-product decomposition would lead to the following C o n j e c t u r e E Let L = L1 * . . . * Lk be a *-product decomposition of an alternating link L. If L is amphicheiral, then (1) k is even, and (2) L i , . . . , L k is grouped into pairs { L I , L i } 1 < i ~ j < k, in such a way that Lj is ambient isotopic to the mirror image of Li. As a consequence of Conjecture E, we have C o n j e c t u r e F Suppose that L is an alternating link and is amphicheiral. Then co(L), the leading coe3~cient of the reduced Alexander polynomial AL(t) of L, is a square of some integer (up to sign). The conclusion of Conjecture F might hold for even non-alternating finks. As supporting evidence for Conjecture F, we can prove Theorem 7.1 below, which is a surprising application of the theorems related to the braid index.

96 T h e o r e m 7.1 [MP] Suppose that L is an alternating link. If co(L) is a prime, L is never amphicheiral. In contrast to the Alexander polynomial, the Jones polynomial is generally unsymmetric: VL(t) ~ Vz(t-1). Since VL(t) must be symmetric for an amphicheiral link, the Jones polynomial detects the amphicheirality for many (but not all) links. Similarly, Pt,(v, z) is partially symmetric for an amphicheiral link L: PL(v, z) = PL(v -1, --z). Although PL(V, z) cannot provide a complete solution to the amphicheirality problem, it can be used to prove the non-amphicheirality of many links. In fact, the proof of Theorem 7.1 depends on the detailed evaluation of max deg~ P(v, z) and rain deg~P(v, z).

References [A] J.W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci., U.S.A., 9 (1923) 93-95. [B] C. Bankwitz, Ilber die Torsionzahlen der alternierenden Knoten, Math. Ann. 103

(1930) 145-161. [FW] J. Frank-R.F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987) 97-108. [FYHLMO] P. Freyd, et al., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 103-111. [GLi] C. McA Gordon - R.A. Litherland, On the signature of a link, Invent. Math. 47 (1978) 53-69 [GLu] C. McA Gordon - J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 3 (1989), 371-415. [J] V. F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335-388. [K] L.H. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395407. [LM] W.B.lZ. Lickorish-K.C. Millett, A polynomial invariaat of oriented links, Topology 26 (1987) 107-141. [Mo] H.R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge. Phil. Soc. 99 (1986) 107-109 [MS] H.R. Morton-H.B. Short, The 2-variable polynomial of cable knots, Math. Proc. Cambridge Phil. Soc. 101 (1987), 267-278.

97 [Mu 1] K. Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387-422. [Mu2] - - , Jones polynomials and classical conjecture in knot theory, Topology 26 (1987) 187-194. [Mu3] - - , Jones polynomials and classical conjecture in knot theory (II), Math. Proc. Cambridge Phil. Soc. 102 (1987) 317-318. [Mu4] - - , On invariants of graphs with applications to knot theory, Trans. Amer. Math. Soc. 314 (1989) 1-49. [Mu5]

- -

On the signature of a graph. C.R. Math. Rep. Acad. Sci. Canada 10 (1988) 107-111. ,

[Mu6]

- -

,

On the braid index of alternating links, (to appear in Trans. Amer. Math.

Sot.). [MP] K. Murasugi-J.H.Przytycki, The index of a graph with applications to knot theory (preprint) [PT] J.H. Przytycki-P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987) 115-139. [R] D. Rolfsen, Knots and links, Publish or Perish Inc (1976). [Thl] M. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987) 287-309.

[Th2}

- - ,

Kauffma.u's polynomial and alternating links, Topology 27 (1988) 311-318.

[Th3]

- - ,

On flypes and alternating tangles, (preprint).

[Wr] P.

Traezyk, On the index of graphs: Index versus cycle index (preprint)

[W]

H. Whitney, 2-isomorphic graphs, Amer. J. Math. 55 (1933) 236-244.

[Y]

S. Yamada, The minimal number of Seifert circles equals the braid index of a link, Inv. math. 89 (1987) 347-356. University of Toronto Toronto, Canada MSS 1A1

MORSE THEORY OF CLOSED I-FORMS

A. V. Pazhitnov Institute of Chemical Physics, Kosygin str. Moscov, 117977, USSR

1. I n t r o d u c t i o n . Let M n be a smooth manifold and w be a real-valued 1-form on M. We say that w is Morse form if it is closed and if locally w = dh, where h is a Morse function. In this paper we consider the analogue of usual Morse theory for such 1-forms. The first step in this direction was undertaken by S. P. Novikov [1,2], motivated by his and I. Smeltzer's research on periodic orbit of some hazniltonian systems [3]. We present here the results of the theory as developed by S. P. Novikov, M. Sh. Farber and the author. Usual Morse theory corresponds to the case w = dr, f : M ~ R 1. Denote by rnp(f) the number of zeros of index p of the Morse function f. The classical Morse inequalities state that rnp(f) > bp(M) (where bp(M) stands for p-th Betti number of the manifold M). One also obtains the stronger inequalities which involve torsion in integral homology : rap(f) > bp(M) + qp(M) + qp-1 (M), where qp is the least possible number of generators in the group TorsHp(M, Z). These stronger inequalities are sharp in the following sense: if a manifold M " is 1-connected and n > 6 there exists a Morse function f on M for which the mentioned inequalities turn into equalities (Smale [4]). Now we try to find corresponding generalizations of the results for the case of closed 1-forms. First of all note that for any zero c of a Morse form w the notion of Morse index of c is well defined; the number of zeros of index p will be called the p-th Morse number of w and denoted by rn~(w). There arise three natural problems: 1) find the suitable analogues of Betti numbers, i.e. find the nontrivial numerical homotopy invariants of (M, w), providing lower bounds for rnp(w). 2) give (if possible) a method for calculating them in terms of usual homology of M. 3) find homotopy invariants, providing sharp lower bounds for rnp(w). Three subsequent sections of the present paper correspond more or s~less to these three problems, and contain partial solutions for them. We present formulations of theorems and the outlines of proofs sometimes omitting the technical details. The presentation of the material is not in historical succession so we make now some historical remarks. Three problems above were partly formulated and solved by Novikov in 1981-82 (cf. [1,2]). He was concerned mainly with the case of the forms w with rational cohomology class [w], and introduced for them the analogues of strong Morse inequalities (see §3). He conjectured that these inequalities are sharp for ~rlM n = Z, Typeset by AAdS-TE~

99 n >_ 6. This was settled by Farber [5] in 1985. The numerical estimates of Morse type for the case of arbitrary cohomology class [w] (see §1) were obtained simultaneously in 1985 by S. P. Novikov [6] and the author [7]. In the papers [6-8] we gave methods of calculation of corresponding numerical invariants (see §2). In 1987 J. C. Sikorav proved an algebraic lemma (§4, lcmma 4.3) which enables one to give the better estimates for rap(w), than discussed in §1. The same year the author [9,10] proved the sharpness of the inequalities for the case ~rlM" = Z "~, n > 6 (and some restrictions, for precise formulation see th. 4.2, 4.3 of §4). 2. Analogues of Betti numbers. In this section we give the simplest lower estimates for rap(w) which are analogues of Morse estimates mp(f) > bp(M) for Morse functions f. W e need some notation. From now on M n will denote connected compact smooth manifold without boundary. Denote by ~ the d e R h a m cohomology class of the Morse form w. For any t E C define the 1-dimensional representation Pt : ~rlM ~ GL(I, C) = C* as following: exp t

w

C*.

Clearly pt depends only on ( = [w]. Consider the corresponding local systems of coefficients on M and denote by H*(M, pt) the cohomologies with coefficients in it. The vector spaces H*(M, pt) are finite dimensional; let flp(~, t) = dim HP(M, pt). We need a simple lemma. The proof we postpone until §2.

LEMMA 2.1. The numbers flp(~,t) do not depend on t for Re(t) large enough. Thus we introduce the notation Bp(M, ~) = limae(t)-.oo/~p(~, t) and now we can state and prove our estimates. THEOREM 2.2. mp(w) >_Bp(M, [w]),

0 _~ p < n.

PROOF. First of all note that standard deRham type argument shows that cohomology with coefficients in the local system pt equals the cohomology of the deRham complex f/*M with perturbed differential dt where dtA = dA + t~ A A, (A E ft*M). The latter is equal to the kernel of corresponding Laplacian At = d~dt + d~d~ (for any given Riemannian metric). It is enough to prove that

rap(w) > flp(~, t) = dim ker(At : fP'(M) --* fl"(M)) for t real and large enough. We proceed further just as in [11], where the case w = df, f : M --* R 1 is treated. All the proofs work equally well in the case of general w. We reproduce the main lines here for the sake of completeness. Let A E f/P(M); the explicit computation shows that t,,;~ = AA + Itlzl~,lzA + tG(A), where A is a usual Laplaciart, A = d*d + dd*, G is a tensor field of type (p, p), not depending on t. Now fix any real A > O and let t --* oo. Then the eigen p-forms of At concentrate in the neighbourhood of the zeros of w.

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Next we turn to to local considerations. Let c be any zero of w of index p. We choose the local coordinates xi in the small neighbourhood Uc of the point c in such a way that the point c represent the origin of the coordinate system and in U we have _--

2

.....

+

zp+ 1 -{-..,

+

We can suppose that the Riemannian metric in coordinates xi is given by gij = 5ij. Then one can check that the p-form

•kc=

(fl e '-')- ~ d x l A . . .

Adxp

k----1

(defined only inside U) satisfies AtAc = 0. We can extend the form Ac to a global form ~c which vanishes outside Uc. Moreover we can do it in such a way that the quadratic form (At .," ) on the space generated by all the ~'c is (for sufficiently large t) less than A { . , . ). Thus we conclude (from minimax principle) that the vector space AP(M), generated by p-eigenforms of At, having eigenvalues less than A has the dimension at least rap(w) (t is sufficiently large). One can also prove that the dimension of AV(M) is exactly rap(w) (this is more difficult and I will not reproduce it here). The spaces Ap(M) form a finite-dimensional subcomplex A*(M) C (~*(M), dr), and its cohomology is equal to H*(~*(M),dt) = H*(M, pt). Now we have rap(w) = dim AP(M) >_dim HP(M, pt). , 3. C o h o m o l o g y w i t h local coefficients and M a s s e y p r o d u c t s . Now we have the analogue Bp(M, ~) of Betti numbers for ~ E H I ( M , R ) . The subject of this section is to compute them in terms of usual cohomology of M and Massey products of the form <x, ~ , . . . , ~). First we need some generalities about the (co)homology with local coefficients. Let k be a field. For a 1-dimensional representation p : 7riM ~ GL(1, k) (i.e. a homomorphism p : ~r~M ~ k*) wc denote by H . ( M , p) the homology with local coefficients in k, determined by p. Every 1-dimensional representation factors through some epimorphism E : 7riM --~ Z z. Fix now E and denote by R(E) the space of all representations, factoring through E; R(E) = (k*) l C k z. Consider the covering 2~ : .A'~---, M, corresponding to the subgroup kerE C ~'IM. The structure group of the covering is 7/. Denote by A the ring k[Z I] = k[t~ 1.... ,t~t]. The homology H.(M,p)is the homology of the complex C.(M, k) with differential dp, which is a regular function on R(E), dp E A. Suppose now that k is infinite. The standard argument shows that for all p ¢ V for some algebraic variety V C (k*) l we have dimk H.(M, p) = dim{n} H.(M, k).

(3.1)

For p ~ V we call H.(M, p) general position homology (g.p. homology) and denote it by Hg.'P'(M, p) and dimHp(M, p) - - by ~'P'(M, E). The same result holds for cohomology: for some algebraic variety V t C (k*) t we have dimk H*(M, p) = dim{A} H*(M, k). We denote general position numbers by bs.p.(U , E). Consider now some examples and specifications.

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1) The local system p is given by a 1-form w (as in section 2). It factors through E : t r i M --+ Z t where I is maximal number of Q-linearly independent periods of w. This number is the rank of im(~ : H I ( M , Z ) ~ R) where ~ = IT], and we will call it the degree of irrationality of ~ (or w). Denote these periods by a l , . . . , at. Then the local system Pt is given by a curve in the space (C*) t with the coordinates p(t) = ( e x p ( a i t ) , . . . ,exp(att)). One checks immediately (using ai • R) that for Re(t) sufficiently large this curve does not meet the algebraic manifold V' and so for Re(t) ~ c~ we have flp(~, t) = ~.p.(M, E). (In particular, we have proved Lemma 2.1). 2) Suppose that l = 1. Then the algebraic variety V consists of a finite number of points in complex plane and they can be explicitly computed in terms._.of homology of corresponding cyclic cover M. Namely, consider the module Hp(M, k) over the principal ideal domain A = kit, t-l], and decompose it as a sum

(A)',,

a/alpha) i=1

where ai are the polynomials in k[t]. To compute the cohomology HP(M, p) which is the same as the homology of the complex Horn ( C , ( M ) , k) (k[t] is a module over A induced by p) we apply the universal coefficient theorem: 0

, (Extl(Hp_l(~r), k)

, HP(M,p)

, Hom(Hp(~r), k) -----+ 0.

Now one easily sees that for general p the left term is zero, the right one has dimension and bp. The dimension of left term jumps up when p(t) is the root of some u_(p-l) i the dimension of right term - - when p(t) is the root of some al p) In particular we see that the jumps of the b.({,t), where ~ = [w] has the degree of irrationality 1, take place in these points t for which e t is a root of some polynomial Pi in the decomposition N

= (Ca)• (ie= 1 3) Using the generic cohomology one can get the estimation of §2 type. Namely suppose that the cohomology class { = IT] of the Morse form ca is the linear combination Aiei of the integral classes ei e H 1(M, Z) = H o m (rrl(M), g). Choose the epimorphism E : rq (M) + Z such that all ei are Z-linear combinations of the coordinate functions of the projections. Then rap(w) >_ ~'P'(M, Z). Now we turn to computation of cohomology with g.p. local coefficients in deRham case, following [6]. Consider as in section 2 the differential in l-/*(M) of the form dt = d + t{ A . We are interested in general t, so we treat it as a parameter and consider the space of fl*(M)[[t]] consisting of the power series, converging somewhere near t = 0, and differential dl on it. Suppose w(t) = wo + w i t +w2t 2 + ... i~ dr-cycle. After simple calculation we get dw0 = 0, { A w0 = - & o l , . . . it means that all Massey products of the type ({, {,...,w0} vanish in ordinary cohomology. Factoring dt-cocycles by d~-coboundaries correspond to factoring by images of Massey products.

102

We can generalize the procedure as follows. Consider the case of m-dimensional representations of 7rx(M) (generally nonabehan) and let dt be a / - p a r a m e t e r analytic deformation of d of the kind dt = d + O(t)A, O(t) E f~l(M) ~) E n d ( C ' ) , ~ = 0. (The corresponding deformation of trivial representation I is given by P,(7) = exp(f~ O(t)). Write O(t) = 0 i t + 02t 2 + . - . , then integrability condition 0 = dO(t) + O(t) A O(t) takes form dO,, + ~,__-x Oi ^ O , - i = 0. The condition for w(t) = Wo + wit + w2t 2 + . . . to n be a dt-cocycle is dw0 = 0,... , ( ~ i = 1 0 i w , , - a ) + dw,, = 0. One sees at once that these conditions mean exactly that all Massey products @ 0 , 0 1 , . . . , 0 0 exist and vanish (precise statements of this kind we give in next section). Now we intend to generalize this procedure to the jnfinlte field k of any characteristic. We can not use the deRham framework any more and we must work directly with chains and cochains. The main difflculty here is to find the suitable formalism, which certainly must be the spectral sequence, converging to H.s'P'(M, k), with differentials expressible in terms of Massey products. The existence of such spectral sequence was conjectured by Novikov (private communication). Actually he conjectured more general thing, involving multidimensional representations: CONJECTURE 3.1. (S. P. Novikov). Let k be any algebraically closed field and suppose we have a representation p: ~rlM --* GL(m, k), which is close enough to trivial one and is in general position. Then(co)horaology of M with local coefficients, generated by p can be computed ezvlicitly in terms of ordinary homology and for rn = 1 it can be done by means of Massev produc~. In [8] we proved this conjecture for m = 1 and now we will formulate the result and give main lines of proof. It appears to be more convenient to work with homology. We will assume for simplicity that the fundamental group is equal to Z" although this restriction can be easily eliminated. Suppose that X is a CW-complex ~rlX = Z", k-infinite field. Consider the space R of representations R = (k*)" C k" and any algebraic curve 7(t) in it such that 7(0) = I = ( 1 , . . . , 1) with polynomial coordinate functions Pi(t) = l+a~it+...+aNitlv; 1 < i < n~ aii E k. First of all we construct a spectral sequence (E.~, 0r), begining from H.(X, k) and converging to H(a)(X, k), where (c0 is a generic point of 7. Namely, consider the ring W = S-lk[t], where S = {P1,..., P,}, and the short exact sequence 0 , W t , W e,k ~ O. Now, tensoring C.(.~,k) with it over A = Z [ t ~ l , . . . , t $ a] and passing to homology, we obtain an exact couple H . ( C . ( X , k) ®h W ) - - ! + H . ( C . ( X , k) ®^ W )

re(x, k) and one easily shows, that the term Eoo of the corresponding spectral sequence equals to/-/,(a)(X, k). For n = 1 and "r(t) = 1 + t this spectral sequence is exactly the same as considered by Milnor in [12].

103

Now we proceed as to compute differentials in this spectral sequence. Denote ~ E

H I ( X , k ) the cohomology class, given by a tangent vector {dPi/dt} to a curve 7 at t = 0. We will prove that dr(x) = (x, ~ , . . . , ~), where indeterminacy of Massey product is reduced suitably with respect to 7- Now pass to the precise formulation. We will need a very special kind of Massey products and I reproduce here from [13] only the definitions for this case. Let R be any commutative ring, ~a, ~2,.. •, ~rEC*(X, R). Denote (-1)l=lx by 5. We say, that ~1, ~2,..., ~r form an incomplete symmetric Massey triangle (i.s.t) if d~l = 0, d ~ = Y]d=a k - 1 -~k-j~j, k > 1 (ab means here a U b); if so, then # = )"]~=a-~r-i+l~i is a cocycle, called symmetric Massey product (~t) r, and if p = d~r+l, we say, than ~1,~2,... ,~r+X form a complete symmetric Massey triangle (c.s.t). If r = co we call this an infinite complete symmetric triangle (~.s.t). If Z l , . . . , z r e C , ( X , R) and (~1,..., ~,) form an (c.s.t) we say, that (zi, ~.i) form an incomk plete quasisymmetric Massey triangle (i.q.t) if dz~ = O, dzk = ~']~i=~zi f3 ~k-i(k > 1) (then p = ~[=1 xi gl ~r--i+l is a cocycle , denote by (x1,~1) r etc.). We arrange this usually as follows ~1, .....

, ~1

~2,...,~2 • . . . . . .

Xl,

~1, .....

, ~1

=2,~2,...,~2 ,•,..

Now I introduce the spectral sequence (Er, dr). LEMMA 3.1. For every infinite symmetric triangle A = (~1,~2,.;.) there exist the spectral sequence (Er, dr), satisfying i) and ii) i) z~ = H . ( X , R ) , d~= = = n 6 ii) if x is a cycle and there exist xl = z, z 2 , . . . , zr, forming together with ~J an (i.q.t) and y = dz + ~ ' x i n ~r-i then x survives up to Er and drx = y. It appears that the above spectral sequence (E~, Or) coincides with (Er,-dr)(A), if we choose A suitably, according to the curve 7. I shall now explain how to choose A. Fix a map f : X ~ T n inducing iso in ra and Ha. Consider some (~.s.t.) ( ~ , ~2,...) on T n and lift it to R". Since R n is contractible, every 1-cocycle is canonically cohomologous to zero, and we have an (~.q.t): hi = 1 , 6 , 6 , . . . h2, ~2,...

where hi = 1 E C°(R"), hi E C°(R"). Denote by ei orthonormal basis in R", by picorresponding Massey eocyele, and by Aij(&) the value (Pi, ei). THEOREM 3.2. For arty polynomial curve 7(t) in a space kn given by 7i(t) = Pi(t) = 1 + axit + . " + agi tN and for any symmetric Massey triangle A = (~1, ~2,...), satisfying ~a = ~t [t=o and Aij( A ) = aij, the spectral sequences ( Er, Or)( Converging to homology with coe~cients in gener/c point of 7) and (Er,-dr)( "Massey spectral sequence') coincide. For the proof see [8].

104

Now we must be sure that infinite symmetric triangles (~1,...) on T n with prescribed values of Aii exist. This is the contents of the following lemma, which should have been known long since, but I couldn't find it in literature. LEMMA 3.3. Let Y be a space and suppose ¢p : R ..--+K is a monomorphism, where K is l etd of cha,'acte,'istic a, d H'(Y,R) H*(Y,K) i, mo,,o. The,, e,,e incomplete symmetric Massey ~riangle A = (~a,... ,~r), consi~in9 of odd-dimensional cochains, can be completed. I'd like to emphasize that the proof relies heavily on the vanishing of torsion in integral homology of Tn = B(Z n) = Bah (X). The idea of the proof is as follows. It suffices to complete over K. We pass to the detLham algebra A*(Y, K), which is strictly commutative, so for odd-dimensional cochains z we have (z) r = 0. Alas, the morphism Pa : A*(Y,K) ---, C*(Y,K) is not multiplicative, but there exist corresponding higher homotopies (cf. [14]) pi, and, finally, the completing vertex is written down explicitly using p~-~(~i), pi (for detailed proof see

[sD 4. T h e s h a r p inequalities. In this section we discuss the problem 3) of the Introduction. The general remark to this section is that all the numerical invariants providing lower bounds for rnp(w), which are known up-to-date come from homology of free abelian covers and the sharpness theorems concern correspondingly the manifolds with free abelian fundamental groups. We begin our exposition of sharpness results with the case of Morse functions, i.e. the forms of irrationality degree zero. We cited already the classical Morse inequalities

my(f) >_by(M) + qv(M) + qv-l(M)

(4.1)

where bp(M) is the rank of the group Hp(M), and qp(M) is the minimal number of generators of the subgroup Tors Hp(M). These inequalities are sharp in the following sense: For any simply-connected manifold M", one can find a Morse function f on M, for which rap(f) equals the righthand side of (4.1). The proof of these inequaities goes as follows. For a given Morse function f one constructs a free chain comlex C.(f) over Z for which the number of generators p(Cp(f)) is equal to rnp(f), and the homology Hp(C.(f)) is isomorphic to Hp(M, Z). When C.(f) is constructed the proof of (4.1) becomes an easy algebraic excrcise. To construct the complex C.(f) we define Cp(f) to be the free abelian group generated by critical points of f of index p, and for such point x define dx -= ~, A(x, y)y, where the sum is taken over all y of index (p - 1) and the incidence coefficient A(x, y) is defined as follows. We choose the Riemannian metric on M and consider all the paths of steepest descent (with respect to f ) going from x to y. To each of them one assigns the number (+1) or (-1) according to some rule which I'll notdiscuss here (see [15]) and then A(x, y) is by definition the sum of these numbers. One checks that (Cv(f) , d) is the chaincomplex and that Hp(C.(.f)) "~ Hp(M, Z). The inequalities (4.1) are not sharp for the non-simply-connected manifolds. The reason is that for any regular cover M -----+ M with the structure group G the Morse function f determines the complex C.(f) of free Z[G]-modules such that pz[a] (Cp(F)) = rnp(f) and the complex C.(f) is simply homotopy equivalent to the chain complex of M, given by some triangulation of M. So the Morse number rnp(f) must be not less then

105

minimal possible number/~(Cp), where C. runs through the free based chain complexes over Z[G] simply homotopy equivalent to C.(M). This number mine. ~(Cp) is generally greater then righthand side of (4.1). The best possible bounds got in this way come from the universal cover M ~ M. Sometimes the inequalities obtained this way are sharp. This holds e.g. for the case of Morse functions f on cobordisms (W"; V0, V1),OW = V0 [J V1, f is constant on both components of boundary, n >_ 6, lr1110 , 7rl W , ~r1111 are isomorphisms, WhOr 1W) = 0 [16]. Now we pass to the forms w of degree of irrationality 1. To obtain the inequalities of type (4.1) Novikov ([1, 2]) constructs the analogue of the Morse complex. Namely, the form becomes exact on some infinite cyclic cover p : M -* M, p*w = df. We have an analog of Morse complex of the function ] with an essential difference: for a given critical point of ] of index p there can exist an infinite number of critical point y of index (p - 1) connected with x by the paths of steepest descent. To overcome the difficulty Novikov considers the completion ,~ = (Z[[t]])[t -~] of the group ring A = Z[Z] = g[t+l], and constructs the complex C,(M,w) with the following properties: 1) C . ( M , ~ ) is a free chain complex over A, UX(C,(M,~)) = m,(~), 2) Hp(C,(M,w)) ~ Hp(M) ®h ~,. Since the ring ~, is the principal ideal domain we obtain

rnp(w) >_bp(M, [w]) + qp(M, [w]) + qp-l(M, [w]),

(4.2)

where bp(i, [w]) denotes the rank of the module Hp(M) ®^ A over .~ and qp(M, [w]) - the minimal number of generators of the module Tors~,(Hp(M) ®A A). If no ambiguity is possible we denote these numbers just bp((), qp(~), where ( = [w]. Note that the cyclic covering M ~ M is uniquely determined by the cohomology class [w] E H ~(M, R) (recall that IT] is a multiple of the integer class) so these numbers bp, qp really depend only on [w]. These inequalities are exact [5] in the following sense. THEOREM 4.1. [5]. For any manifold M ~, n >_ 6,~rlM" = Z, there exists a Morse form w, representing a generator in H~(M,Z), ~uch that rnp(w) i~ equal to rigMhand ~ide of (4.2). Now I will say few words about the proof. One particular case of the problem was considered in 60's already by Browder-Levine [17].They ask when the manifold M n, 1riM n = Z can be smoothly fibered over the circle. (Note that the forms of degree of irrationality 1 are up to the constant just the Morse maps into the circle, so in our language the problem is following: when does there exist a smooth Morse form w, IT] ¢ 0, without critical points, i.e.m.(w) = 0?) The answer is that for n _> 6 the necessary and sufficient condition for existence of the smooth fibration is that the fiber of the map M r* * S 1, inducing iso in zrl, is homotopy equivalent to finite CW-complex. This is easily checked to be equivalent to the condition bp(M, [w]) = qp(M, [w]) = 0 for allp. The Browder-Levine's argument is as follows. Consider the arbitrary smooth map f : M --* S 1, representing the generator ~ E Hi(M, Z) and denote by V a regular inverse image of c e S 1. The manifold W = M \ T u b ( Y ) , where Tub(V) stands for a small open tubular neighbourhood of V in M, has the boundary OW consisting of two components:

106

O W = OoWUO1W, OiW ~-. V. The infinite cyclic covering M is obtained as follows: take an infinite number of copies of W (denoted by Wn, n E Z), and glue them together, identifying the components of the boundary: 8o(W,) m V ~ O(W,-I). Note that Z acts freely on M and if we denote the generator of Z by t then tWn-1 = W,, t(OoWn-1) = 00Wn. We can suppose that ~(t) = - 1 . Now att~hing handles to V inside W we can modify V in such a way that 7rlV = 0 = ~rlW, H . ( W , V ) = 0. (We need a finite number of handles since M has the type of finite CW-complex.) Afterwards the application of Smale's theorem finishes the proof. Farber's proof of his sharpness theorem goes the same line, but is more technically complicated. He shows that after attaching to V the finite number of handles inside W we construct a manifold V, for which 7rlV = 0 = 7rlW and Betti numbers by(W, V) and torsion numbers qp(W, V) are equal to corresponding Novikov numbers bv(W, [w]), qp(W, [w]). He finishes applying Smale's theorem to the cobordism (W; 00W, 01W). We will not reproduce here the details of the proof of theorem 4.1, since many of them appear again in the proof of the theorem 4.2 below. Suppose now that rk H1M is greater than 1. There is no reason to expect (4.2) to be sharp for all classes ~ E H 1(M, R) of irrationality degree 1 (see the above discussion of Morse functions). Still, this holds for the classes of general position. That is the subject of the following theorem. THEOREM 4.2. [9,10]. Suppose 7rlMn = Z m, n > 6, universal covering M n is ~connected. Then there exist a finite number of integer hyperplanes Fi C H i ( M , Z) such that any nonzero integer cohomology class ~ G H I ( M , Z ) \ [.JiPi is represented by a Morse form w with mv(w ) equal to the righthand side of (4.e). Although the formulation says nothing about the forms of degree of irrationality > 1, the proof requires them urgently, and, in turn, gives an sharpness result for them. So I pass now to these forms. First of all, we need the analog of the ring ~, above. Denote A the ring Z[Z k] = Consider the abelian group Z[[t~x,...,t~l]] of all formal power series ~n t~ 1. For each element ,k = ~ ,~It z of this group we define supp,~ C Z k to be a set of indices I with ,~I # 0. Now let ~ be a homomorphism Z k ~ R. Novikov ring A~" consists of all power series ,k, for which the set supp,~ N(~ < c) is finite for any c G R. Return now back to the forms. Let w be a Morse form on a manifold M, ~ = [w]. Suppose p : M ~ M is a free abelian covering with the structure group Z k, such that p*w is exact: p*w = df. In this case ~ determines a homomorphism Z k ~ R which we'll denote by the same letter ~. The consideration, similar to above leads us to a definition of the Novikov complex C.(M, w) which has the following properties: 1) It is a free chain complex over A~- and p(Cp('M,w)) = mv(w ). 2) Its homology is isomorphic to H , ( C , ( M ) ®A A~-). We'll be particulary interested in the case when degree of irrationality of ~ = [w] is equal to rkHI(M,Z), which we denote by m. These cohomology classes (and corresponding forms) will be called totally irrational. The corresponding homomorphism : Z m ~ R is then a monomorphism. The following lemma enables us to define in this case the corresponding numerical invariants - Betti and torsion numbers. It is due to

107

J. C. Sikorav (autumn 1987, private communication; the full proof can be found in [9,

I0]). LEMbiA 4.3. If ~ : g t - , R is a monomorphism, the ring A~ is a principal ideal

domain. Using this lemma one defines for a totally irrational class ~ E H a(M, R) the numbers bp(~), qp(~) as follows. Take a maximal free abelian cover ~ n , M. Homology H , ( M ~) is a A = z[zmJ-module; each module Hp(M n) ®^ A~" is the finitely generated over the principal ideal domain, and we define b,(~) and q,(~) as its rank and torsion number. Considering the Novikov complex we get again the inequalities (4.2). Note that the full proof of (4.2) using the Novikov complex presents technical difficulties. See [10] for the complete proof appealing only to Morse theory for compact manifolds with boundary. Now we need a little more algebra. Let C, be any free finitely generated complex over A = g[gk]. Then for totally irrational ~ : gk _., R the ranks bp(C,, ~) and the torsion numbers qp(C,, ~) of the homology are defined. One can show that Hv(C, ®A A~-) = Hp(C,) ®A A~'. The next lemma was also communicated to the author in autumn 1987, for the complete proof see [9, 10].

LEMMA 4.4.0. C. Sikorav). The number bp(C., does not depend on There exists a finite set of integer hyperplanes Fi C Hom(Zk, R ) = R t, such that numbers qp(C,,~) are constant in each connected component of the complement Hom(gk, R ) \ UiI'i. Using this lemma we can correctly define the numbers b,(C.,¢), qp(C., ~) for any E Hom(gk, R ) \ Ui Pi, setting b.(C., ~) = b.(C., ~'), where ~' is totally irrational and sufficiently close to ~ (the same definition for q.). For a manifold M in consideration we will denote these numbers Bp(~), Qp(~). For our further purposes we replace the ring A~- by a suitable localization S~-IA. Namely, let se = {1 + XlX e A, suppA C (¢ < 0)}. We need to know that S~'IA is also a principal ideal domain for ~ totally irrational (see [9]). For the case m = 1 the corresponding localization was introduced and used by Farber [5]. The algebraic part of the Theorem 4.2 is given by THEOREM A. For any manifold M, IrlM = Z m there exists a finite set of integer hyperplanes Hi C Hl(M,R)(including Fi from lemma ~.~ and maybe something el~e), such that I) for any nonzero ~ E H1(M,R) \ [-JiHi, the module S-'HF(M,Z) is isomorphic to

Qp(~) (S~-IA)Bp(~) @( ~

S~-1 A/aj(p)S~-1 A),

0_ 6, 7rz(M) = w3(M) = rr4(M) = 0, and for the class ~ E H I ( M , Z ) the eondition(~.S ) i~ fulfilled. Then there ezi~ts a splitting mainfold V such that V is regular and

Hp(V-)(~K~(K)4C" @(q~'A"Ia~P'K) P

(4.5)

j=l

Now we'll show how to deduce Theorem 4.2 from Theorem B. Suppose that (4.5) holds. Then the homology of the pair ( V - , t V - ) is easily computed using the exact sequence of the pair and the fact that for any P-module N we have N / t N ~ S - 1 N / t S - 1 N . We have (by excision) qp(e)

(4.6) .i=1

Consider now the cell complex of the pair (W, tV). It is a finitely generated free R-complex. A purely algebraic argument using the representation (4.6) shows that it is homotopy to a complex C,, having in each dimension p exactly bv(~) -t- qp(~) -t- qp-x(() generators. Since bi(~) -- qi(~) = 0 for i -- O, 1, n, n - 1 we can apply results of [16] and realize the complex C. as a Morse complex of some Morse function f on the cobordism

109

(W, tV), constant on the upper and lower boundaries. Glueing together V and t V we get the map M ~ S 1. The proof of Theorem B goes by induction, with the help of procedure similar to that of [5]. Instead of Smale's theory of minimal Morse function on a simply-connected manifolds we use the Sharko's theory [16]. There arises the obstruction in dimension n - 3. The situation here is similar to considered by Farrell [18]. At present we can cope with it only when the situation is essentially the same as in [18]. Namely we need mp(~) = mp+l(~) = mp+2(~) = 0 for some p with 2 < p < n - 4 (where m.(~) stands for b.(~) + q.(~) + q.-l(~)). (It means that we don't expect any critical points of these indices.) This holds for example if p = 2 and the condition of theorem B is fulfilled. Then the FarreU's arguments work and (recall that the corresponding obstruction sits in C ( Z [ Z " - I ] ) = 0) we are done. Now I'll formulate the exactness result for the cohomology classes ~ of any degree of irrationality. We say that the set U C R N is conical if z E U =~ tx E U for t > 0. THEOREM 4.5. Let r i M " = Z m, n > 6, 7r2(M) = r s ( M ) = ~r4(M) = 0. Then there exists a open conical subset U C H a ( M , R ) = R m, ~uch that any nozero element ~ E U is represented by a Morse form w, having the minimal number of zeros of all indices, namely, rap(w) = Bp(~) + Qp(~) + C2~-,(~). This theorem follows form theorem 4.2 immediately. Indeed, take any rational ~ [.-]i Hi and a Morse form w : [w] = ~. Then one easily shows that in sufficiently small neighbourhood of ~ in H 1(M, R) any class ~ is represented by a small perturbation w I of w, so that w ~ is Morse form and mv(w' ) = mv(w ). Passing if necessary to a smaller neighbourhood we have Bp(~') = Bv(~) , Qv(~') = Qv(~), but recall ~ ~ [-Ji Hi, hence Bp(~) = bp((), Qp(~) = qp(~) and since rap(() is given by the righthand side of (2.2), we arrive at the conclusion. REFERENCES [1] Novikov S. P. Multivalued functions and functionals analogue of Morse theory. Dokl. AN SSSR, 1981, v.270, N 1, p. 31-35 (in Russ.). [2] Novikov S. P. Hamiltonian formalism and multivalued analogue of Morse theory. Russ. Math. Surveys, 1982, v.37, N 5, p. 3-49 (in Russ.). [3] Novikov S. P., Smeltzer I. Periodic solutions of the Kirchhof type equations for the free motion of the solid body in the liquid and the eztended Lusternik-Schnirelman-Mor~e theory (L-Sch-M) I. Funkz. anal. i pril. 1981, v.15, N 3, p .54-66 (in Russ.). [4] Smale S. On the structure of mainfolds. Amer. J. Math. 1962, v.84, p. 387-399. [5] Farber M. Sh. The ezactne~s of Novikov inequalities. FuseE. anal. i pril. 1985, v.19, N 1, p. 49-59 (in Russ.). [6] Novikov S. P. Bloch homology. Gritical points of functions and closed 1-forms. Dokl. AN SSSR, 1987, v.287, N 6, p. 1321-1324. [7] Pazhitnov A. V. An analytic proof of the real part of Novikov's inequalities. Dokl. AN SSSR 1987, v.293, N 6, p. 1305-1307. [8] Pazhitnov A. V. Proof of Novikov's coniecture on homology with local coefficients over a field of finite characteristic. Dokl. AN SSSR. 1988, v.300, N 6, p. 1316-1320.

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[9] Pazhitnov A. V. On the ezactness of Novikov type inequalities for 7raM = Z m and Morse forms within the generic cohomology classes. Dokl. AN SSSR. 1989, v.306, N 4, p. 544-548. [10] Pazhitnov A. V. On the ezactness of Novikov inequalities for the manifolds with free abelian fundamental group. Mat. Sbornik 1989, v.180, N 11, p. 1486-1523. [11] Witten E. Supersymmetr~ and Morse theory. Journal of differential geometry, 1982, v.17, p. 661-692. [12] Milnor J. W. Infinite cyclic coverings. In: Coafer~ce on the topology of Manifolds (edited by J.G. Hocking) Prindle Weber & Sehrnldt 1968, p. 115-133. [13] Kralnes D. Higher order Massey products. Trmasactions of American mathematical society 1966, v.124, N 5, p. 431--439. [14] Bousfield A. K., Gugenheim V. K. A. M. On PL deRham theory and rational homotopy type. Memoirs of the American Mathematical Society, 1976, v.8, number 179. [15] Milnor J. W. Lectures on the h-cobordism theorem. Princeton 1965. [16] Sharko V. V. K-Theory and Morse ~heo~ i. Preprint Kiev Inst. of Math. AN SSSR 1986, N 86.39. [17] Browder W., Levine J. Fibering manifolds over a circle. Comment. Math. Helv. v.40, 1966, p. 153-160. [18] Farrell F. T. The obstruction to fibering a manifold over a circle. Indiana Univ. Math. 3ourn. 1971, v.21, N 4, p. 315-346.

Morava K-Theories: A survey Urs Wiirgler Mathematisches Institut der Universitgt Bern CH 3012 Bern For any prime p, the Morava K-theories K(n)*(-) , n a positive integer, form a family of 2(p ~ - 1)-periodic cohomology theories with coefficient objects

If(n)* = 7r_.(K(n)) = Fp[v,~,v~-l], where [vn I = - 2 ( P ~ - 1).They were invented in the early seventies by J. Morava in an attempt to get a better understanding of complex cobordism theory. Morava's work used rather complicated tools from algebraic geometry and, unfortunately, it seems that no published version of it exists. So topologists interested in this subject were very pleased to see the paper [18] of Johnson and Wilson where a construction of these theories together with many of their basic properties were carried out in more conventional terms. In the period after the appearance of [18] the importance of the Morava K-theories for algebraic topology and homotopy theory became more and more obvious. First, in the work of Miller, Ravenel and Wilson (see [33]) it was shown that making use of a theorem of Morava, the cohomology of the automorphism groups of these K-theories is strongly related -via the chromatic spectral sequence- to the stable homotopy groups of the sphere. Then, in their paper [50], lZavenel and Wilson demonstrated the computability of the K ( n ) ' s by calculating K(n)*(-) for Eilenberg-MacLane spaces. From this paper it also became clear that the K(n) constitute a useful tool for the problem of describing the structure of BP.(X), an idea, which has found further applications in the papers of Wilson and Johnson-Wilson [60],[19]. More recently, from the work of Devinatz, Hopkins and Smith (see [11], [15]) it becomes appearent that the Morava K-theories also play a very important r61e in stable homotopy theory. The purpose of this paper is to give a brief survey of some of the basic properties of the K ( n ) ' s with the aim to help a non-specialist to get quickly informed about some important aspects of this topic. Clearly, the choice of the material we are presenting here is mostly dictated by personal taste and we pretend by no means to be complete. In the first section we indicate where the Morava K-theories come from and sketch a method how they can be constructed. Section 2 contains a description of the stable operations in K(n)*(-) and in the third section we study some connections with other B P - r e l a t e d cohomology theories. In 4. some K(n)-computations are reviewed. Section 5 contains some properties of the connected cover k(n) of K(n) and in 6. we treat uniqueness questions. Finally, in section 7 we make some comments concerning the significance of M o r a v a / f - t h e o r i e s for certain topics of stable homotopy theory.

112

1

T h e origins of M o r a v a K - t h e o r i e s

One of the key motivations which led J. Morava to the construction of his K-theories was certainly a remarkable theorem of Quillen [43] relating the theory of formal groups with complex cobordism theory. Let MU* ( - ) denote complex cobordism theory. Then

MU*

TM

Z[xl,x2, ...], xi E MU -2i

and M U * ( - ) is a complex-oriented theory, i.e. there is an element y E M U 2 ( C p ~ ) such that MU*(CPoo) "~ MU*[[y]], MU*(CPoo × CPoo) ~ MU*[[y ® 1, 1 ® y]]. The classifying map m : C P ~ x C P ~ --, CPoo induces a power series

FMU(Yl, Y2) = rn*(y) = ~

ai,j yl®yJ2

i,j

with the three properties

FMU(X,y) FMu(FMu(x,y),z) FMU(X,O)

= = :

FMU(Y, X) FMU(Z, FMU(Y,Z) x

commutativity associativity identity

We define a formal group law G over a commutative ring A to be a formal power series G(x, y) E A[[x, y]] having these three properties of FMU. Quillen's observation was

T h e o r e m 1.1 The formal group law FMu over MU* is universal in the sense thai for any formal group law G over any commutative ring A, there is a unique ring homomorphism 0 : MU* --* A such that G(x,y) = ~ O(ai,i)ziy j = O.FMtr. The universal group law FMrj may be described rather explicitly: For any formal group r over a torsion free ring A define its logarithm logF(x) E A ® Q[[x]] by

logF(x) -- fo

OFdt (t 0)" ayk

'

Then logF(F(z, y)) = logF(x)+logF(y), i.e. logF is an isomorphism over A ® Q between F and the additive formal group law and F is determined by its logarithm. A theorem of Mischenko [40] asserts that xn-[- 1

IOgMU(X)

n~LLCr'nJn+l' X"~r n_>O

where [CP,~] denotes the element of MU* determined by the complex manifold CPn • A formal group law over a torsion free ring is called p-typical with respect to the prime p, if its logarithm is of the form logF(X) = ~'~i>0 lixP" This definition may be extended to rings with torsion, see e.g. [14]. A theorem of Cartier [10] asserts that every formal group law F over a torsion free Z(p)-algebra is canonically isomorphic

113

to a p-typical formal group law F t~p in the sense that if logF(x) -~ ~i>_o aiz', then pi logf,u,(x) = ~'~i>o % , x . Applying this result to FMU over MU* ® Z@), Quillen was able to construct a multiplicative and idempotent natural transformation ep: MUZ(*p)(-) -~ MUZ(*p)(-) whose image is represented by a ring spectrum BP, which is called the Brown-Peterson spectrum (see [9] for the original approach). On homotopy, Cp is determined by [CP,] 0

ep([CPn]) = This implies that the logarithm of FBp

ifn=pi-1 otherwise

= "P'YP MU =

= E

[CPP'-I] i>>_o

(ep),FMu

is given by

li=p' 6 BP* ® Q[[x]].

i>_o

Moreover, FBp is universal for p-typical formal group laws over Z(p)-algebras. The BP-spectrum bears as much of informations as MUZ@), and, because homotopy theory is essentially a local subject, homotopy theorists concern themselves mostly with the smaller spectrum BP. If G is a formal group law over A and if f, g 6 A[[z]] are power series without constant term, we define f + c g = G ( f ( x ) , g(x)) and for any positive integer n we set = =

+o =. T1

The following theorem of Araki [2] is very useful and shows that it is possible to find generators of BP* which behave well with respect to the formal group law FBp. Another (and equally useful) set of generators was earlier found by Hazewinkel, see [14]. T h e o r e m 1.2 Let p be any prime. There is an isomorphism of Z(p)-algebras

BP,

TM

Z(p)[vl, v2, ...]

where the generators vi 6 BPu@~-I) may be chosen to be the coefficients of x W in the series [plF.p(x)

=

v,=,'.

i>0 Now the construction which leads to the formal group law FMU applies to every complex-oriented cohomology theory: For example, the formal group law associated to H * ( - ; R ) is Ga(x,y) = z + y, the additive formal group law, and the group law associated to complex K-theory K * ( - ) is the multiplicative formal group law Gm(x,y) = x + y + txy where t 6 K* ~ Z[t,t-1]. In general, one may ask if given a (graded) commutative ring A and a formal group law G defined over A there exists a complex-oriented cohomology theory which realises (A, G) in the sense indicated above. In this generality, an answer to this question is not known today. However, one may try to realise special types of formal groups.

114

A formal group law F over a commutative Fp-algebra A is of height n (n > 0) if the series [PIE(x) has leading term ax p" with a ¢ 0. If [p]F(x) = O, F is of height co. Consider the ring homomorphism 0n : BP* --~ A defined by 0,~(v,) = 1 and On(vi) = 0 i f i ¢ n, and put Fn(x, y) = (O),FBF. From theorem 1.2. we see that -fin is of height n. Now a theorem of Lazard [30] (see also [13],[14]) asserts that over a separably closed field K of characteristic p > 0 any formal group law G of height n is isomorphic to F,~. In view of this theorem it is certainly interesting to try to realise the formal groups Fn resp. the graded versions of them. T h e o r e m 1.3 Let p be any prime. For all integers n >__ 1 there is a multiplicative, 2(p n - 1)-periodic and complex-oriented cohomology theory K ( n ) * ( - ) with coefficient ring = 1] where v,~ is of degree [v,,I = -2(p'* - I) and whose associated formal group law Fr,(x, y) satisfies the relation [v]F°(x) =

If p is odd, the product on K ( n ) * ( - ) is commutative, for p = 2 it is non-commutative. The theories K ( n ) * ( - ) of this theorem are named after Jack Morava who proved a version of 1.3. (he did not know the K(n)'s to be multiplicative) in the early seventies in a paper which never appeared in print. The first published reference concerning the K ( n ) ' s is the paper [18] of Johnson and Wilson. It may be interesting to notice that K(1) has a rather familiar interpretation: Let K * ( - ) denote complex K-theory. As Adams showed (see, e.g. [2]), K* (-)(v) decomposes into a direct sum of copies of a cohomology theory G * ( - ) which is periodic with period 2 ( p - 1). Then there is an isomorphism K* ( - ) ~ G* ( - ; F p ) . To construct the K(n)'s one uses (co)bordism theories of stably almost-complex manifolds with singularities, see [3]. Very briefly, the idea behind the construction of these theories is as follows. By a singularity type }2 we mean a sequence {P0, P1, ..., Pn} of closed stably almost-complex manifolds Pi of dimension Pi and with P0 = *. A n-decomposed manifold is a manifold M together with a sequence {OoM,..., OhM} of submanifolds of codimension 0 of the boundary OM of M such that OM = OoM U • .. U OhM. Baas defines a manifold of singularity type E (a E-manifold) to be a family V = {g(w)lw C {0.1 .... ,n}} of n-decomposed manifolds V(w) with OiV(w) = 0 for i 6 w together with a system of diffeomorphisms (the structure maps) fl(w,i) : OiV(w) ~-~ V(w,i) x Pi, i q[w which satisfy certain compatibility conditions (see [3]). The E-boundary 6 z V of a Emanifold V is defined by 6~V = {6zY(w)} where 6~V(w) = OoY(w) = Y(w, 0). 6~Y is a E-manifold with structure maps Oi6zV(w) = OiV(w,O) p(,~,o;i) V(w,i,O) x Pi = 6 z V ( w , i ) x Pi for i • w U {0}.Notice that dim(6zV) = dim(V) - 1 and that 6~V = 0. Using this concept of manifolds, Baas was able to mimick the usual construction of a bordism theory to get for any singularity type E a homology theory M U ( E ) . ( - )

115

(this is also known as the Baas-Sullivan construction). These theories are representable by spectra MU(P~) which are module spectra over the ring spectrum MU. If ~ is a singularity type, we denote by Zi the singularity type which results from P. by deleting the i th entry of ~. The following theorem relates bordism theories based on manifolds of different singularity types: T h e o r e m 1.4 ([3]) For each i there is a natural exact sequence

• ..--* MU(I]i),(X) ~ MU(]E,),(X) -~. M U ( ~ ) , ( X ) -~h M U ( ~ , ) , ( X ) --+... where the natural transformations Oi,qi and 6i are of degree Pi, 0 and -(Pi + 1), respectively. Oi is given by multiplication with [Pi]. If the sequence {[P1],...,[Pn]} is regular, i.e. if for all i = 1 , . . . , n , zero-divisor in MU./([P1], ..., [Pi-a]), this implies that

[Pi] is not a

MU(P.), ~ MU,/([PI], ..., [Phi). In this way one can kill off any regular ideal in MU,, and, by passing to the limit, even ideals with infinitely many generators. For example, one can kill the kernel of the map MU, ~ BP,. After localizing at p this produces Brown-Peterson theory. One may continue this process by killing generators of BP, to obtain ,for example, theories P ( n ) , ( - ) , k ( n ) , ( - ) or B P ( n ) , ( - ) with coefficients

BP(n), P(n), k (rt) ,

~- Z(v)[Vl,...,v, ] ~ Fv[v,,vn+1,... ] ~: F p [V]n •

The spectrum k(n) is the (-1)-connected version of the spectrum If(n) of Morava K-theory. Using k(n) one defines K(n) by

K(n) = holim{E-2i(P"-l)k(n) Y-~ k(n)}. Similarly, one defines (periodic) spectra E(n) = holim{E-21(P"-l)BP(n) ~" BP(n)} resp. B(n) = holim{E-21(P"-l)P(n) Y-~ P(n)} with coefficients E(n), = Z(p)[vl,...,v~,v; 1] resp. B(n), = v z l p ( n ) , . By the construction of these spectra, one has canonical morphisms B P ~ P(n) ~ K(n) etc.. Moreover, for different n, the P(n)~s are related by stable cofibrations

°", P(n) 0", P(n + 1) 0 . The question whether (co)bordism theories of manifolds with singularities are multiplicative is a delicate one. Using geometric constructions on E-manifolds, Mironov [40], Shimada-Yagita [57] and later Morava [36] constructed good products for a large class of such theories. Using purely homotopy theoretic methods, products for theories like P(n), g(n) etc. were constructed in [62], see also [66] for the case p = 2. Where they apply, these homotopy theoretic methods also give uniqueness results. In this context it is interesting to remark that the methods of Sanders [56] and unpublished work

116

of Margolis show that for example the spectra k(n) and K ( n ) may themselves be constructed by homotopy theoretic methods, so many of the questions we are discussing here are in fact independent of the theory of manifolds with singularities. Let F(n) denote one of the spectra P(n), k(n) or g ( n ) . By their construction, the F(n) are canonically module spectra over the ring spectrum B P and the natural map IZn : B P -+ F ( n ) is a map of B P - m o d u l e spectra. T h e o r e m 1.5 1. Suppose p is an odd prime. There is exactly one product m n : F ( n ) A F(n) ---* F(n) which makes F(n) a BP-algebra spectrum compatible with the given BP-module structure . This product is associative, commutalive and has a two-sided unit.

Z. Suppose p=2. There are ezaelly two products m , , ~ : F ( n ) A F ( n ) --* F ( n ) which make F(n) a BP-algebra spectrum compatible with the given BP-module structure • Both are associative and have a two-sided unit. m~ and-ran are related by the formula -~, = m , o T - - m , + v , m , ( Q , _ l A Q n _ l ) where Qn-1 is a slable F(n)-operation of degree 2n - 1 satisfying the relalion 2 Qn-1 = 0 (a Bockslein operation). In particular, this theorem settles the question about products in the K(n)~s in a satisfactory manner.

2

Operations and cooperations

To apply the K ( n ) ' s in concrete situations it is clearly important to know something about (stable) operations. There is a duality isomorphism

K ( n ) * ( K ( n ) ) ~- H o m g ( ~ ) . ( K ( n ) , ( K ( n ) ) , K ( n ) , ) , so one may consider as well the algebra K ( n ) , ( K ( n ) ) . Now from Adams [1] we know that if E is a ring spectrum such that E , ( E ) is a flat E,-module, E , ( E ) is a Hopf algebroid and E , ( - ) takes values in the category of E , (E)-comodules. This assumption is true for the spectra P(n) and K ( n ) , so one should try to describe the structure of their cooperation Hopf algebroids. The basic information needed to compute them is contained in the following theorem [1], [43]: T h e o r e m 2.1 There are elements ti E BP2(p~_D(BP), to = 1, such that

B P , ( B P ) ~- BP,[tl,t~, ...] as a BP,-algebra. The counit e salisfies e(1) = 1, e(ti) = O, i > O, and the conjugation c resp. the eoproduet ¢ are given by the formulas

t.c(t

)p" = 1,

n,j>_O

¢(t,) = i>_o

F.. i,j>o

t, ®

pi

117

The behaviour of the right unit ~IR on lhe generators of B P . is defined by

i,j>O

i,j>O

The last formula concerning the action of 7/R on the vl is due to Kavenel [45], it is extremely useful, especially for computational purposes. Combining the above theorem with work of Baas-Madsen [4] concerning H . ( P ( n ) ; Zp) , the fact that the ideals In = (vo,...,Vn-1),n >_ 1, v0 = p, are invariant with respect to stable B P operations and the stable cofibrations

~ 2 ( p . _ l ) p ( n ) v., P ( n ) " " , P ( n + 1) 0 % Z 2 p . _ l p ( n ) one can prove (see [62] for the case p odd and [26] for the ease p = 2) T h e o r e m 2.2 For any prime p, P ( n ) . ( P ( n ) ) is a (commutative) Hopf-algebroid over P ( n ) , . I f p is odd, there is an isomorphism of left P ( n ) . - a l g e b r a s

P ( n ) . ( P ( n ) ) ~ P ( n ) . ®BJ'. B P , ( B P ) ® E(ao, el, ..., a n - l ) where E(ao,al, ...,an-l) is an exterior algebra in genera$ors ai of degree 2p i - 1 and for p = 2, P ( n ) . ( P ( n ) ) ~- P(n),[ao, ..., a n - l : Q , t 2 , ...]/jn where J , = (a~ - ti+l : 0 < i < n - 1). Modulo the generators at, P ( n ) , ( P ( n ) ) is for all primes isomorphic to ~he Hopf-algebroid B P . ( B P ) / I ~ and the coproduct resp. the conjugation are given on the generators ai by the formulas k

On(ak) = E

2i+x

ai ® ak_i_ 1 + 1 ® ak

i=O k-i

cn(ak) = --ak -- 2--~ c~[al)ak-~-1 i=0

for p = 2, with lhe obvious changes for p odd. Observe that there is again a duality isomorphism

P( n *) (P( n ))"~ = H o r n *p(n). (P( n .) (P( n )) / P ( n ) *). Under this isomorphism, the generators ai correspond to Bockstein operations Qi of degree 2p i - 1. In particular, Q n - 1 = T/n o On. To get from theorem 2.2. to the structure of K ( n ) . ( K ( n ) ) one may use Landweber's exact functor theorem [29]. Let B:Pn denote the category of P ( n ) . ( P ( n ) ) - c o m o d u l e s which are finitely presented as P ( n ) , - m o d u l e s (we set P(O) = B P and v0 = p). Then T h e o r e m 2.3 Let G be a P(n).-module. The funclor

M ~ M ®P(n). G is exact on the category B'Pn if and only if mulliplicalion by Vn on G and for each k > n, multiplication by vk on G/(vn, ...,vk-1) is monte.

118

For n > 0, this theorem has first been proved by Yagita [68]. The canonical map "~n : P(n) --+ K ( n ) makes K ( n ) , a P ( n ) , - m o d u l e for which Landweber's theorem clearly applies. One then gets a natural multiplieative equivalence

P ( n ) , ( X ) ®p(,). K ( n ) , -% K ( n ) , ( X ) . This equivalence is the mod In version of the theorem of Conner-Floyd. In particular, it produces an isomorphism of Hopf algebroids

K ( n ) , ( K ( n ) ) ~- K ( n ) , ®P(n). P ( n ) , ( P ( n ) ) ®P(n). K ( n ) , . Combining this with theorem 2.2. and Ravenel's formula of theorem 2.1. one then obtains (see [70], [631) T h e o r e m 2.4 Let p be any prime. There is an isomorphism of left K(n),-algebras

K(n).(K(n))

_~ K(n).[tl,t2,...]/(vntf" - v~'ti) ®

E(ao,al,...,an_l)

for p odd and K ( n ) . ( K ( n ) ) ~- K(n).[ao, ..., a . - t , t l , t 2 , ...]/J. for p = 2, where Jn = (v.ti2" - v . 2~t,,. a 2i - t i + i ) . Right and tef nit agree i . K(n).(g(n)) and the coaction map en resp. the conjugation cn may be described on the ti by the formulas E Fn t.c(tj)P" = 1, .j>o

E Fo ¢(t,) = E F. t, ® ikO

i,j>O

and on the generators aj as in theorem 2.2.. The intimate relation between the structure of the tIopf algebroids considered above and the respective formal group laws may be expressed in a slightly different manner. Recall that a groupoid is a small category in which every morphism is an isomorphism. Let k be a commutative ring and let Ak be the category of k-algebras. By a groupoidscheme over k we mean a representable functor G : .4k ~ ~ from .Ak to the category of groupoids. Here representable means that the two set-valued functors A ~-~ ob(G(A)) and A ~-+ mor(G(A)) are representable. For all A we have morphisms (natural in A)

mor(G(A))

TM

Hom.ak (C, A) ~ gom.ak ( S, A) ~- ob(G(A))

which are induced by the maps source, target and identity of the category G(A). These morphisms give rise to homomorphisms of k-algebras '/n, 7]L : B --+ C and e : C ---* B. Furthermore, the composition of morphisms in G(A) is represented by a map ¢ : C -+ C ®B C and all these data together make (B, C) a Hopf algebroid. Let n >_ 0. For any Fp- algebra (Z(p)-algebra if n = 0) A consider the set TIn(A) of triples (F, G, ¢) where F,G are p-typical formal groups of height >_ n over A and ¢ : G -+ F is a strict isomorphism. Tin (A) is a groupoid in an obvious sense and we get a functor T i n ( - ) : A~ ~ 6. One then has the following theorem of Landweber [28]:

119

T h e o r e m 2.5 T / n ( - )

is a groupoidscheme over Fp (resp. over Z(p) / f n = 0) which is represented by the Itopf algebroid (BP,//n, BP,(BP)/I,).

Using theorem 2.5. it is easy to describe the group of multiplicative automorphisms of K(n). In this context it is important to consider also the Z~-graded version of g(n)*(-) which we define by

K(n)'(X)

; @~-~K(n)2i(X)

if- = 0 = 1

l

where q = ion . Let Mult(K(n)*(-)) resp. Mult(K(n)'(-)) denote the groups of multiplicative automorphisms of g(n)*(-) resp. of K(n)'(-). Let SAutF.(Fp) resp. •SAutaFr(K(n),) denote the groups of strict automorphisms of the formal group law F , considered as an ungraded power series over Fp resp. as a graded power series over Fp[vn, vnl]. Then T h e o r e m 2.6 For all primes p and all n > 0 there are isomorphisms

Mult(K(n)" (-)) _~ SAut~: (K(n),) Mult(K(n)'(-)) ~- SAutE. ( r p ) . This theorem was first proved by Morava (unpublished), see also [44], [67], [65]. Now in fact, for each n there is an isomorphism A,

SAut~(K(n),) ~ $I C Zp, where S1 denotes the group of p-adic units congruent to 1 mod (p), (see [67]), and so the elements of Mult(Z(n)*(-)) may be considered as some sort of (stable) Adams operations. In the Z2-graded case the situation is more interesting. A theorem of Lubin and Dieudonn6 (see [14], [13]) asserts that if k is a field of characteristic p containing Fq where q = p", then the endomorphism ring of Fn over k is isomorphic to the maximal order E , of the division algebra D , with center Qv and invariant ~. 1 More explicitely, E , may be obtained from the Witt ring W(Fq) by adjoining an indeterminate S and setting Sn = p and Sw = was for w 6 W(Fq), where ~ denotes the lift of the Frobenius automorphism of Fq to W(Fq). Let S~ = {1 +

w,S'lw, e W ( r q ) } i_>l

be the group of strict units of E , . Then there are isomorphisms

where Fp denotes the algebraic closure of Fp. In [5], A. Baker showed that the element 1 + S 6 S . determines a multiplicative operation

[I+S]:K(.)

,

V ~eZ/@--1)

120

which satisfies the relation [1 + S](y) = y +rn YP E K(n)'(CPoo). Putting r , = (pn _ 1)/(p - 1) one can in fact decompose [1 + S] as

[1+,.9]-1= ~

ea

a~ZIr~ where the 0a : K ( n ) ~ E2"@-t)K(n) are stable operations. The 0 a satisfy the product formula

and one has (oa,t~} = (--1)k~a,k; 1 < k < p'~ -- 1. Baker then obtains the following theorem: T h e o r e m 2.7 The indecomposables of K ( n ) * ( K ( n ) ) have a basis QO, 0 o, 01,0 p ' ..., OP"-' over If(n)*, where QO E K ( n ) X ( K ( n ) is the 0 th Bockstein. In [5], this theorem is stated for odd primes, but in fact it also holds for p = 2. Using r~venel's calculation for the 2-line of K ( n ) . ( K ( n ) ) [44] it is possible to describe the relations amongst these indecomposables. An interesting family of stable operations arises also by considering the duals Oi of the elements ai of theorem 2.4.. We will make some comments on these Bockstein operations at the end of the next section. Let us also remark that in [59], Steve Wilson determines the unstable K(n)-operations by computing their dual K ( n ) , ( K ( n ) ) * ) as a Hopf ring where K ( n ) , = {K(n),} denotes the f2-spectrum representing If(n).

3

Relations with other cohomology theories

A very important aspect of the Morava K-theories is the fact that they are strongly related to BP-theory and complex cobordism via several types of intermediate spectra. For example, consider the diagram

/a~ P(n) P(n + 1)

l. ~ v ~ P ( n )

=

B(n)

121

where In means localization with respect to vn. The triangle is exact and determines a Bockstein spectral sequence. Assuming that we know P(n+I),(X) for some X, then the Vn-torsicn of P(n), is determined by P(n + 1),(X) and the behaviour of this spectral sequence, whereas the vn torsion-free part of P(n),(X) passes monomorphically to B(n),(X). If X is finite, this is a finite process: There is an n such that if m > n, then P(m),(X) ~- H , ( X ; F p ) ® P(m), and the m - th Bockstein spectral sequence collapses. Now the point is that in fact B(n),(X) is determined by K(n),(X): There is a natural isomorphism

B(n),(X)

g(n).(X) ® Fp[Vn+x, Vn+ , ...]

(see [18] for the existence of such an isomorphism and [61] for the fact that it is natural), so in particular B(n),(X) is a free B(n),- module whose rank equals the rank of K(n),(X) as a K(n),-module. Because K(n),(X) is in many cases computable and P(O) = BP, this process can he used to get information about BP,(X) in terms of the K(n), (X). A beautiful example how this works in a concrete case is the Ravenel-Wilson proof of the Conner-Floyd-conjecture (see [50],[58]). In fact, the relation between the two homology theories B(n),(-) and K(n),(-) is even more close as indicated above. B(n),(K(n)) may be considered as a left B(n),(B(n))-and a right K(n),(K(n))-comodule and using results of [32] one can prove the following (see [63], [] denotes the cotensor product) T h e o r e m 3.1 There is a natural equivalence

B(n),(X) ~- B(n),(K(n))OK(,).(K(,))K(n),(X ) of homology theories with values in the category of B(n),(B(n))-comodules. This is of some importance if one observes that the Bockstein spectral sequences considered above are in fact spectral sequences of comodules. In analogy to the splitting of MUZ(e ) into a wedge of suspensions of the BrownPeterson spectrum BP one may ask if there is a similar splitting of B(n) into a wedge of suspensions of K(n). Unfortunately, because the formal group laws Fn and FB(n) are not isomorphic over B(n)., this is not the case (see [64]). However, such a splitting is possible if one completes B(n) suitably. This problem was studied in [64] and, in a more general way, in [7]. First, we should explain what we mean by a "suitable completion". Let R be a commutative ring and let m n. In fact, there is a much more conceptual and elegant way to formulate the theorem above (see [50]): The IIopf ring K ( n ) , ( K , ) is the free K(n),[Z/(p)]- Hopf ring on the Hopf algebra K(n),(K1). Let us also mention that in [50], these results are used to compute v Z x B P , ( g ( Z / ( p ) , n)) which, applying the methods briefly mentioned at the beginning of section 3, allows them to prove the Conner-Floyd conjecture. Observe that there is an isomorphism

l i m j K ( n ) , ( K ( Z / ( p J ) , q)) ~ K ( n ) , ( K ( Z , q + 1)), so, by the Kiinneth isomorphism, K ( n ) . (BG) is known for all finitely generated abelian groups G. It is interesting to observe that through the eyes of Morava K-theories, the Eilenberg-MacLane spaces for finite abelian groups appear as finite complexes. If G is an arbitrary finite group one has the following general result of Ravenel [48]: T h e o r e m 4.3 For any finite group G, K(n)*(BG) is finitely generated as a module over K(n)*. If n = 1, K(1) is a summand of mod p complex K-theory and Atiyah's description of K*(BG) in terms of the complex representation ring may be used to show that the rank of 1((1)* (BG) is the number of conjugacy classes of p-elements in G (see [23],[48]). In [23], N. Kuhn has proved the following generalisation of this: T h e o r e m 4.4 Let G be a finite group with an abelian p-Sylow subgroup P, and let W = N a ( P ) / C a ( P ) . Then

rankg(,,). K(n)*(BG) = ]P'~/W[, the number of W-orbits in pn. The question of finding the group-theoretic significance of the rank of K(n)* (BG) is clearly a very interesting one and actually several people are working on this problem. Among other things, the interest in this question is stimulated by the fact that although the Morava K-theories are fairly well understood today, one does not know

128

any good model for the spaces representing them. One then hopes that a better understanding of K(n).(BG) in terms of G might furnish some ideas in this direction.Let us also mention in this context the following result of Hopkins, Kuhn and Ravenel (see [24]): For topological groups F, G let Hom(r,G) denote the space of continuous homomorphisms. Letting act G on itself by conjugation this becomes a left G-space. Let G be a finite group. Then Hom(Z~, G) is the set of n-tuples of G generating an abelian p-group. One now has the T h e o r e m 4.5 Let G be a finite group. Then

dimK(,,). K(n)~'er*(BG) - dimg(n). K(n)°dd(BG) :

IHom(Z'~,G)/G].

There are a lot of other spaces X where K(n).(X) is known. As examples, let us only mention the computation of K(n),(f22S ~+~) by Yamaguchi [71] , the recent description of K(m).(~22SU(n + 1) by Ravenel in [49] and the work [16], [17] of J. Hunton where (among a lot of other things), he develops a method for computing the Morava K-theories of classifying spaces of wreath products G ~Cp, Cp a cyclic group on p elements.

5

The connected

cover of

K(n)

In this section we will review some properties of k(n), the connected cover of K(n). k(n) is a ring spectrum (non-commutative ifp = 2) with coefficient ring k(n). = Fp[vn] and vnlk(n) : g(n) and there are cofibrations • ..---~ E:P"-2k(n) ~", k(n) '~, HFv "if", E 2 P " - l k ( n ) ---~ .-.

(3)

where 7rn : k(n) ~ HF v denotes the Thorn map. Let A*(p) denote the mod p Steenrod algebra. Then (see [4]) ~r,, induces an isomorphism H*(k(n); r p ) ~- A*(p)/.4*(p)Q, and so Qn : rcnQn, where Qn e .A*(p). Because k(n)* is a principal ideal domain, k(n)* (X) decomposes as a k(n)*-module into copies of Fp[vn] and of the quotients Fp[v,~]/(v~), where s >_ 1. The free part of k(n)*(X) is detected by g(n)*(Z) while the torsion part is analyzed by the Bockstein spectral sequence { E r , d r ] associated to the exact triangle 6.1.. One has E1 = H * ( X ; Fp) and dl : Q,, (resp. dl : Sqa"+ 1 if p : 2) where Q,, denotes the Milnor operation which is inductively defined by Q0 = fl and Qn = 7)P"-'Q,,-1 - Q n - l P p"-~ • Let

T*(X) = ker'{v: : k(n)'(X) --~ k(n)*(X)} and set

T*(X) : (_] T;(X). r>_l

T*(x) is the torsion part of k(n)*(X). Then E ~ ~- k(n)*(X)/ (T*(X) + v.k(n)*(X))

129

and there is a short exact sequence o ~ ,7,-~k(~)*(X)/v[,k(~)'(X)

--. E* -~ T; IT;_~ ~ o.

The spectral sequence {Er,dr} is a spectral sequence of algebras and it can be identified with the Atiyah-Hirzebruch spectral sequence for k(n)*. A detailed study of k(n)*(X) and the associated spectral sequence appears as the main tool in the paper [20] of R.M. Kane where he proves that for a connected, simply connected rood 2 finite H-space X, Q~"e"H*(X;F2) = 0 where QH*(X;F2) denotes the module of indecomposables. Another application of this Bockstein spectral sequence appears in [71] where k(n).(~2S 2~+1) is calculated. The algebra k(n)*(k(n)) of stable k(n)-operations has been studied by Yagita in [67] and by Lellmann in [31]. To describe it, one needs to define some algebras associated to it. For any spectrum X, define

Z*(X) = ker{Q,~ : H'(X;Fp) --+ H ' ( X ; F p ) } B*(X) = im{Q,~ : H*(X;Fp)~ H*(X;Fp)} and 7t*(X) = Z*(X)/B*(X). Z*(k(n)) inherits an algebra structure from ~4*(p) with respect to which 0r,,), is a homomorphism of algebras. Let kP(n),P(n)k

=

k(n), ®P(n). P(n),(P(n)) ®p(,). k(n),.

kP(n),P(n)k inherits a Hopf algebra structure from P(n),(P(n)) and we define L*(n) as the dual k(n),-Hopf algebra. The canonical map P(n), (P(n)) ---+k(n), (k(n)) factors to give a map ~: kP(n),P(n)k ~ k(n),(k(n)) and we write n for the composition ~ : k(~)*(k(~)) -~ H o m ~ ( . ) . (k(n).(k(~)), k(~).) '-L L*(~). Using these notations one then has (see [31]): T h e o r e m 5.1 There is a surjective algebra homomorphism ~r, : L*(n) ---* Tl*(k(n))

whose kernel is the ideal of vn-divisible elements and the diagram . L'(n)

k(~)*(k(~))

(~.), z*(k(~))

~

. ~*(k(~))

is a pullback diagram of algebras. This has been proved in [31] for p odd but it also hohts for p = 2, see [26]. In [67], Yagita described k(n)*(k(n)) by generators and relations These may also be deduced from the theorem above. Notice that the structure of the algebra k(n).(k(n)) iz also known (see [67] for the case p odd and [27] for p = 2).

130

We say that a spectrum X has k(n)*-exponent < e , expk(,). ( X ) _ 0 and q : 2(p" - 1). Thus k(n) [rq] is again a (commutative) ring spectrum and ~ri(k(n)[rq]) = ~ri(k(n)) if i < rq and ~ri(k(n)[ rq]) : 0 if i > rq . In particular , k(n)[ °l = HFp. Using the fact that the Postnikov factors of k(n) are related to the Bockstein spectral sequence the following splitting theorem for k(n) A X may be proved: T h e o r e m 5.2 Let X be a locally finite connective spectrum and suppose e > 1. Then lhe following are equivalent:

1. ezpk(,~). ( X ) 1, let A be the sum of n copies of Z[(s]. We form the semi direct product group: (2.1)

P = A x G.

F is a crystallographic group of rank 4n with holonomy group G. We are going to prove that H*(Z/2Z; Wh~'V(Mr)) ~ O. First of all, for any crystallographic group F, we have the following calculation from [CK2]:

Wh~'V(Ur) = ~ Wh(Nr(H)/H)

(2.2)

H

where H runs over a set of conjugacy classes of those finite subgroups of P for which rk ZA(H) > rk ZA(K) if H C K. This last inequality is the algebraic way of specifying the isotropy groups ofF. In the present case, IHI = 1 or 5, and if H ~ 1 then Nr(H)/H -~ ZA(H) = 1 so that (2.1) here reduces to

(2.3)

Wh pl,p a (Mr) = W h ( P ) .

The "forget control map" Wh(F)c --* Wh(r) (where Mr/G is the control space) is an isomorphism in this case by results of the Ph.D. thesis of G. Tsapogas [T]. This follows from the fact that A contains no one-dimensional G-submodules. So now we apply the spectral sequence of F. Quinn ([Q1],[Q2]) which computes Wh(r)c. We have: E~q = H a ( M r ; / ( q ( Z G , ) ) . Since /~q(ZGz) = 0 if Gz = 1, we get: E~q = Hv(Mra;/~q(ZG)). By [CEll, Lemma 2.2, each component of M ra is a torus of dimension equal to rk ZA(G). Since ZA(G) = O, Mra is discrete and we get: E~q = 0 if p ~ 0. Since Wh(G) = Z, and the algebraic involution on Wh(G) is trivial (by Milnor [M])we get:

(2.4)

Wh(r) ~ Wh(r)c = H0(M~; W h ( G ) )

~- H0(M~;

Z),

and the involution is trivial on Wh(r). According to [CK1], the number of components of Mra is equal to IHI(G; A)[ = IZ(~5)/(1 - ~5)1n = 5 n. So from (2.3) and (2.4) we get isomorphisms: 5 •

(2.5)

Z(Wh(G))i ~ Wh(F) ~ Wh~'V(Mr). i= l

By [CL] section 2, these maps preserve the involutions if the left hand group has trivial involution. This uses the fact that dim Mr is even and the fact that the algebraic involution on Wh(G) is trivial. Hence we obtain an isomorphism: s~ / t i ( Z / 2 Z ; Whg'"(Mr)) = ~ / ~ i ( z / 2 z ; i=1

(

0

if

i is odd

if

i is even

Wh(G)) = ~ an F2 vector space (

of dimension 5 n

141 I n particular:

(2.6)

~ ° ( z / 2 z ; Wh~'"(Mr)) # 0.

This is the non-vanishing result we sought.

§3. An example where (*)r fails. Let G be the cyclic group of order four acting on the Gaussian integers, Z[i], via multiplication by powers of i. Let n be a postive integer. Let A be the direct sum of n copies of Z[i] mad two copies of Z, the trivial G-module. Set F = (A >~G) × K where K is the fundamentM group of the Klein bottle. T h a t is to say, K = T x~ T where ~ : T ~ T is the n o n trivial automorphism. The holonomy group of F is G r = G × G' where G' is the cyclic group o£ order two. We are going to prove that / t * ( Z / 2 Z ; Wh~Pr'P( Mr) ) ¢ O . Let N be the monoid of positive integers: {1, 2, 3, ...}. According to [CdaS], the nil-K theory,

N K . ( R ) is a Z[N] module in a n a t u r a l way, for any ring R. In the present case, we claim there is an isomorphism of Z[N] modules:

(3.1)

6 : F2[N] ~ N K o ( Z G ) .

To see this observe that the b o u n d a r y m a p in Nil-K-theory of the MeyeI-Vietoris sequence of the cartesian square:

zv J, z[z/2z]

~ -~

z[i] ,L F~[Z/2Z]

provides an isomorphism:

(3.2)

d: NKI(F2[Z/2Z]) ~- N K o ( Z G ) .

because the groups NKj(Z[Z/2Z], NKj(Z[i]),j = 0, 1 vanish, d preserves the algebraic involutions so the involution on Z G acts trivially on NKo(ZG). By a result of Sass-Murthy (IBM], 7.6):

N KI (F2[Z /2Z]) "~ NU(F2[Z/2Z]) . But NU(F2[Z/2Z]) = {1 4- elvx + e2vx 2 + ...] ~, = 1 - t, ei = 0 or 1} where 2'/2Z = {1,t}. The action of an element s e N on a unit p(x) e N U ( R ) sends p(~) to p(x'). The m a p of Z[N] modules: F~[N] ---* NU(F~[Z/2Z]) which sends 1 to 1 + vx is easily seen to be an isomorphism, and this provides the isomorphism of (3.1). By [Sw], [C] and [CdaS] /£,(ZG) = 0, i < 0 and N K _ a ( Z G ) = 0. This implies that (3.3)

/ ( 0 ( Z G × T) = N K o ( Z G ) • N K o ( Z G ) ,

142

(3.4)

,K'-I(ZG x T) = 0

and the automorphism a = (1 x ~), : K0(ZG x T) ~ /~0(ZG x T) summands of (3.3). Therefore

(3.5)

interchanges the two

coker(1 - a) ~- N K o ( Z G ) . The exact sequence of [FH1]:

Ko(ZG

x

T) --*/4o(ZG x T) --+

/(o(ZG × T x~ T) N K o ( Z G x T , a ) @ N K o ( Z G x T , a -~)

K _ I ( Z G x T)

together with (3.4) implies that the inclusion map (3.6)

0 ~ coker(1 - a) ---* K0(ZG x T x , T)

induces an isomorphism of 'rate cohomology groups. Therefore by (3.1), (3.5), (3.6) we get, for any i (3.7)

/~i(Z/2Z; K0(ZG x K)) ~ F2[N] .

Now we turn to W h ~ f ( M r ) . Since N r ( H ) / H is free abehan if H is a finite subgroup of order 4, or if H is an isotropy group of order 2, the formula (2.2) yietds: Wh~rP(Mr) = Wh(F) . But this time Wh(F)¢ = 0. To see this, note that E 2 term of Quinn's spectral sequence vanishes because h'q(ZH) = 0 if q < 1 and H = {1} or Z/4Z (Carter [C], Swan [Sw], Milnor[M]). Now, according to [CK2], in this case we have:

e: w h ( r ) --- w (r)/wh(r)o ~- Wh~P'P(Mr).

(3.s)

The isomorphism in (3.8) preserves the involutions because F has even rank (see[CL], section 2). The split monomorphism j :TxGxK---* A>4GxK where T goes to a trivial summand of A, yields a split monomorphism of Whitehead groups (3.9)

j,:/~(Z/2Z;

W h ( T x G x g ) ) ~ / ~ i ( Z / 2 Z ; Wh(r)) .

The fundamental theorem of algebraic K-theory [B] then yields a split monomorphism: (3.10)

i: [(o(G x K ) --* W h ( T x G x g ) .

By combining (3.8), (3.9) and (3.10) we obtain a split monomorphism: (3.11)

e,j,i, : / t ' ( Z / 2 Z ; W h ( T x G x K)) ~ / t ' ( Z / 2 Z ;

Wh~Pr'P(Mr))

for all i.

143

By 3.3,

e.j.i.&: F2[N] --* /:/i(Z/2Z; WhortOp,P(Mr)) is a split monomorphism. In particular: (3.12)

H~(Z/2Z; Wh~Pr'P(Mr)) # O.

This is the non vanishing result we were seeking.

§4. Geometric Consequences of the Calculations. Here we show that S(F) ¢ 0 and SPt(F) ~ 0 for certain F, as explained in §1. These examples are all coming from the nonvaaishing of relevant Whitehead torsions. Examples of a rather different flavor, due to the nonvanishing of relevant UNil groups, are also possible, as has been pointed out by S. Weinberger [W]. To begin, we give a careful definition of the structure sets we are using. Let (/~/, F) be a topological manifold with a properly discontinuous F-action for which /~//F is compact and for which each fixed set .QrH is a contractible, locally flat submanifold in any bigger fixed set ZT/K, K C H. The standard example is (-~/r, F) = R n with the isometric action. According to [CK1] there is a F-map, unique up to equivariant homotopy:

We write M for ~I/A, Mr for-~/r/A; J induces a G-homotopy equivalence J : M ~ M r whose torsion can be measured in Wh~P'P(Mr). If J can be chosen isovariant, we say (/t~/, P) is a crystallographic manifold. S(F) is the set of equivariant homeomorphism classes of crystallographic manifolds whose torsion, in Wh~P'P(Mr) is zero. If we wish to drop the torsion condition, we write Sh(F) for the set of equivariant h-cobordism classes of such manifolds. If we wish to consider PL-manifolds and PL-actions, up to PL homeomorphism we write Spl(F), this time requiring the torsion to vanish in Wh~'P(Mr). We will be using the following two exact Rothenberg sequences of structure sets:

(4.1)

S(Mr x I) ~ Sh(Mr × I) ~ H ° ( Z / 2 Z ; Wh~P'P(Mr)) --, S(F) --. Sh(F) --, H I ( Z / 2 Z ; Wh~P'P(Mr))

(4.2)

Spl(Mr x I) ~ S$z(Mrh

x I) ~ /~°(Z/2Z; Wh~"(Mr))

---, Spt(F) ~ S~I(F ) ~ H I ( Z / 2 Z ; Wh~'P(Mr)). For a proof of exactness of (4.1) see [CK2]; the proof of the exactness of (4.2) follows in a formally identical manner. Here S(Mr x I) means the G-structures on M r x I which are homeomorphisms over M r x OI ;other structure sets are defined similarly.

144

First suppose that r is the group defined in §2. By (2.5) and (4.2) either Spl(F) or Sit(Mr x I) is non trivial. If Sit(Mr × I) ~t O, then an easy application of Farrell's thesis [F] implies that Sht(F x T) # 0 and it also yields an exact sequence: 0 -~ s i , ( r × T) -~ s p , ( r × T × T)

So either Spt(F) or Sw(F × T × T) is # 0. Next suppose I' is the group discussed in §3. By (3.8), (4.1) and the argument in the previous paragraph, either S(F) or S(F x T 2) is ~t 0. These are the failures to the rigidity conjectures mentioned in §1. §6References [B] Bass, H.:Algebralc K-Theory. New York: W.A.Benjamin Inc., 1968 IBM] Bass, H., Murthy, P.: Grothendieck groups and Picard groups of Abelian group rings. Annals of Math.(2)86,16-73 (1967) [C] Carter, D.: Lower K-theory of finite groups. Comm. Algebra 8 1927-1937 (1980) [CdaS] Connolly, F., daSilva, M.:NIKo(Z~r) is a finitely generated Z N i module for any finite group ~r. (to appear) [CK1] Connolly, F., Kolniewski, T.: Finiteness properties of classifying spaces of proper F actions. Journal of Pure and Applied Algebra 41, 17-36 (1986) [CK2] Connolly, F., Kolniewski, T.:Rigidity and Crystallographic Groups, I. Inventiones Math.99 25-49 (1990) [CK3] Connolly, F., Ko~.niewski, T.:Rigidity and Crystallographic Groups, If. (in preparation) [CL] Connolly, F., L/ick, W.: The involution on the Eqnivariant Whitehead Group. Journal of K-Theory,(to appear, 1990) [F] Farrell, F.T. :The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J. 21,3125-346 (1971) [FH1] Farrell, F.T., Hsiang, W.C.: A formula for KI(R,~[T]). Proc. Syrup. Pure Math. vol. 17 (1970) [FH2] Farrell, F.T., Hsiang, W.C.: Topological Characterization of flat and almost flat manifolds, M",n ~t 3,4. Amer. Jour. Math.105,641-672 (1983) [HS] Hsiang, W.C., Shaneson, J.: Fake Tori. In: Topology of Manifolds. Chicago, Markham 1970 pp. 18-51 [M] Milnor, J.W.: Whitehead Torsion. Bulletin of the Amer. Math. Soc. 72, 358-426 (1966) [Q1] Quinn, F.: Ends of maps II. Inventiones Math.68,353-424 (1982) [Q2] Quinn, F.: Algebraic K-theory of poly-(finite or cyclic) groups, Bulletin of the Amer. Math. Soc.12, 221-226 (1985). [St] Steinberger, M. : The eqnivariant topological s-cobordism theorem. Inventiones Math. 91, 61-104 (1988) [StW] Steinberger, M., West, J.:Equivariant h-cobordisms and finiteness obstructions. Bulletin of the Amer. Math. Soc.12, 217-220 (1985) [Sw] Swan, R.: The Grothendieck ring of a finite group. Topology 2, 85-110 (1963)

145

[Ts] Tsapogas, G. : On the K-theory of crystallographic groups, Ph.D. dissertation, University of Notre Dame, 1990. [W] Weinberger, S. : Private communication

Frank Connolly * Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556, USA

Tadeusz Ko/niewski ** Instytut Matematyki Warsaw University PKiN IXp, 00-901 Warszawa, Poland

* Partially supported by NSF Grant DMS-90-01729 ** Partially supported by Polish Scientific Grant RP 1.10

SUR LA TOPOLDGIE DES BRAS ARTICULES 3ean-Claude HAUSMANN

(0.1) Consid~rons l'application ~a : (sk-1)n-"~ Rk d~finie par ~a(Zl,...Zn) = Z ai.zi

(a i ~ 0).

Nous appellerons ~a le bras artieul~ dans Rk, de longueur n e t de type a = = (a1,...,an). Dans cet a r t i c l e nous d~montrons quelques r~sultats concernant les points critiques de ha ainsi que sur les pr~-images Ba1({q}), q ( R k. I I est possible que ce genre d'information soit u t i l e en robotique (voir [Coi et 2]~ En tout cas, on verra que c'est la source d'exemples et de probl~mes int~ressants de topologie diff~rentielle.

L'application ha est transverse ~ ~x{O} (voir (1.3) ci-dessous). Nous appelerons Va la pr~image de ce rayon : Va = ha (RFox{O}), qui est donc une sous-vari~t~ de codlmension k-1 de (sk-1) n. On denote par Ya : Va-'~R la premiere composante de ha i Va" L'application 7a est le bras articul~ a extr~mit~ c~ullissante de longueur n, de type a, dans Rk.

,

'r,,

j

(0.2) RemaEques a)

CommeBa est la restriction ~ (sk-1) n d'une application l i n 4 a i r e de (Rk)n

~ Rk,

les espaees B~1({q}) sont 1'intersection dans (Rk)n d'un sous-espace affine de codimension k avec le produit de spheres (sk-1) n. En p a r t i c u l i e r , ce sont des ensemble alg4briques r4els de (Rk)n. Nous nous eontenterons cependant de consid4rer

147 ~a1({q})- commeespace topologique ou, dans le eas o~ q est une valeur r~gulibre, oomme vari~t~ diff~rentiable. b) L'espaee Ba1({q}) est llespace de configurations du systbme articul~ suivant dans Rk •

0 Ces espaces ont ~t~ ~tudi~s, en t o u t cas pour k = 2 et n ~ 5, par W. Thurston, [TW], K. Walker [Wa], A Wenger [We]. Pour une ~tude du point de vue de la g~om~trie alg~brique voir [GN]. 1

SYMEIRIES ET ACTIONS

(1.1)

Pour M ~ O, on a

~a(Z)

= ~-~a(Z). I I en r ~ s u l t e que les p r o p r i ~ t ~ s de Ba

qui nous i n t ~ r e s s e n t seront les m~mes que c e l l e s de ~Ba" On ne r e s t r e i n t done pas l a g ~ n ~ r a l i t ~ en supposant, par exemple~ que ~l~ment du simplexe standard ~ n - 1 .

~a

i = I. Le vecteur a est a l o r s un

(Nous noterons les ooordonn~es de ~ n - 1

de I

n au l i e u de l a oonvention h a b i t u e l l e qui est O , . . . , n - 1 . ) (1.2)

Le groupe symetrique ~ - n

coordonn~es. Si O ~ n ,

a g i t sur (sk-1) n

et s u r ~ n - l ,

par permutation des

on a

~oa(OZ) = ~a(Z) Par exemple, sl aI = a2 . . . . .

an, l'espace Ba1({q}) est un sous-espace de (sk-1) n

invariant par permutation de ooordonn~es. Cela donne d'int~ressants exemples d'actions alg~briques du groupe sym~trique. (1.3)

Consld~rons d'une part 1'action standard de SOk sur Rk et d'autre part celle

de ee m~megroupe sur (sk-1) n, diagonalement, par 1'action standard sur ohaque facteur. Pour ~ ~ SOk, on a ~a(~.z)

=

a.~a(Z)

Soit 0 ~ q ~ R k et solt z0 ~ ~1({q}). On eonsid~re l'applieatlon SOk---~(sk-1) n envoyant ~ sur ~z O. Sa composition avec Ba envoie ~ sur :q. Cette derni~re appllcatlon est une submersion sur la sphere de rayon Hql(. On en d~dult que ~a est transverse aux rayons de R k e n

partieulier au rayon R~ox{O},

d~finitlon de V a au paragraphe O.

comme annonc~ dans la

148

(1.4)

Si 0 ~ q 6 Rk. I d e n t i f i o n s l e s t a b i l i s a t e u r de q dans SOk avee SOk_1. Cela

donne une a c t i o n de SOk_I sur ~a1({q} ) :

.,./.. ~~_~0~_,

o~./" En p a r t i c u l i e r ,

on a une a c t i o n de SOk_I sur Va t e l l e

que ya(mZ) : ya(Z).

3. LES POINTS CRITIQUES DE ~a Dans ce paragraphe, nous d~terminons 1'ensemble C r i t ( ~ a ) des p o i n t s c r i t i q u e s d'un bras a r t i c u l e Ba dans Rk a i n s i que 1'ensemble C r i t ( y a) de ceux du bras a r t i c u l e e x t r e m i t ~ c o u l i s s a n t e associe Ya" On demontre que Ya cst une f o n e t i o n de Morse. (3.1) Th~or~me

Supposons que a. ~ 0 pour t o u t i . Le p o i n t z = ( Z l , . . . , z n) est un

p o i n t c r i t i q u e de ~a : (sk-1)n Z ' ~ R k s i et seuiement s i z i = + z j , pour t o u t i , j . Le th~or~me (3.1) implique que z e s t

un t e l p o i n t c r i t i q u e s i et seulement s i ~a(Z)

est une c o n f i g u r a t i o n a l i g n ~ e :

~aL~3 Preuve :

Soit c ~ Tz(Sk-1)n , repr~sent~ par une courbe t t--w z ( t ) . L'image de e par

1 ' a p p l i c a t i o n tangente ~ ~a en z e s t

Tz~a(C)

=

Z

ai ~i (0)

s o i t q = Ba(Z). L'espace tangent TqRk est naturellement i d e n t i f i e identification,

~ Rk, Via c e t t e

1'image de TzBa est, vu la formule ci-dessus, l e sous-espace

v e c t o r i e l de Rk engendr~ par la r6union des supp14ments orthogonaux au vecteurs z i . Le th4or~me (3.1) en d~coule imm4diatement. Le r ~ s u l t a t pour les points c r i t i q u e s de 7 : V ) ] 0 , I ] est analogue. Consid~rons . 0 + k-~ a la sphere S = {_I} comme incLuse dans S , en i d e n t i f i a n t ±1 ~ ( ± 1 , O , . . . , O ) . Si p = ( ± I , ± I , . . , ± I ) ( ( s O ) n , on d~note par ind(p) l e nombre de composantes ~gales ~ +I.

149 (3.2) Th~or~me Ya : V a ' ~ ] 0 ' 1 ] est une fonction de Morse avec Crit(y a) = VaO (sO)n. L'indlce du point critique p = ( ± I , ± I , . . , ± I ) est ~gal ~ (k-1)(ind(p) - I). Ce th~or~me a ~t~ obtenu par K. Walker pour dans le cask = 2 [Wa]. En f a i t , la demonstration de Walker n'est pas vraiment complete; e l l e ne t i e n t pas compte par exemple du cas 2 ci-dessous. Notre preuve repose sur un principe different. Preuve :

I1 r~sulte de la dSflnition de Va que, pour z ~ Va, 1'application tangente

ha se factorlse (y = f ( z ) ) Tz~a

:

TzYal~TzBalvzVa

:

:

TzVa ~ VzVa

i

Ty(RXO) O Ty(OxRk-1))

Comme ha est transverse ~ ]0,1]xO, l ' a p p l i c a t i o n Tz~alVzVa

•

VzVa--~Ty(OxRk-1))

est surjectlve. On a donc Crit(Ya) = Crit(~a)t~ Va, Par le th~or~me 3.1, on a Crit(~ a) /% Va = Vat~ (sO)n. Soit p 6Crit(Ya). La d~monstration que p e s t un point critique non-degenerese f a i t par r~currence sur la longueur n du bras. Cas I

n = 2 :

Posons a = (A,B). Comme¥a(p) est une position align~e, un syst~me de

coordonn~es (carte de Va) au voisinage de p e s t donna par (z1,z2)~-~x = (x1,...,Xk_ 1) ~ Rk-l, ou x est la projection sur {O}xRk-1 du point AZl, 1'unique articulation de Ya" Les trois cas de figure possibles sont :

On a donc, pour ces coordonnees : Ya(X)

=

±

4 A2 - Ilxu2'

± / B 2- llxll 2

Les d ~ r l v ~ e s p r e m i e r e s : ~Ta ~x.

s'annulent

~ =

x|

~A z -II×ll z

~

x. ]

~/-B z _ iI×ll z'

pour x i = O, q u i sont l e s coordonn~es de p. Les d ~ r i v ~ e s secondes en ce

point valent

:

150

1

La m a t r i c e h e s s l e n n e en p :

.~

( B2ya ~.(0)~ 3 i

(ya,p) =

est donc diagonale. Son d~terminant ne p o u r r a i t s'annuler que si les signes sont opposes dans la formule donnant 7a e t

s i A = B. Mais alors ya(p) = 0 ce qui c o n t r e d i t

l e f a i t que p ~ Va. Cela prouve que p e s t non-d~g~n~r~ dans l e c a s n = 2. Cas 2

: n ~ t 3 e t a i = aj pour t o u t i , j

et z = ( I , - 1 , 1 , - I , . . . , - 1 , 1 )

: On est dono

dans l a s i t u a t i o n :

o roCrl Comme t o u s l e s ai sont egaux, on a ya(Z) = 7a(OZ) pour t o u t o ~

~n'

oG l ' a c t i o n du

groupe symetrlque sur Va est c e l l e donn~e en ( 1 . 2 ) . Pour demontrer que l e p o i n t c r i t i q u e p e s t non-d~g~n~r~, i l

s u f f i t donc de d~montrer que o.p l ' e s t ,

ou o est l a

t r a n s p o s i t i o n ( 2 , 3 ) . La c o n f i g u r a t i o n pour 7a(Op) est :

On e s t donc p l a c ~ dans l e cas ] c i - d e s s o u s . Cas

3 : Ce s e r a l e cas o~ n ~ 3

Les a r t i c u l a t i o n s

e t o6 l e s c o n d i t i o n s

de i a c o n f i g u r a t i o n

p e u t donc d~composer I e bras a r t i c u l ~

de cas 2 ne sont pas v ~ r i f i ~ e s .

y a ( p ) o c c u p e n t au moins 3 p o i n t s

7a en deux s o u s - b r a s de l a manl&re s u i v a n t e

a = (a I . . . . . an) ,

a' = (a I . . . . as ) , a" = ( a s + 1 , . . . , a n)

z = (Zl,...,Zn)

z'

e t supposer que y a , ( p ' ) 1)

O~-Ta,(P')

2)

Ya(p) < Ya' (P')

,

= (z 1 . . . . z s)

~ 0 ~ 7a(p).

~

#o' (~ •

• •

On :

, z" = (Zs+ 1 . . . . . z n)

T r o i s cas s o n t ~ d i s t i n g u e r

o ~

C 7a(P)

distincts.

r~l~

:

r.{r) v

~' (r')

Comme i l s se t r a i t e n t de fagon s i m i l a i r e , nous ne donnons que les d ~ t a i l s du oas I . En u t i l i s a n t les sections locales usuelles de SOk p o i n t z dans un voisinage de p :

P Sk ' l ,

on peut t r o u v e r pour t o u t

151 - Une unique r o t a t i o n : ' ( S O k t e l l e que ~ ' ( ~ a , ( Z ' ) )

.......

~ ]O,l]x{O}

~'~SOk telle que ~'(7a(Z)-~a,(Z')) ~ ]0,1]x{O}

L'application z ! )(~'(z'),~'(z"),X(Ba,(Z')), ou a' et ~' agissent diagonalement sur les composantesde z' et z" et x d~signe la projection sur OxRk-l, donneun diff~omorphisme d'un voisinage de p dans V sur un voisinage de (p',p",O) dans Va, x Va,, x Rk-1.

a

Choissisons des coordonn~es u et v au voislnage de p' et p", dans Va, et Va,,. On v ~ r i f i e que dans les coordonnees ( u , v , x ) , forme

la matrice hessienne de 7a en p e s t

de l a

:

(Ya,P) :

i

~(Ta,,p,) I 0 " i (7a"iP")

oG A = 7a,(p') et B = 7a,,(p"). On peut supposer, par hypothbse de r~currence, det~Ya,,p') ~ 0 ~ det~(Ya,,,p"). On en deduit que det~(Ta, p)#O et done p e s t un point critique non-d~g~n~r~. La formule pour l'indlce de p se d~montre aussl par recurrence, utilisant 1'expression de ~ ( 7 a , p) ci-dessus.

4. PREIMAGES SPHERIQUES ET ACTIONS

Soit ~a un bras articul~ de longueur n, dans Rk. (4.1) P r o p o s i t i o n

Si I - 2min{a i } ~ ~u~l ~ I a l o r s ~a1({u}) est diff~omorphe ~ l a

sphere standard S ( n - 1 ) ( k - 1 ) - 1 . conJugu~e ~ l a r e s t r i c t i o n

L ' a c t i o n de SO. . sur ~ - l ( { u } )

sur l ' a c t i o n S ( n - I ~ - I ) - I

est d i f f ~ r e n t i a b l e m e n t

~e l ' a c t i o n diagonale de SOk_I

sur l e p r o d u i t de ( n - l ) copies de Rk-1. Preuve :

On a Ba1({u}) = y ~ 1 ( { q } ) , ou q = I1uI~. 11 s u f f i t

de demontrer l e r ~ s u l t a t

pour les pr~images de 7. S o i t b = 1 - 2 m i n { a i } . Observons que s i 7a(Z) alors z # (1,1,...,I)

]b,1[,

e t de plus aucun des z I ne peut ~ t r e ~gal ~ - I . Par la

p r o p o s i t i o n (3.2) l ' i n t e r v a l l e q ~ q' ~ ] b , l [

~.

] b , 1 [ ne c o n t i e n t a l o r s que des valeurs r e g u l i e r e s . Si

on peut c o n s t r u i r e un diff~omorphisme SOk_1-~quivariant de 7a1({q})

sur 7 a 1 ( { q ' } ) en suivant les t r a j e c t o i r e s du champ de veeteurs grad7_ convenablement _1d On peut done supposer que 7a ( { q } ) est dans un

normalis~. ( v o i r [Mi2, Th~or~me 3 . 4 ] ) .

voisinage convenable de p = ( 1 , 1 , . . . , I ) .

152 Comme p e s t

un maximum non-d~g~n~r~, par la Proposition ( 3 . 2 ) ,

le lemme de Morse

[ M i l , Lemme 2.2] assure l ' e x i s t a n c e d'un syst~me de eoordonn~es (x) au voisinage de p t e l que Ya(X) = I -

II x l l z

Les surfaces de niveau de cette fonction sont des spheres standard. On en d~duit dono s(n-1) ( k - I ) - I que ~a1({u}) est diff~omorphe Pour trouver un diff~omorphisme SOk_1-~quivariant, on proc~de de la mani~re suivante : on suppose que y~1({q}) ~ U, o~ U est le voisinage de p constitue des points z E Va tels que chaque composante de z a sa premiere coordonnee strictement positive. Dans une t e l l e configuration, chaque ar~te est dirigee vers la droite. L'ouvert U est le domaine d'une carte de Va X : U~O z l

- C

R( k - l ) ( n - 1 )

~ x(z)

oG

x.(z)z = projection sur OxRk-1 de Y(al,a2 , . . . , a i ) ( z l , z 2 , . . . z i ) .

.

. . . . _ _ .

La carte X est SOk_1 ~ q u i v a r i a n t e pour, sur R ( k - 1 ) ( n - 1 ) ,

. - = - - -

l'action

diagonale de

SOk_I . L'image par X de 7 a l ( { q } ) est une s o u s - v a r i e t e de R ( k - l ) ( n - l ) SOk_1-invariante.

a c t i o n que 1'on peut q u a l i f i e r

de q u a s i - l i n ~ a i r e ;

L'application t:

une

on ne peut pas, en g~n~ral

deduire qu'une action q u a s i - l i n e a i r e est d i f f ~ r e n t i a b l e m e n t giquement) conjugu~e ~ une a c t i o n l i n ~ a i r e Soit x ~.

qui est

(Un t e l plongement d'une sphere p r o d u i t sur e e l l e - c i

(ou m~me topolo-

(Voir [Ha]).

; Y a ( t X ) est ~ d~riv~e striotement p o s i t i v e .

Donc

X(yal({q})) est transverse ~ ehaque rayon de R( k - l ) ( n - 1 ) . On en d~duit que la p r o j e c t i o n r a d i a l e de 7a1({q}) sur une sphere standard dans ~ est un diff~omorphisme SOk_lequivariant. Cela ach~ve l a demonstration de la p r o p o s i t i o n ( 4 . 1 ) .

153 (#.2) Remarque :

Dans le c a s k = 3 (bras a r t i c u l e s dans l'espace) on a B~l({q})

diff~omorphe ~ S2n-3 et 1'action de SO2 = SI conjugu~e a l ' a c t i o n standard. On a donc SIX~a1({u}) = CPn-2. 0bservons que chaque o r b i t e de y~1({q}) a un unique repr~sentant z = ( z 1 , . . . , z n) ( Va t e l que zn = ( X l , 0 , x 3 ) , avec x 3~ 0. Les points du quotient correspondant ~ x I ~ - I constituent une c e l l u l e ouverte attaoh~e sur n-3

CP

1

-I

= S ~y, , ( { q - a } ) . On v o i t i c i apparaltre la d~composition \ ~ a 1 , . . . , a- i ~ n c e l l u l a i r e classique desnespace p r o j e c t i f complexes.

_

-)

r~- e

~P''"

'1 .t~- S"

o

o

o. : ~ g'

,Cr"

c

~- P

On peut de m~me consid~rer le c a s k = 2. On a alors 7a1({q})- est diff~omorphe

Sn-2"

On d~montre de m~me que l a r ~ f l e x i o n par rapport ~ l ' a x e horizontal est conjuguee par ce dlff~omorphisme a l ' a p p l i c a t i o n

antipodale. Le m~me argument que ci dessus f a i t

appara~tre l a d~composition c e l l u l a i r e

bien oonnue de RPn-2.

5. PREIMA£.£5 OF ZERO

Les pr~images de 0 pour un bras a r t i c u l ~ sont d~termin~es par les pr~images d'un bras a r t i c u l ~ de longueur n-1. Pour v o i r cela, s o i t T : (R 0 )n "--~(R ) 0 )n-1 l ' a p p l i c a t i o n d~flnle par 1

T ( a l , o . . , a n)

= ~

(al'°'''an-1)

La projection de ( s k - 1 ) n - - - ~ S k-1 sur le ne facteur donne, par r e s t r i c t i o n -1 ~ a l ( { 0 } ) un ftbr~ de f i b r e 7T(a)({an}) et de groupe s t r u c t u r a l S0k_ 1 (aver 1'action de 50k_ 1 sur les pr~images de 7T(a) donn~e en ( 1 . 4 ) ) .

154

On peut caract~riser ce fibre en disant q u ' i l a m~me S0k_1-fibre principal associe que le fibr~ tangent unitaire T15k-1 ~ Sk - l . Cette assertion est ~quivalente au r~sultat suivant :

(5.1) Proposition

Pour a : ( a l , . . . , a n ) ,

6a1({0})

on a -1

S0k XSOk_l YT(a)({an})

=

On volt q u ' i l est de premiere importance de conna~tre 1'action de S0k_1 sur les pr~images de YT(a)' comme cela a ~t~ f a l t dans un eas p a r t i c u l i e r au paragraphe 4. -I On d 6 f i n i t une application f : S0k x yT(a)({an})

Preuve :

f(~,Zl,...,Zn_

1)

=

~ ~a1({0}) par

(~z 1 . . . . ,~Zn_l,-U)

oO

u

1

=

B T ( a ) ( e Z l , - ' ' , ~ Z n _ 1)

6T(a)(#Zl . . . . . CZn_ 1) I i est c l a i r que f(~ ~',z) = f(~,~'z) pour ~' ~ S0k_l, d'o~ une application f

-1

:

S0k XS0k_l 7T(a)({an})

)

B~I({o}).

En u t i l i s a n t l ' a c t i o n de S0k suc 6a1({0}), on v ~ r i f i e facilement que f est un hom~omorphisme, et un diff~omorphisme st an e s t une valeuc r~guli~re de 7T(a).

6.

CLASSIFICATIONDES PR£IMA(I_SDE Ta

Soit Ya : (sk-1)n

~0,I]

(a 6 /kn-1) un bras ~ extr~mit~ coulissante de

longueur n. Pour d ( [0,1], considSrons 1'ensemble H ( n , d ) ( : A n-1 qui est l'unlon des hyperplans de ~ n - 1 d'~quation ~ l - a i

une composante connexe de ~ n - 1

_ H(n,d).

= d, avec ~ i (

{ ± I } . Appelons ¢hambre

155 (6.1) Proposition Soit d ~ 0 e t a ~ n - 1

Alors

a) d est un niveau critique de 7a si et seulement si a E H(n,d). b) Si a, a ' ~ I n t ~ n-1

sont dans une m~mechambre, les vari~t~s diff~rentiables

7al({d}) et ya~({d}) sont canoniquement diff~omorphes. Preuve : Nous avons vu dans le paragraphe 3 que 1'ensemble des points critiques de 7a est constitu~ par les positions align~es {z ~ (sk-l) n I zi = ±i}. Un ~l~ment d

[0,1]

est donc un niveau critique pour Ya si et seulement si 11 existe ~ i ~ {±I} ( i = 1 , . . . , n ) tels que ~ E i ' a i

= d. Cela d~montre a).

Si a e t a ' sont dans la m~mechambre C, i i existe v7 0 t e l que u(t) = ta + ( 1 - t ) a ' ( C pour t ( ]-v,1+v[. L'application diff4rentiable (sk-1) n x ]-v,1+v[ (z,t)

I.

est alors transverse ~ cobordisme Wentre

• Rk x ]-v,1+v[ ~ (Bu(t)(z),t)

{(d,O)}x]-v,1+v[.

La pr~image de cet intcrvalle donne un

7al({d}) et 7a?({d}). La projection sur ]-v,1+v[ est une fonction

g : W~]-v,1+v[ sans point critique t e l l e que ya1({d}) = g-1(I) et 7a~({d}) = -1 g (0). Les trajectoires du champ de gradient de g convenablementnormalise donnent un diff6omorphisme de 7a1({d}) sur ya~({d}) (volr [Mi2,Th~or~me 3.4]). Remarquonsque ce diff~omorphisme commute avec 1'action de SOk_I • On a l e m~mer~sultat pour c l a s s i f i e r les espaces Ba1({O}). On u t i l i s e pour le d~montrer la proposition (6.1) ci-dessus et les r~sultats du paragraphe 5. L'~nonc6 precis est :

(6.2) Proposition a) 0 est une valeur critique de ~a

b) Si a, a'~ i n t ~ n-1

si et seulement si a £ H(n,O).

sont dans une memechambre, les vari~tes diff~rentiables

Ba1({O}) et Ba~({O}) sont canoniquement diff~omorphes. Remarquons que H(n,d) ne depend pas de k. Le nombre de pr~images diff~rentes des bras articul~s de longueur n dans Rk ne d~pend donc que de n. I i est possible de construire le syst~me d'hyperplan H(n,d) par un proc~d~ de rdcurrence surn que nous allons d~crire maintenant. Soit H(n) la famille d'hyperplans du prlsme ~ n - 1 x [0,1] d'~quation ~

~ i . a i = d, avec ~ i

~ {±I}. On

a H(n,d) = H(n) h ~n-1 x {d}. Le proc~de de recurrence conslste en deux operations : I)

H(ntO) d~termlne H(n I :

Soit

A = {(al,...,an+ 1)~ ~n-lJ avec ~ i

~r ~ . -1 a1.

= d}

G {±1}, un hyperplan de H(n,O). A est 1'intersection de deux hyperplans A±

156 de H(n), d ' ~ q u a t i o n ~ i - a d ' ~ q u a t i o n ai = 0 s i ai = 0 s i ~ i

i = ±d. L ' h y p e r p l a n A+ i n t e r s e c t e l a face de ~ n - 1

,mi = - I e t A

i n t e r s e e t e l a face de ~ n - 1

= I . On o b t i e n t a i n s i l a f a m i l l e H(n) ~ p a r t i r

x {I}

x {I}

d'~quation

de H(n,O).

La d~termination des pr~images de Ya ~ p a r t i r de eelles de ~ I ( { 0 } ) n'est pas complete. Toutefois, le f a i t que Ya est une fonetion de Morse donne quelques indications sur ces pr~images. II)

H(n) d6termine X(n+1,0) :

h

o. *°' "°°

:

~n-1

consid~rons l'isomorphisme l i n ~ a i r e par moroeau

x [ 0 , 1 ] - ~ { ( a 1 , . . . , a n + I) ~ ~ n I an+1 ( I/2}

,oo ,01:

, oI , Ill ,

..... a,0

h((al,...,an),1) = (a1/2,...,an/2,1/2)

~-J

On v~rifie ais~ment que H(n+1,0) = h(H(n)) SJ {an+ I = I/2}. La pr~image d'un point a ~ 11~ I/2} est vide. Celle d'un point h(a,d) o~ a est un point d'une de la ehambre { y;1({d}), par la ehambre C de proposition (5.1).

- X(n,d) est diff~omorphe a S0k xS0k_I

Nous allons i l l u s t r e r les pas I e t

I I ci-dessus. I I est c l a i r que X(2,0) consiste en

l'hyperplan {a I - a2 : O} de ~ I , c'est-~-dire le point {a I = a2 = I / 2 } . On a ~ ; I ( { 0 } ) = ~ si a ~ H(2,0) et ~ ; I ( { 0 } ) = Sk-1 si a : (1/2,1/2).

#

z

¢

~Hlt,o) En appliquant le pas I ) , on trouve, pour X(2) :

J

I,

4

|

#

157

On applique maintenant le pas II) pour obtenir H(3,0). Comme1'action de SOk_1 sur ~k-2 est l ' a c t i o n standard (proposition (4.1)), la pr~image non-vide g~n~rique de 7a sera le fibre tangent unitaire TISk-1 ~ la sphere Sk-1.

rF

o)

H

Le pas I) appliqu~ ~ cette figure donne, pour H(3), une famille de 6 hyperplans de ~2 x [0,1]. Nous avons dessin~ ci-dessous quelques familles H(3,d) correspondantes. On observe un point t r i p l e en ((1/3,1/3,1/3),I/3), e'est-~-dire que pour a = (I13,113,113),

7~I({I/3}) oontient 3 points critiques. Pour les autres points doubles

(a,d), 7;1({d}) contient deux points critiques. Les dessins des espaces 7;I({d}) pour les niveaux critiques d sont ceux pour le c a s k = 2 o~ l'on obtient les graphes : 7; l({niveau critique} )

(k = 2) J

,,

o

js

f

1 point critique

I

I

I

~'

2 points critiques

•

!

3 points critiques

i

158

QUELQUES FAMILLES H ( 3 , d )

:

~/=J.

/

0

,t=,,,~

0 J= O,E

.c~S~-~

d=O

159

BIBLIOGRAPHIE

[GN]

GIBSON C.G.-NEWSTEAD P.E. On the geometry of the 4-bar mechanism. Aota Applic. Math. 7 (1986) 113-135

[Gol]

GOTTLIEB D.

[Co2] GOTTLIEB D.

Robotsand f i b r e bundles Bull. Soc. Math de Belglque 37 (1987) 219-223 Topologyand the robot arm Acta Applic. Math. 11 (1988) 117-121

[Ha]

HAUSMANN J-C1. Action quasi-lin~aires sur les spheres. A para~tre.

[Mill

MILNOR 3.

Morse Theory. Ann. of Math. Study 51, Princeton Univ. Press 1969

[Mi2]

MILNOR 3.

Lectures on the h-cobordism theorem Princeton Univ. Press, 1965

[TW]

THURSTON W.-WEEKS J. The mathematics of the three-dimensional manifolds S c i e n t i f i c American, July 1986, 94-106

[Wa]

WALKER K.

Configuration spaces of linkages Bachelor Thesis, Princeton 1985

[We]

WENGER A.

Etudes des espaces de configurations de certains syst~mes articul~s. Tray. D£pl$me, Univ. de Gen~ve, 1988.

POST SCRIPTUM : La construction et 1'usage de fonctions de Morse sur l'espace de configurations de bras articul~s (dans S3) ont ere r~cemment d~velopp~s pour ~tudier l'espace des SU(2)-repr~sentations de hi(V), o~ V e s t une 3-sphere d'homologle seifertlque. On peut u t l l i s e r ces r~sultats pour calculer l'homologie de Floer de V vla les techniques de F£ntushel-Stern. VoIr : A. KIRK et E. KLASSEN, Representation spaoes of Selfert fibered homology spheres, preprint. Section de Math~matlques Unlversit~ de Gen~ve

SEMICONTRACTIBLE

LINK MAPS

AND THEIR SUSPENSIONS

Ulrich Koschorke* Mathematical Sciences Research Institute, Berkeley, and Mathematik V, Universit/i.t GH, D5900 Siegen

Introduction. Given dimensions Pl,. -. ,p, and m, a map /=fzH...Hf,-

: Sa" I I - . . I I S J " ' - - ' * 6 *n

whose components have paJrwise disjoint images ( i . e . / i ( S p') N fi(SPs) = 0 for 1 < i # j 1; c+ o r e _ equals 1}, together with (X - X - l , 0 ) , from a ][-basis of the kernel of fold O fold (see also 2.7 and 2.8). This has the following two interesting corollaries. First, there is an infinite number of additively independent elements both in ~r0(£+ n £ _ ) and in ~r1(£) which all become trivial in the corresponding base point free setting as well as in LM~, 2. In paxticular, the kernel of the track homomorphism in 2.3 contains an infinitely generated free abelian subgroup. Secondly, the full ~r-invariant depends crucially on the base point and level preserving aspects of the track of [p] E ~%(£). The only possibly new additive invariant (besides Kirk's original o-invariant) induced by ~ in the base point free setting or on LM~,a, takes values in a cyclic group. At present, there is no other invariant known which might detect nontrivial elements in the kernel of a. On the other hand, it is hard to believe that a should be injeetive on LM~,2, since this would imply e.g. that every link homotopy class remains unchanged if we precompose one or both of the component maps with a reflection.

169

References.

[D] U. Dahlmeier, "Gewisse Verschlingungen und ihre 3in-Suspensionen," Diplomarbeit, Universitaet Siegen, 1989. [FR] R. Fenn and D. Rolfsen, Spheres may link homotopieally in 4-space, J. London Math. Soc. (2) 34 (19s6), lZT-ls4. [J]G. T. Jin, Invaraints of two-component links, Thesis, Brandeis University (1988). [Kil] P. Kirk, Link maps in the 4-sphere, Proc. Siegen Topology Symp. LNiM 1350, Springer Verlag (1988). [Ki2] , Link homotopv with one eodimension 2 component, Trans. AMS, to appear. [Kol] U. Koschorke, Link maps and the geometry of their invariants, Manuscr. Math. 61 (1988), 383-415. [Ko2] , Multiple point invariants of link maps, Proc. Second Siegen Topology Symposium 1987, Springer LNiM 1350 (1988), 44-86. [Ko3] , On link maps and their homotopy classification, Math. Annalen, to appear. [Ko4] ~ Link homotop~ with many components, Topology, to appear. [M] J. Milnor, Link groups, Ann. of Math. 59 (1954), 177-195. [P] C. D. Papakyriakopoulos, Dehn's lemma and asphedeity of knots, Ann. of Math. 66 (1957), 1-26. [It] D. Rolfsen, "Knots and Links," Math. Lect. Series 7, Publish or Perish, 1976. [S] G. P. Scott, Homotop~l links, Abh. Math. Sere. H~-nburg 32 (1968), 186-190.

The KO-Assembly Map and Positive Scalar Curvature JONATHAN ROSENBEKG Department of Mathematics, University of Maryland College Park, Maryland 20742, U.S.A. Abstract. We state a geometrically appealing conjecture about when a closed manifold with finite fundamental group lr admits a Riemannian metric with positive scalar curvature: this should happen exactly when there are no KO.-valued obstructions coming from Dirac operators. When the universal cover does not have a spin structure, the conjecture says there should always be a metric of positive scalar curvature, and we prove this if the dimension is ~ 5 and if all Sylow subgroups of ~r are cyclic. In the spin case, the conjecture is closely tied to the structure of the a~embty map KO.(Bzc) ---* KO°(R~r), and we compute this map explicitly for all finite groups It. Finally, we give some evidence for the conjecture in the case of spin manifolds with 7r = Z/2. §0. INTRODUCTION This paper is a continuation of my previous papers [111], [1t2], and [113], but with an emphasis on manifolds with finite fundamental group. In other words, I shall try to answer the following question: given a smooth closed connected manifold M " with finite fundamental group 7r, when does it admit a metric of positive scalar curvature? A few very partial results on this problem were given in [R2] and [R3], and some further cases were studied in [KS1] and [KS2]. Extrapolating from these and other cases, I would like to make here a somewhat audacious but intuitively appealing conjecture: CONJECTURE O. 1. A closed manifold M " with flnite fundamental group admits a metric of positive scalar curvature if and only if all (KO.-valued) index obstructions associated to Dirac operators with coe~cien~s in fiat bundles (on M and it covers) vanish, a~ least fin>5. The rest of this paper will be devoted to explaining exactly what are the obstructions described in the Conjecture, and to proving that the Conjecture is valid in m a n y cases. As explained in [GL2] and in [R2], the problem naturally splits into two cases, depending on whether or not w2(/~/), where M is the universal cover of M , vanishes. If w2(-~/') ¢ 0, so that M (and afortiori M) doesn't admit a spin structure, then there are no Dirac operators with coefficients in fiat bundles defined on M or on any of its covers. Thus the Conjecture reduces to: CONJECTURE 0.2. If M " is a closed connected manifold with ~nite fundamental group r , and if w2(/~lr) ¢ 0 and n _> 5, then M admits a metric of positive scalar curvature. Section 1 will be devoted to the proof of an interesting case of Conjecture 0.2. I would like to thank the referee for some corrections to the proofs and improvements in the exposition. By the way, the condition in Conjecture 0.2 that ~r be finite cannot be Partially supported by NSF Grants DMS-8400900 and DMS-8700551. This paper is in final form and is not merely an announcement of work to appear elsewhere.

171

omitted in general, as shown by the example in [GL3, p. 186] of C P 2 # T 4. (The reader concerned about the fact that this example has the exceptional dimension 4 can easly replace it by (CP 2 x $ 2 ) # T 6 . ) The rest of the paper, §§2 and 3, will deal with the spin case, that is, the case where w 2 ( M ) = 0. Section 2 actually involves no geometry, only pure algebraic topology and algebra, and may have some independent interest because of its parallels to known results about assembly maps in L-theory and algebraic K-theory. Theorem 2.5 was proved during a visit to /~rhus in 1985, and I would like to thank Ib Madsen and Gunnar Carlsson for helpful suggestions at that time. The concluding section, §3, returns to the problem of positive scalar curvature. Here Conjecture 0.1 is restated in the spin case, using the language of §2, and we give some evidence for the Conjecture in the "hard case" of spin manifolds with fundamental groups of even order. We also briefly indicate how to interpret the Conjecture when w 2 ( M ) = 0 but w 2 ( M ) ~ O, though there are substantial technical difficulties in getting any good results for this case. §1. POSITIVE SCALAR CURVATURE THE UNIVERSAL COVER IS NON-SPIN

WHEN

The object of this section is to give some evidence for Conjecture 0.2 above. In fact, this conjecture was proved in [R2, Theorem 2.14] in the case where ~r is cyclic of odd order, and this result was strengthened in [KS1] to cover the case of any group of odd order with periodic cohomology (or equivalently, with all Sylow subgroups cyclic). One of the technical advances in [KS1] was Corollary 1.6 of that paper, which showed that the conjecture holds for a finite group ~r if and only if it holds for all its Sylow subgroups. However, as is clear from [R2], [l:t3], [KS1], and [KS2], it is much harder to prove results for even-order groups than for the odd order case. Thus the following theorem is in a way much more convincing evidence for Conjecture 0.2. THEOREM 1.1. If M n is a closed orientable connected manifold with cyc//c finite fundamental group ~r, and i£ w2(h~/") ~ 0 and n >_ 5, ~hen M admits a metric of positive scalar curvature.

COROLLARY 1.2. I f M " is a dosed orientable connected manifold wlth a finite fundamen~al group lr, all of whose Sylow subgroups are cyclic, and if w2(/t~/) ¢ 0 and n _> 5, then M admits a metric of positive scalar curvature. PROOF OF COROLLARY: This follows immediately from the Theorem and from [KS1, Proposition 1.5]. | PROOF OF THEOREM: Because of the results of [R2] and [KSl] just quoted, it's enough to consider the case where our cyclic group has order a power of two. We begin with the key case where 7r is of order 2. By [R2, Theorem 2.13], it is enough to exhibit an oriented Riemannian manifold X n of positive scalar curvature, together with a map X " -4 RP °°, in every class in ~n(RP°°), for all n >_ 5. For this we use the well-known isomorphism of [S, pp. 216-217]:

Q,(RP °°) ~ fl,= E) ~,,-z.

172

The summand of f/n corresponds to the case where X is simply connected (or at least the map X " --* RP °° is null-homotopic), so this case is handled by [GL2, Proof of Theorem C]. So it remains to deal with the summand ~l,-1. Suppose y , - 1 represents a class in 9 ~ - 1 . By the analysis in [S, pp. 216-217], the corresponding element of ~ ( R P °°) is represented by f : X'* --~ RP °°, where Y is the submanifold of X of codimension 1 which is dual to the line bundle defined by f . Note that Y doesn't determine (X, f ) uniquely; however, the class of (X, f ) in a , ( R e ¢°) is determined up to an element of ~,, (which we can "subtract off" by what we already know). Now given the manifold y , - 1 , if Y is orientable, we can simply orient Y and take X = Y x S 1, with f factoring through S 1 and inducing a surjection on ~rl. If Y has a metric of positive scalar curvature, we can give X a product metric, and then X will have positive scalar curvature as well. So suppose Y is not orientable, and let ]2 be its orientable double cover, which carries a canonical orientation-reversing involution r. Let cr be the orientation-reversing involution on S 1 defined by complex conjugation on the unit circle in C. Then v × a is an orientation-preserving involution on 1 / x S 1, so X = (Y x S1)/(T x a) can be oriented, l~urthermore, there is a map ~rl(X) ---* Z/2, and thus a map f : X ~ RP °°, associated to this construction of X, for which Y is the dual submanifold. Finally, if Y has a metric of positive scalar curvature, we lift the metric to 1> and give Y x S 1 the product metric, and this descends to a metric of positive scalar curvature on X. Hence to complete the proof for the case where ~r has order 2, it will suffice to construct additive generators with positive scalar curvature for 9~,, for all n > 4. In fact, since the property of positive scalar curvature is preserved under taking products, it's in fact enough to find m u l t i p l i c a t i v e generators for 9l. with positive scalar curvature. But by the structure theory for unoriented bordism (see for instance [S, pp. 96-98]), 9l. is a polynomial algebra over the field F2 of two elements, with generators represented by even-dimensional real projective spaces and by hypersurfaces of degree (1, 1) in products of pairs of real projective spaces. These manifolds all have natural metrics of positive scalar curvature (cf. [GL2, p. 43]), so this completes the first part of the proof. Now we have to go on to the case where the order of ~r is any positive power of 2. The key fact we need, which is proved in IS, pp. 209-212 and 233-236], is that the oriented bordism spectrum is Eilenberg-MacLane at 2, and thus that for l r a 2-group, the Atiyah-Hirzebruch spectral sequence

a.) collapses, and

(1.3) Note that the natural map ~,(B~r) ~ H,,(B~r, Z) corresponds to projection onto the (p = n, q --- 0) summand. In order to facilitate future improvements of Theorem 1.1, we first prove the following:

173

LEMMA 1.4. Let 7r be a finite 2-group and let M be a closed connected oriented nmanifold with fundamental group ~r such that w2(2V/) ~ O, n > 5, and the bordism class of M maps to zero in Hn(Bzr, l ) . Then M admits a metric of positive scalar curvature. PROOF OF LEMMA: We need to produce enough manifolds of positive scalar curvature to generate the summands in (1.3) other than the (p = n, q = 0) summand. These are of two types, copies of Hp(Brc, Z) in bidegrees (p, q) with q > 4 divisible by 4, and copies of Hp(Blr, l / 2 ) in bidegrees (p, q) for which flq contains a l / 2 summand. The summands of the first type are no problem, since they correspond to oriented bordism classes (over Blr) of the form N p × Y 4 t ++BTr, where ¢ only depends on the first coordinate, where y4t is a generator for a torsion-free summand in ~4~, where g p L BTr generates a cyclic summand in Hp(B~r, l ) , and where p + 4t = n > 5. Since t > 1, then by [GL2, Theorem C], we may choose y4t to have positive scalar curvature, and then so does N p x y4t for suitable product metric. Consider now the summands of ~,(B~r) coming from H.(zr, l / 2 ) . If a class in H.(Tr, -l/2) is the reduction of an integral class, it can be realized by some N p ~ BTr with N p a closed oriented p-manifold, and as before, the corresponding classes in ~.(B~r) are represented by N p x Y ~ BTr, where ¢ only depends on the first coordinate, where Y is a closed oriented manifold giving a 2-torsion summand in f/.. Since all such Y's can be chosen to admit metrics of positive scalar curvature [GL2, Theorem C], so can N x Y. So it remains to deal with classes in H.(zr, l / 2 ) which are n o t reductions of integral classes. Such classes only occur in even degree and c a n n o t be represented by oriented manifolds mapping into BTr. They can, however, be represented by n o n - o r l e n t a b l e manifolds, since !Yl.(B~r) surjects onto H.(~r, l / 2 ) . Thus consider a class in ~/.(BTr) corresponding to ¢.([Y]) x Y, where g p ~--~ BTr, N is non-orientable, IN] is its Z/2fundamental class, and Y is an orientable manifold giving a Z/2-torsion class in ~'/.. Fortunately, we can construct an oriented manifold mapping into BTr and defining the same bordism class. Namely, observe that the metrics of positive scalar curvature on the standard generators of the torsion classes in fl., the Dold manifolds appearing in the proof of [GL2, Theorem C], admit orientation-reversing (not necessarily free) involutions. If we choose such an involution a ~ on Y and let a be the orientation-reversing free involution on the oriented double cover N of N, then a x a' is free and orientation-preserving, and we have a fibration Y

--,

x

x

--, g .

The composite (N x Y ) / ( a x a') ---, N L BTr now represents our class in ~2.(Bzr) by an oriented manifold of positive scalar curvature. This completes the proof. ! PROOF OF THEOREM 1.1, CONTINUED: Suppose now that 7r is a cyclic 2-group. By the lemma, it's enough to exhibit an oriented manifold of positive scalar curvature

174

corresponding to each cyclic summand in H.(~r, Z). But lens spaces obviously do the trick. | In fact we can improve Corollary 1.2 considerably by allowing a much greater variety of Sylow 2-subgroups. The following two theorems give saznple results along these lines. THEOREM 1.5. I f M " is a closed orientable connected manifold witla fundamental group ~r = Q, ~Ae quaternion group of order 8, and if w2(/171r) # 0 an(:/n > 5, then M admits a metric of positive scalar curvature. PROOF: By Lemma 1.4, it is enough to exhibit an oriented Riemarmian manifold X " of positive scalar curvature, together with a map X " --+ BQ, in every class in H , ( Q , Z), for all n > 5. So we only have to worry about the case of manifolds of the form N n ~ BQ` generating a cyclic summand in H,,(Q, Z). By [CE, pp. 253-254], such summands occur only for n odd. If n = 3 (mod 4), there is only one such summand, generated by a quaternionic lens space, which can be given a metric of constant positive sectional curvature. If n - 1 (mod 4), there are two such summands, each of order 2, and since one can be taken to the other by an automorphism of Q, we only have to worry about one of them. Such a summand is represented by a submanifold of codimension 2 in a quaternionic lens space S4't-1/Q`, dual to a flat complex line bundle. Note that Q` 1, we obtain, t h a t Be~ = el E M (9 N ,

t(S) = ~ f ( B - ' SBei, el) = i----1

= ~ f((B-'pSei, ell + (B -~ (1 - p)Sel, el)). i----1

(1 - p)Sei E N H ~[ = sparta { h o , . . . , h_m+l }, hence B -1 (1 - p)Sei E spanA { C o , . . . , e-m+1 } and the second s u m m a n d vanishes. Also B -1 limp = id, so

t(S) = ~ f(pSel, el) = t(pSp). • i=l

COR.OLLAItY 4.6. If in 4.5 M H-K is orthogonal to N, and {hi} is an A-orthobasis

of M H N, then t(S) = ~7, f(Shi, hi). i=l DEFINITION

4.7.

Let F : HA -+ H,4 be an A-Fredholm o p e r a t o r a d m i t t i n g an

adjoint,

o) F~ HA = Mo H N o

, Mx H N1 =HA

(D)

a corespondent decomposition (see [7]), So, S, are from 3 x E n d ~ H A such t h a t the diagram F

HA

) HA

1So Is, HA

F )

HA

200

commutes. Let us define

,o=(0

onMo ,,:(0

So

on No '

Sl

on MI on N1

and

L ( F , S , D ) = t(~o) - t(.~x). LEMMA

4.8.

Let HA = Mo (D No --~ MI (D N1 = H A ,

(D)

HA = ~o elVo ---+ ~ , m NI = gA

(D)

be two decompositions for F. Then L(F, S, D) = L(F, S, 1')).

PROOF. 1Choose such A-inner products, that/14o and No, M1 and N1 are orthogonal, F : Mo ~ M1 preserves product. Let po : Mo ~ No --+ No, go : ~ro @ No --+ No, pl : M1 @ N1 "-+ N1, J~l : M.1 @ N1 -+ N1 he projectors. T h e n

L ( F , S , D ) = t(SoPo) - t(S~p~), L ( F , S , D ) = t(SoPo) - t(S,p,) = = t(SoPopo) + t(SoPo (1 - po)) - t(Sl~flP, ) - - t(SlPI (1 -- p , ) ) = = t(SoPo) + t(SoPo(1 - Po)) - t(Slp,) - t(S,p~ (1 - p~)), since Po on No = impo is the indentity operator, as well as Pl on N1. So

L(F, S, D) = L(F, S, D) + t(SoPo (1 - po)) - t(S, p, (1 - p, )). W i t h o u t loss of generality (see the proof of 4.5 ) we can t a k e Mo -~ H a and let { e l , e 2 , . . . } be its A-orthobasis. T h e n { F e , , F e 2 , . . . } is an A-orthobasis of M1 and by4.6 t(S~p~(1 -pl)) = ~

f(SlplFe,,Fel)

=~

y(SiFpoei,Fel)

=

i=1

=

i=1

f(FSopoel, Fei) =

i=1 OO

= ~_, (f(FpoSo~e,, Fei) + f ( F ( I - po)Sopoe,, Fei)) = i=l

= ~ (:(pl FSo~o~,, Fe,) + f(O -po)So~e,, e,)) = i----1

= ~ f(So~o~,, e,) = t(So~(1 - p o ) ) . . i=1

201

4.9.

LEMMA

Let HA = (Mo e g o ) e / t o

-----, (M, e g~) @ g~ = HA

(D,)

, M~ • (N1 • K~) = HA

(D~)

and H a = Mo • (No • Ko)

be two decompositions for F. Then L(F, S, D~) = L(F, S, D2). PRoov. We hnve to verify, that t(Sopo) = t(Sap~), where Po : HA ~ No and p~ : HA ~ N~ are the projectors. Choose the metrics such that M s ± Ni ± K i (j = 1, 2) and F : Mo • No ~ M1 (9 Nl preserves A-inner product. Let No C sparta(el,..., e,,,), then N~ C sparta ( F e l , . . . , Fe,,,), where {el } and {Fei } are a-orthobases of Mo • No and M i e N 1 (we assume them to be isomorphic to HA as above). Let qo : HA ~ MoeNo and ql : HA ~ M1 @ N1 be projectors. Then

t(Slpl) = ~

f(SlplFei,Fe,)= ~

i=1

•= ~

f(SIFpoei,Fe,)=

i=1

f(FSoPoei, Fel) = ~ (f(FqoSopoe,, Fei) + f ( F ( 1 - qo)Sopoel, Fei)) =

iffil

iffil

= ~_~ (f(qoSopoei, el) + f((1 - ql )FSopoei, Fel) = i=1

f(Sopoei, el) = i=1

=tCSopo). . LEMMA 4.10.

Let n a = Mo e No

~.M~ e NI = HA

(D)

HA = M-'-'o$ ' ~ o

, -~1 • ~

(-~)

and = HA

be two decompositions for" F. Then L(F,S,D) = L(F,S,-D). So L does not depend on D and we denote it by L(F,S). PROOV. Choose a free module Vo with A-orthobasis e l , . . . ,er, No C Vo, and a free V1 with A-orthobasis h i , . . . , h~, N1 C V1. Choose an arbitrary ¢ > 0 and a projective finitely generated module Ko C/14o, Mo = Lo (9 I(o, such that It(1 - PKoONo )Iv, It < ~o =

I1(1 --PlQ~Na )Iv, II < ~, =

2rl{Soll Ilfll' 2m11S1H 1If11 '

where K1 = F(Ko), LI = F(Lo),

PKo*~o : Lo ~ Ko ~ No --, Ko ~ No, PKI~N~ : L1 ~ K1 @ N1 --* K1 ~ NI

202

are projectors. Consider the decomposition Lo (9 (Ko (9 No)

, L~ (9 ( g l (9 N~).

(DK)

By Lemma 4.5 L(F, S, D) = L(F, S, OK). Take Q1 = PK,¢N,

Qo = p -° No

Ro = Mo N (Ko (9 No),

(N'I),

R1 = M1 N (K1 (9 N1).

When 6 is small (and Ko is "large") w

Qo "V go~

Q1 ,,~ gl~

HA = Mo (9 Qo, Ko (g No = Ro (g Qo,

HA = MI (g Q1, K1 (g N1 = R1 (9 Q1,

where Ro and R1 are projective finitely generated modules. We have

F : Ro ~- *R1,

F : Qo "~ Q1.

Indeed, F : Mo ~ M1 and F : Ko (9 No ~ K1 (9 N1. So

F : Ro = Mo N (Ko (9 No) - , i ~ N (lil (9 N1) = R~. Let x E Qo, then x = Pl~osNo (Y), Y E "No, and

F(x) = FPKo$No (Y) = PK,¢N, F(y) e Q,, since F(y) E "NI. Let

( Lo @ Ro ) (9 Qo

~ ( L1 (9 RI ) (9 Q1

(D,)

and m

Mo (9 Qo ----* M1 @ Q1. Then by 4.9 L(F, S, D1 ) = L(F, S, D), and by Lemma 4.8

L(F, S, Du) = L(F, S, 01 ) = L(F, S, D). Let Po : Mo @ No --* No,

Pl : M1 @ NI --* N I ,

qo : Mo • Qo ~ Qo,

ql : M1 @ QI --* Q1

be projectors. We can assume that sums on the first line are orthogonal.

L(F, S, D2) - L(F, S, D) = t(Sopo) - t(Soqo) - t(S,p, ) + t(S,q,).

(D2)

203

Since Vo± C M'o, V~± C M't, we have Po]vo~ = qolvo~ = O, p~ [v J. = q~ Iv~ = 0 and

L(F, S, D2 )-L(F, S, D) =

i=1

i=1

j=l

+ ~_, f(Slqlhj, hi) = j=l

= ~ f(So(po-qo)ei, e i ) - ~ f(S~(p~-q~)hj,h~) = i=1

j=l

= xZ, (f(So(Po -- qo)poel, ei) + f(So(po -- qo)(1 -- po)ei, el))+ i=1

-- ~ (f(S1 (Pl -- ql )Pl hi, hi) + f(Sx

(PI - qx )(1 - pl ) h i , hi)).

j=l

Since (1 -po)ei E -Mo, (1 - pg)hj E -M~ and qol~, = Pol~'o = 0, q11~'2 = PII~'2 = 0, then the second term in each brackets vanishes. Al:~o (Po - qo)po = (1 - qo)po, (Pl - ql )px = (1 - (h)pl. By the estimate in the begining of the proof we have IIL(F,'S, D2) -

L(F,S,D)II 1.

For n = 1 this yields an easy proof of a result of Olum [9], see (1.15). For n >_ 2 the description of the group (6) gives us the result of Sieradski [11], see (2.8). Our computation of the homotopy category P n / -~ , n >_2 , also solves a problem of Barratt [1], compare the remark following (2.14). In the first two sections § 1, § 2 we describe the main results of this paper. In section § 3 we recall some basic facts on crossed chain complexes which are the crucial tools in our proofs in section § 4. In particular we derive from the tensor product for crossed chain complexes (due to Brown-Higgins [6] ) a formula for the crossed chain complex of the James construction J(X) of a CW--complex X , see (3.5). This formula is essential in our computation of (4) and (5) above, see (4.5) and (4.9). The author would like to acknowledge the support of the Max-Planck-Institut fiir Mathematik in Bonn.

§ 1. The homotopy category of pseudo projective planes

Pseudo

projective

planes,

Pf=M(#/f,1),

are

the

most

elementary

2---dimensional

CW-complexes. They are obtained by attaching a 2-cell e 2 to a 1-sphere S 1 by an attaching map f: S 1 ~

(l.i)

S 1 of degree f_> 1, that is

Pf = S 1 [,If e 2 = D / ~ f .

Here D is the unit disk of complex numbers with boundary S 1 = 0D and with basepoint * = I. The equivalence relation ~f is generated by the relations x ~f y {=4 x f = yf with x,y E S I. Clearly P2 = ~P2 is the real projective plane. Let P be the category consisting of pseudo projective planes Pf and of cellularmaps. We consider the quotient functors

(1.2)

_p__

__-,

209

where we use 0-homotopies (-~) running through cellular maps and homotopies (~) relative * . Moreover, there is a canonical functor (1.3)

r : Pair([]) ~ P

where pair([]) is the category of pairs in the monoid [] of natural numbers. Objects are elements f E [] and morphisms f---og are pairs (~,T/) E [] x [] with g~ -- r/f. Let If,g] be the set of such morphisms (~,~/) : f---og. The functor r carries f to Pf and (~,~7) to the map r e : Pf---~ Pg with r~{x} = {x ~} for x E D, see (1.1). Theinduced homomorphism

(1.4)

~'l(~,r/) = a'l(r~) : ~'1(Pf) = ~[f---~ ~rl(Pg) = I/g

on fundamental groups is given by the number ~/= g~/f which carries the generator 1 E ~]f to ~/. 1 E #/g. Clearly r above is a faithful functor. We now introduce the natural equivalence relation ~ on Pair([]) which is generated by the relations

(~,,fl " (,~' ,,7')

¢=~ ~],r/t - 0 m o d g ,

• i(~,~) = ~'1(,~',,f) = O. (1.5) Theorem: The functor r induces faithful functors

1": Pair([]) >---~ P_J~,

and

r : Pai r([])/~ >---~ PJ_~.

The image category of r in P_J-~ is the subcategory of principal maps in the sense of (V.§3) in Baues [3]. We now define a category tt which is actually a simple algebraic model of the category P__]-~. (1.6) Definition: The objects of the category R are the elements f E []. A morphism ,~ E R(f,g) is an element ,~ E #[#/g] for which there is ~/E ~ with g. e(,~) = f. ~1. Here e : ~[~/g] ~ ~ is the augmentation of the group ring. Composition ~ o/~ for /~ E R(h,f) is defined by (1) ,~ o / z -- ,~ • ,~#(/z) where the right hand side is a product in the group ring ~[#/g]. The homomorphism ,~#[x] = [~/x] is induced by the homomorphism ~#: ~[~/q ---, ~[Z~/gI with

~'1(~) = ,7: ~/f---, ~/g. Let (2)

of= [xJ xE#/f

be the norm element in H[#/f]. We introduce a natural equivalence relation -~ on the category R as follows (,~,/~ E R(f,g)) :

210

(3)

~ ~/~ ~ rl(,~) = 7rl(#) and 3 3 E #[~/g] with 2 - # = A # ( 0 f ) • ~,

(1.7) Theorem: There are isomorphisms of categories p:P__/~ ~ , R , a n d

p:P~

~ "l' ~

/ ~ "

Various results of Olum [9] and Rutter [10] are immediate consequences of this theorem. For to E Hom(~/f,Y/g) let [f'g]to and [Pf'Pg]to be the set of all morphisms in [f,g] and [Pf, Pg] respectively which induce ~ on fundamental groups, see (1.4). By (1.5) the function (1.8)

r : [f'g]la --~ [Pf'Pg]~

is injective for ~, ~ 0 and is identically 0 if ~ = 0. The group of integers ~ acts freely on [f,g] by ( ~ , ~ 7 ) + k = ( ~ + k f , T/+kg) and [f,g]~ is the orbit of (~,~/) with r l ( ~ , y ) = ~ . On the other hand the coaction Pf ~ Pf V S2 induces an action + of the cohomology group (1.9)

E v = t]I2(pf, v*~2Pg) = r2Pg/(r2Pg)- V#0f ,

on the set [Pf, Pg]~ which is transitive and effective. The group 7r2Pg can be described by each of the following equations

(1.1o)

2Pg = H2 g = kernel (e:

= {x e

~[ZZlg]-~

I0g. = o}

~)

= ([0l - [1]) • #[#IS].

Let t : # ~ Eta be the homomorphism mapping 1 to the class of t = f . [ 0 ] - ta#0f E kernel (e) = ~r2Pg. (1.11) Proposition: r in (1.8) is t-equivariant or equivalently ~-(~,~1) + k ' t

= T(~ + kf, ~1+ kg) in [Pf, Pg].

This result follows easily from (1.7). We next derive from (1.7) a result on the erouu_ of homotouv eouivalences_

Aut(Pf)* , in the

211 category P__]_~.Let I be the ideal generated by the norm dement Of in #[#]f] and let Uf be the group of units in the quotient ring #[#/t]/I. Moreover let 0f be the group whose elements are those of Uf but with a multiplication

{~} o {~,} -- {~.~#(~,)}. Here {,~} denotes the class of ,~ E #[#/i~ modulo I . (1.12) Proposition: There is azLisomorphism of groups Aut(Pf)*

T

~

0f.

Proof: Let F,(f) be the group of equivalences of the object f in R_J_. Then we have {~} E E(f) iffthere is # with ~# ~- [0], #,~ ~- [0]. This is equivalent to #.(##,~) ~ [0] and lrl# = (rl,~)-l. This is the ease iff 3/~ with ,u'(##A) = [0] + ~ • # # 0 f ~:4

38 with (,1##). ,~ = [0] + (,~#Z) " 8f

Since the composition in

Aut(Pf)*

corresponds to the composition in

R, we get the

isomorphism for Aut(Pf)*.

// We do not know whether the functor

~'

admits a splitting where ~ splitting of the homomorphism

,,

rCyc

is the category of finite cyclic groups #If, f E ~. However a

~rl: Aut(Pf)* ~= ~ f

** Aut(#/f)

can be constructed as follows. For this we consider the commutative diagram

212

[~,gl

If,g]

(1.13) Hom(#/f,#/g)

t~

g [f,g]l~

Here r is defined for (~,~) E [f,g] by

F(~,~/)=

~ [j-~l]E#[#/gl

j=0 with ~ = lrl(~,r/) and q is the quotient map. (1.14) Lemma: The function P induces a function ~ such that (1.13) commutes. Proof: We have to check that lrl(~,~) = r l ( ~ ' , r / ' ) = ~ implies F(~,r/) ~_F ( ~ ' , r f ) :

~+ f-1 r(~+f,~+g) j=0

f+ f-1 [j. ~11 = r(4,~) + ~ [j. ~1] j=~ f--1

[j. 1]

= F(~,~/)+ ~#([~.1] •

j=0

= r(~,,l) + ~,#([~. 1l • ~#of

// (1.15) Proposition: Let U~ be the group of units x in the quotient ring e(x) = 1. Then we have the split short exact sequence of groups 1

0 ---* U~

~f ~

#[#/f]/I

with

Aut(#/f) ~ 0

where I~f_~ Aut(Pf)* by (1.12). The splitting carries 1oE Aut(~/f) to (---1)#~(~) . (1.16) Remark: Proposition (1.15) is proved by different methods in (3.5) of Olum. It is known that there is an isomorphism of abelian groups

u~ ~ #x • #/f

213

where X is the number of all i E ~, 1 _(i _~ f/2, for which i is not a divisor of f. It is, however, a deep number theoretic problem to determine the action of Aut(E/f) on ]~X • ~/f in terms of basis dements. This action is defined by the splitexact sequence (1.15).

//

§ 2. The homotopy category of suspended Pseudo projective planes

We consider the suspensions

(2.1)

L'n-lpf = M(~/f,n) = Sn Uten

of pseudo projective planes, n ) 1, which are Moore spaces of cyclic groups. Let P----n be the category consisting of the spaces L~n-lpf, f _) 1, and of cellular maps. In section 1 above we studied the category P -- P1 of pseudo projective planes and its homotopy category P__]=. We here compute the suspension functor E : P__n/=---, P_n+l/=, n )_. 1, which is an isomorphism of categories for n )_ 3. For this we consider the commutative diagram of functors Pair (IN)

r

p__]= E p "

(2.2)

~r I

~~p..~3/=

=2/-

-

H2

H3

FCvc where H 2 and H 3 are the homology functors. The next result seems to be new; recall that [f,g] is the set of morphisms f---~ g in Pair(IN), f,g, E IN, see (1.3). (2.3) Theorem: Let ~ E Hom(///f,~/g). Then there is a unique element ~ = B2(~) in the image of ~.T: [f,g] --~ [Epf,Epg] with H2~ = ~. Moreover there is a unique element ~ = B3(~) in the image of ~ 2 : [PpPg]

with H3~ = ~o.

, [E2ppE2pg]

214

(2.4) Corollary: The functors Hn (n = 2,3) in (2.2) admit a splitting functor B n : FCyc --'* Pill-~ with HnB n = 1 This follows immediately from (2.3) since the definition of Bn(la) is compatible with compositions. The splitting functor Bn, however, is not additive; below we describe the distributivity law for Bn( ~ +ior). The functors Hn in (2.2) are part of the following commutative diagram in which the rows are split linear extensions of categories (compare IV. § 3 and V § 3a in [3] )

P21~

E2 + >

,,

E3 + >

~ P__3/~

H2 , ,

FCyc

(zs) H3 ,, FCyc

Here E n is the bifunctor on FCyc given by

(i)

zn(~/f,~/g) =

Ext(~/f,rl2z/g)

where r nI is Whitehead's functor r for n = 2 and the functor

-~/2

for n )_ 3 . The group

(1) is a cyclic group of order (f,2g,g2) for n = 2 and (f,g,2) for n _Y 3 where the bracket (...) denotes the greatest common divisor. The natural transformation or. in (2.5) is induced by the surjection

(2)

® El2

compare (IX. 4.4) [3]. The action of E n on P__n/~ is given by the well known central extension of groups

(3)

Ext(~/f,~n+lU) >i__~ [En-lpf, u]

which is known as the ' ~ r u , ~ # a ~ ~ ~ ' For U = E n - l p g

we have

~ n Hom(i//f, rnU )

compare Hilton [7] or (V.3a) in Banes [3].

~n+l U = Fnl(~/g ). The splitting B n gives us an identification

(n)_ 2)

(4)

HomC~/f,E/g) x En(E/f,~/g) = LEn-lpf, En-lpg]

215

which carries formula

(5)

(~,a)

to

B n ( ~ ) + i(a). The composition in P__n]= then satisfies the simple

(~,~) o (*,Z) = ( ~ * , ~ , ~ + **B).

This indeed yields a very simple algebraic description of the category P_n/=. The suspension functor in (2.5) is given by E(~,a) = (~,~.a). We now consider the image category of the functor E : P_J= ~ _P2]=. Recall that 1 E Elf denotes the canonical generator. (2.6) Definition: For maps u,v : Pf ~ Pg in P we set u - v if Ef_~ Eg. Whence the quotient category P__.]-= is the same as the image category E(P_]=). For morphisms ,~,tt E R(f,g), see (1.6), we set ,~ - ~ if rl(,1 ) = r l ( # ) and if for some fl E E[~/g] with - ~-

~ ( ~ - l,)[0] = ([01 - [1]) •

the greatest common divisor (f,g2,2g) divides g. e(fl). //

The following result shows that the image category E(P__]=) is surprisingly small. By (2.3) we know that the image category EE(P_]=) is isomorphic to FCvc. (2.7) Theorem: The isomorphism p in (1.7) induces an isomorphism of categories

p: sCe_/=) = e_J_=

--,

~--.

Moreover one has a split linear extension of categories

+ >

* ~ FCyc

~ P~/-=

~'1 where ~ is the quotient of E 2 above with ~(E/f,g/g) = g-Ext(Y/f,F(E/g)). This group is g/2 if (f,g2,2g) = 2g and in 0 otherwise. The splitting is given by B 2 in (2.3). We derive horn (2.6) and (2.7) the following commutative diagram in which the rows are split , extensions of groups. Here Aut(X) denotes the group of homotopy classes of basepoint preserving homotopy equivalences of X .

216

(2.s)

EAut(Pf)* =~

~/f >

Aut(EPf)*

t ~l(f,2) >

",

Aut(E2Pf)*

Aut(~/0

,,

Aut(E/f)

*, Aut(~/f)

Using different methods the split extension for Aut(r.Pf)* was obtained by Sieradski [11]. The morphism sets [r.pf,Epg] in P2/~ are groups since the suspension r.pf is a co-H-group. As pointed out in (2.4) the splitting

(2.0)

B2: Hom(~/f,~/g) ~ [EPf,EPg]

is not additive. We now describe the distributivity law for B2( ~ + ~ ) . Let

A: HomCiZlfJZlg) ,, HomCiZlf, lZlg) ~ Ext(~/f,r~/g) ~ ~l(f,2g,g 2) be the linear map which carries the pair (to,~ ~) to the element

A(~,io')= (fCf-1)/2)~1"~~ • 1 where ~(1) = ~11, ~'(1) = ~I. Then we get (2.10)Theorem: B2(~+~' ) -- B2(io) + B2(~' ) % A(~,~') The splitting Bn, n > 3, satisfies the addition law

Bn(~,~ ) = Bn(~) + Bn(~ ) + ~r,A(~,~). This followsfrom (2.5).The formula for A yields the followingproperty. (2.11)Lemma: Let f= 2af0, g = 2bg0 where f0 and go are odd. Then we have A ~0 iff a-b_Yl or a = b + l _ Y 2 andwehave ~,A~0 iff a = b = 1 . Using the identification (2.5)(4) we can describe the group structure [L~-IPf,EU-Ipg], n _Y2, by the formula

(2.12)

(v,~) + (v',~') = (v + v',~ + ~' +

nCv,v

))

+

of the group

217 where An = A for n = 2 and An = ~.A for n >_3. This formula describes completely the additive structure of the category P n / -~. Since A(~,~ ~) = A(tot,~/ we see that also the group [~pf,r, pg] is abe]Jan for all f,g 6 ~/. The cyclic summands and explicit generators of this group are described in the next result. (2.13) Corollary: Let f = 2af0 and g = 2bg 0 where f0 and go are odd. Then the homomorphism H2: [EPf,EPg] ~ Horn(Elf, E/g) = E/d has an additive splitting of abelian groups if and only if (a,b / ~ (1,1/. Moreover for the greatest common divisors d = (f,g) and c = (f,g2,2g) one has

I~/d • ~/c for (a,b) $ (1,1), [r.Pf, lEPg]

! LE/2d • El(c/2) for (a,b) = (1,1).

The generator of the first summand is (~0,(f/4)1) if a > b = 1 and (%,0) otherwise where ~0 is a generator of Hom(ll/f,E/g I. The generator of the second summand is (0,1) if (a,b) ~ (1,1) and is (0,2.1 / if (a,b / = (1,1). Here we use again the identification in (2.5)(4). (2.14) Addendum: The homomorphism

H3:

[r.2Pf, r,2pg]~

Hom(i~If,~/g)

=

~/d

has an additive splittingif and only if (a,b) ~ (1,1).Moreover for e = (f,g2,2) one has [v~pf ,.~ ,~ v.?,pg] =

][E/d • E/e for (a,b) ~ (1,1), LEI2d for (a,b)= (1,1I.

The generator of the firstsummand is (~0,01 and the generator of the second summand (0,I).

E/e is

Remark: The result in (2.13), (2.14) is due to Barratt [2], (table 2 in 10.6). Barratt uses Whitney's tube system for proving this result; his arguments are highly geometrical and totally different from our method. Hilton (p. 1251 presents a different approach for the stable groups

__[L~2Pflr'2pg] and

points out that a more simple minded proof of Barratt's result is needed. A

218

further improvement in the results above is the fact that we describe explicitly generators of the cyclic summands. The algebraic description of the category P n / -~ by (2.5)(5) and (2.12) solves a problem of Barratt [1] who used generators and relations for the description of __.n/P ~ , n )_ 3. Our p ] -~ is simpler and also available for n = 2. algebraic model of __n

// Proof of (2.13): H 2 has a splitting if and only if there is a such that (~0,a) has order d. By the group law in (2.12) we obtain the formula d--1

(1)

(~O,a).d = (O,a-d + ~ A(~o,t~o)). t=1

We choose the generator io0 = ~rl(~,~/) with ~1= g/d, see (1.4). Then we have

(2)

h(~0,tto0) = (f(f-1)/2)T/.tz/. 1

and therefore we have (~0,a).d = 0 iff (3)

ad = (fCf-1)/2)r/r/(dCd-1)/2). 1.

If a > b = 1 we see that ~] and d/2 are odd. Thus a = (f/4)l satisfies the equation (3). Otherwise a = 0 satisfies (3) for (a,b) ¢ (1,1). For (a,b) = (1,1) we have ad = 0 for all a, however, the right hand side of (3) is a non trivial element of order 2 in this case. This proves the proposition. If we reduce equation (3) modulo 2 then both sides of (3) are zero for a > b = 1. This shows that B3(~) yields an additive splitting for if (a,b) ~t (1,1).

// Finally we consider the group structure of the homotopy groups with coefficients in ~/f. As in (2.5)(3) we have the central extension of groups

(2.14)

E/f® n+l U >---, [En-lpf, u] ~rn;' Hom(E/f, nU)

where we identify Ext(E/f, r n + l U ) = E/f @ ~n+l U. This extension is completely determined by the following proposition which completes the partial results on the extension (2.14) in Hilton [7] (page 125-128). (2.15) Proposition: For x,y E [~Pf, U] we have the commutator rule (v = f(f-1)/2)

-- x -- y q- x -}- y = v l @ [i*x,i*y]

219

where i : S 2 C I2Pf is the inclusion and where [i*x,i*y] E r3U is the Whitehead product. Moreover let ~ be the subgroup of Hom(~/f, xnU ) generated by an dement ~o. Then there is a function T : ~ - - ~ [L~U-Ipf~u] with rnT(X ) = x for x E E~ and with (r,s E/~) -T((r+s)~) + T(r~o) + T(sla) = rtvl ® (r/*~l)) where r/: Sn + l --~ Sn is the Hopf element. Proof: The property of the commutator follows from the definition of the Whitehead product and the lemma on the reduced diagonal A : P f - - - - ~ Pf A Pf in (4.10) below, see for example II.l.12 in Baues [4]. Next let E/g be the cyclic group generated by ~1) in ~rnU. Then we can choose a map F : E n - l p g ~ U with i*F = ~(1). Moreover an dement t~ E E~ corresponds to a homomorphism t~a : ~/f--~ ~/g. Now we define T in (D.22) by T(t~) = F.Bn(t~) where Bn is the splitting in (2.4) and (2.10).

//

§ 3 Crossed chain complexes

Let CW be the category of CW--complexes X with X0 = * and of cellular maps. Our main tool for the proofs of the results in § 1 and § 2 is the functor p : CW

(3.1)

, II

which carries X to the crossed chain complex p(X). Here H is the category of totally free crossed chain complexes which are called homotopy systems in Whitehead 1-12], compare also (VI. § 1) [3] where we set D = * and G = 0. The crossed chain complex p(X) is given by the sequence of boundary homomorphisms

...

d4

|

~r3(X3,X2)

d3

!

x2(X2 X 1)

d2

Xl(X 1)

with dn_ld n = 0. The cells of X form a basis of the totally free crossed chain complex pX, that is ~rl(X1) is a f.tee group generated by the 1--cells of X and d 2 is a free crossed module generated by the 2-cells of X , moreover xn(Xn,xn-1), n > 3, is a free ~rl(X)-module generated by the n-cells of X . There is a notion of homotopy -~ for morphisms in H such that p induc¢~ ~ functor p : C ~ , . , ...... ~ ~[-~ between homotopy c ~ o r i e s

Here we use basepoint

220 o preserving homotopies for maps in CW denoted by f-~ g. Let f ~ g through cellular maps.

be a homotopy running

(3.2) Theorem: The functor p induces equivalences of categories

[

p:cw

/o N

f,

p : cw2/

H2/

where CW 2 and It.__2 denote the full subcategories of 2--dimensional objects. Moreover p : CW/~ ~H / ~ induces the map

[X,Y]

p:

, [pX,pY]

between homotopy setswhich is a bijectionif dim X _~2, X,Y E C W . This theorem is an old result of J.H.C. Whitehead [12], it is as well proved in chapter VI of [3], (compare (VI.3.5) and (VI.6.5)). We use the theorem as the main tool in the proofs of § 4. We shall also use the tensor product of Brown-Higgins [6] which gives us a functor ® : H x H ~H H_ such that there is a natural isomorphism

(3.3)

p(XxY) = p(X) @ p(Y).

Here XxY is the product with the CW-topology given by product cells exf. The crossedchain complex A ® B is generatedby elements a O b , a ® * , * e b where a E A , b E B with the followingdefiningrelations(plus,of course,the laws for crossedchain complexes): (1)

]a®b I =

]a[ + ]b[,

la®*[

= [al, ]*®b] = ]b].

(2) (*@b)*@t--*®(b t) for ]t]=l and (aOb)*@t = a®(bt) for It]--l,]b])2, (a®*)s®* = (aS)®* for ]s]=l and (aOb)S®* = (aS)Ob for ls[=l, ]a]_>2.

(3) ( a + a t ) ® * = a ® * + a t

,

0.

(a+a')®b=a®b+a'Ob

for

[a[ >_2,

(a+a~)®b=(a®b)a~®*+a~Ob (4)

*®(b+b')=*@b+*®b' a®(b+b')=a®b+a®b' a®(b+b')=a®

b' +(a®

for

]a] = 1 .

, for Ib[>__2, b) *®b'

for

]b] = 1 .

221

(s)

d(a ® *) = (da) ® *, d(* ® b) = * ® (db) and d(a ® b) =

"-a®*-

* ®b+

a ®* + * ® b

[ - (* ® b ) a ® * +

. ® b] -

(da) ® b + ( - 1 )

lal [ -

(da) ® b +

(-1)[a[a

for l a [ =

a 0 db

(a ® . ) * ® b + a ® * ]

[b[=

for l a[ = 1 ,

Ib[ > 2,

for [a[ > 2

[b[=l,

,

for [al >_2 ,

® (db)

1

]b[ _>2.

Moreover A ® B is totally free if A and B are totally free, a basis of A 0 B is given by the elements * ® b , a ® * , a * b where a and b are basis elements of A and B respectively. The isomorphism (3.3) carries e×f to e@f. Next we consider the James construction which is a functor J : CW ~CW given by the direct limit JX = lira JnX where Jn x = (X x ... x X)/~ is given by the relations (Xl,...,Xn_l,*) ~ (Xl,...,Xt_l,*,xt,...,Xn_l) for t = 1,...,n. It is a classical result of James [8] that there is a natural homotopy equivalence (3.4)

J(X) -~

for X in C W . Using (3.3) we obtain a functor isomorphism (3.5)

J:H

~H

together with a natural

pJ(X) = J p ( X ) .

Let A be a crossed chain complex. The crossed chain complex JA is generated by all words al...a n (a i E A, i = 1,...,n and n > 1) with the foUowing defining relations (plus, of course, the laws of crossed chain complexes). Let u,v be such words or empty words ~ and let a,a r,t E A. Then (a) denotes the word given by a E A .

(i) (2) (3)

l a l . . . a n l - lal[ +.-. + [anl. (uav) t = u(at)v for I tl = 1, l al-~2 and (a)t=(a t) for [ t [ = l . n(a + a')v = uav + u a ' v for l a[ )_ 2 at u a ' v + (nay)

for la[ = 1, lu[ ) 1

(uav) al + u a ' v

for [a[ = 1 , Iv[ )_1

[

[(a) + ( a ' )

for u = ~ = v

222

(4)

d(a) = (da) and d(uv)= u-v+u+v

for lul = Ivl = 1,

U+v-u(dv) (du)v +

for lul =1, Ivl >2,

(_1)I u I (_uV+u)

for lul >2, Ivl = 1,

(du)v + (-l) lu In(dv) For a map

F:A---~B

in

tt

for lul >-. 2, Iv I _> 2.

the induced map

JF:JA----~JB

is defined by

(JF)(al...an) = (Fal)...(Fan). There is a well defined natural map (5)

,u : JA ® JA ~

JA

given by /z(uev) = uv,/~(*@v) = v, p(u®*) = u. Therefore JA is an example of a 'crossed chain algebra'. // One can check that JA is totally free if A is totally free. In fact, if Z is a basis of A then Mon(Z) - * is a basis of JA: Here Mort(Z) is the free monoid generated by Z. As an application of (3.2) we get: (3.6) Corollary: Let

X,Y

be CW--complexes in

CW

with dim(X) _~ 2. Then one has the

binatural isomorphism of groups [EX,EY] '~=[X ,JY] ~= [pX ,JpY] where the group structure in [pX,JpY] is induced by /~ in (3.5) (5). As a special case one gets x3(EY ) = r2(JpY ). A more detailed study of the James construction of crossed chain complexes can be found in Baues [5].

§ 4. Proofs

We here prove the main results of § 1 and § 2. Using (3.2) and (3.6) these proofs turn out to be purely algebraic. This indeed is an advantage compared with the longwinded sequence of geometric arguments of Barratt [2]. For a more detailed discussion of the following proofs see Banes [5].

223

We first observe that the group r2(PflS 1)-

is abelian. Let e 2 be the 2-cell of Pf and let e be

the 1-cell of P f . Then the elements commutators satisfy the formula

e ne 2 ,

d2:

r2(Pf, S1 ) . The

ne me ne me / ne mex / me emex - e 2 - e 2 + e 2 ÷ e 2 = ~e2 ,e2 ] - ~ e 2 ' 2 2"

(4.1) Here

n E ~ , generate the group

<x,y> = - x - y + x + y 8x is the Peiffer commutator which is trivial in the crossed module

~r2(Pf, S1)

,

~-I($1).

(4.2) Proof of (1.7): Since ~r2(Pf, s l ) is abehan we have an isomorphism

h2: r2(Pf, S 1) ~ # [~/f] , this follows from a result of J.H.C. Whitehead [12], compare for example (VI.l.12) in Baues [3]. As a special case of diagram (3) in (VI.I.14) [3] we obtain the commutative diagram

~.2(Pf,S1 )

zz[~/ ]

d2

, r l ( S 1) =

d

'~[ /f]

where d(x) -- x • 8f is given by the norm element 8f in (1.6). The boundary d describes the cellular chain complex of the universal covering of Pf and h 1 is a (~ ~ ~/f)--crossed homomorphism. Using the isomorphism h 2 we can identify the crossed module d 2 = p(Pf) with the map f. • where • is the augmentation of the group ring # [#]f] . We now restrict the functor p in (3.1) to the subcategory P C C W . The functor p carries Pf to the totally free crossed module f. • and carries a map F : Pf--------~ Pg to a map (~,T/) : f. e ~ g. • which is given by a commutative diagram

[~/g]

g .~'

We identify the full subcategory of H._2 consisting of the objects f. e , f E ~ , with the category R

defined in (1.6). The identification carries the morphism

(~,T/) in H 2 to the morphism

224

A = ~

rl]

in R . This proves that

p :

P=./~ ~ D~ is the restriction of the first equivalence in

(3.2). A homotopy a : (~,r}) = (~/,r//) in 2H is an r/--crossed homomorphism a : ~ ~ ~[~/g] which is determined by an element o(1) = ~ as in (1.6) (3). The equation - ~ + ~/ = a(fe) is equivalent to the equation

--~ + A/ ------~[I]+ ~'[1] = a(f) ----a(1) • (~7#Of) which is equivalent to the equation in (1.6) (3).

II (4.3) Proof of (1.5): The functor p in (1.7) carries 1"~ to the element where [0] is the unit of the ring ~[~/g]. By (1.7) we know

• [o] e R(f,g) C [2Zlg]

~'(~,r/) ~_ 7-(~',~/') {=~ la = ~rl(~,r/) = ~rl(~',r/') and 3B e 2z[zz/g] with (~-~')[o] = ~. ta#(•f).

This implies r / - r/' = e(/3).g. We now observe

O)

,#a~---~ [~]=t. xE~/f

X [y] ye~2z/f

where t is the number of elements in the kernel of ~a. For /3 =

~

ay[y] in ~[~/g] we have

yEZZlg (2)

= t • ~ ay[y+v] , yE~/g, vEioI//f

/~"la#Sf

--t.

~

(

~

ay)[u].

uEg/g yEu+ia/Tfl Now ~" io##f = (~'-~)[0] with la= ~'lr~ = ~rlr~, implies

(3)

o=

~

,y ~or u e R/g, u ~ o.

yEu+~a~/f The number of elements in taR/f is g/gcd(T/,g). If gcd(r/,g) < g we add up the equations in (3) for u E U - - {x.1;l

The map • is a fibretion with the fibre isomorphic to the homogeneous space Gl(n,

F)/U(n, F) (see Dieudonn~ [6]). Moreover, since the orbit spaces ~,~,m(F)/u(n, F) and N~,m(F) are separable analytic manifolds, they have the homotopy type of a countable CW-complex (see McCleary [141, Thm. 4.4 or Spanier [191). In 1985 Huynh Mui conjectured that ./is a homotopy equivalence. In other words, on the level of the homotopy theory of the space of reachable linear systems, the topological invariants of the space S~,m(F) are "good enough" in a certain sence. Furthermore, the study of S.,m(F) is not only more convenient, but also (as can be seen later) gives us a simple canonical form. The first purpose of this paper is study the topology of the space S~,m(F) and the second one is to prove the conjecture of ttuynh Mui in certain cases. To study the topology of Sr~,m(F), we use the method of cell decompositic,ns. The paper is divided into 4 sections. Section 1 deals with a canonical form and complete invariants of reachable symmetric linear systems, in section 2 we prove that the canonical projection P : S,~,m(F) ~ S,~,m(F) is a principal U(n,F)-bundle. In section 3 we give a cell decompositions of S,~,,~(F) and show that the integral (rasp. modulo 2) homology group of S,,,,.r,(F)is isomorphic to those of the Orassmann manifold G,~,,,+,~,_a(F)for F = C (rasp. R). Finally, for the case F = C we prove in section 4 that the conjecture of Huynh Mui is true. For convenience, we write .~,,,,~, S.,,,,, ~,,,m, ~.,,,, and GL(n) instead of .~,,,.,(F), S,,,,.(F), ~.,,,.(F), E . , , . ( F ) and CL(n, F). Acknowledgements. I wish to acknowledge my deep gratitude to my teacher, Prof. Dr. Huynh Mui for his inspiring guidance and constant encouragements. I would like to express my warm thanks

237

to DR U. Helmke for his helpful suggestions. It is the author's pleasure to acknowledge Prof. Dr. D. Hinrichsen for his many valuable comments and suggestions. Finally, very special thanks are due to the referee and for helpful suggestions for correcting my English. 1. I n v a r l a n t s a n d c a n o n i c a l f o r m s o f s y m m e t r i c l i n e a r s y s t e m s . First we recall that the scalar product on the vector space F n is defined by

(=,y) := f i =i i i=1

where z = ( z l , . . . ,z,,) E F '~, fl = ( y l , . . . ,z,~) E F '~ and Yl denotes the conjugate of Yl. The Grazn-Schmidt orthogonalization process defines an analytic map from GL(n) into U(n),

where W1

Wj -- E i<j (Wj, Vi)V i

Hinriehsen and Pr~tzel-Wolters gave in [10] the Hermite canonical form for reachable linear systems under a similar action of the general linear group GL(n). Using their method, we introduce in this section a canonical form, which will be called the Hermlte canonical form, for reachable symmetric linear systems under a similar action of the unitary group U(n). Let (A, B) E ,~,,,,, be a reachable symmetric linear system. Let B = [b~, b 2 , . . . , b,~]. Since rank [B, A B , . . . , A~-IB] = n hence in the set of nm vectors Aibj (0 < i < n - 1, I < j < m) we can choose n linearly independent vectors by the following elimination procedure: Going from the left to the right through the n x m-matrix

[bl, Abl,..., A ~-161, b2, Ab2,..., A ~-1 b2,..., b,~, Ab,~,..., A ~-1 b,~] delete all the column vectors which are linearly dependent upon their predecessors. By reachability, the remaining vectors from a basis of F'L They can be ordered in the following way

H = H ( A , B ) : = [bl,Abl,... , A k ' - l b l , . . . ,b,~,Ab,.,,... ,Ak"-lbr~] where kl, k 2 , . . . , k~ are non-negative integers satisfying kl + k2 + - " + k,~ = n and the vector Aki bj is lineary dependent on the system "[bl, Abl,..., A ~'-1 bl,..., bj, Abj,..., At~J-lbj} for 1 < j < m. As in [10], k(A, B) := ( k l , . . . , k,~) is called the list of Hermite indices of (A, B). Obviously k(A, B) := k(TAT -1, TB) for each (A, B) e -~,~,,n and for every T e U(n). So we have DEFINITION 1.1. Let Kn,,,~ := { ( k l , . . . , k,~) e Z '~, kj > 0 and ~ j=l kj = n}. For each k e K~,,,~, the set H(k) := .[(A, B) e S,~,,~k(A, B) = k} is called the Hermite stratum of the space Sn,,~ corresponding to k.

238

I n t h e c o n t r o l t h e o r e t i c l i t e r a t u r e t h e following definition t a k e n f r o m B i r k h o f f a n d Maelane [2](1977) are standard.

DEFINITION 1.2. Let a bc a group action of a group G on a space X , i.e. a : G x X ~ X , ( g , z ) ~ g . z such t h a t e . z = z a n d g ( g ' . z ) = g g ' . z for all z E X a n d g,g' G G, h e r e e d e n o t e s t h e n e u t r a l e l e m e n t of G. G r o u p a c t i o n a i n d u c e s an equivalence r e l a t i o n on X b y z -,, y ¢* 3g E G

such t h a t

y = g • z.

(i) A m a p f : X ~ Y ( Y is a set) is called a n i n v a r i a n t for a if z ,-, y i m p l i e s f ( z ) = f(y) for all z , y E X . I n t h e case of Y = F t h e n f is called a s c a l a r invariant. (ii) A n i n v a r i a n t f is called c o m p l e t e if it s e p a r a t e s t h e o r b i t s of a , i.e. z ~ y ~=} f ( z ) = f(y)

z , y e X.

for all

DEFINITION 1.3. A c a n o n i c a l f o r m for a g r o u p a c t i o n ct on X is a m a p c : X --4 X which satisfies for all z , y E X : c ( = ) ~ ~ a n d • ~ v ¢* c ( ~ ) = c ( v ) .

The following theorem gives the canonical form and the complete invariant family for the similar action of U(n) on a Hermite stratum•

T H E O R E M 1.4. Suppo, e (A,B) ~ if(k), k = (kl,...,k~,) ~ K .... Then (A,B) i, similar by the unitarv group to ezactly one couple ( A k , B~) of the form all

*

0

"

...

•

,

o

a21

N2

0

Bk =

Ak

* aml

N,~

:

:

0

0

0 ...

0

where ~j2 a j2 =

°.°

°°

°°°

°°°

•°°

°•°

°,°

bjk =

aft

row kl + k2 + . . . + kj-1 + 1

239

z j l , . . . ,z./kj are real numbers, aj2,... ,ajkj are positive real numbers• Moreover, ajl is positive if kj >_ 1, ajl = 0 if kj = 0 and the entries * in Bk are elements o f F . Here we set ko := O. The couple (A~, Bk) is called the Hermite canonical form of the couple (A, B). PROOF. First we prove the uniqueness of (A~, Bk). Suppose (Ak, Bk) and (A~,, B~) are of the form as in Theorem 1.4 and they are similar to another, i.e. (SAj, S -1 , SBk) = (Ak,Bk) for some S E U(n). Let I

t

H(A~,Bk) := [b~,A~,b~,

,~,-l~k

• " " ' ~Ak

wl

' " " " '

b~,Akb~,

ak,,,-l~k

" " " ' ~ k

vrttJ"

It can be verified that H(Ak, Bk) and H(A'k, B'~) are the n x n-upper triangular matrices, whose entries on the diagonal are positive real numbers, and H ( S A k S -1, SBk) = S H(A'k,B'k) for every S e U(n). Since ( S A k S - 1 , S B k ) = (A~,B~), so we have H ( S A k S - 1 , S B k ) = SH(A'k,B'k). Because the set of all n x n-upper triangular matrices whose entries on the diagonal are positive real numbers form a subgroup of the general linear group GL(n), hence S is an upper triangular matrix with positive real numbers at diagonal entries, too. Furthermore, S E U(n), we get S = In. The uniqueness is proved. Now we will show that (A,B) is similar to a couple (Ak,Bk). Let x

:= [x[,..

0

•

" " ,Xlrb,"

•

. , "rk~ T I ~ -11 J

be the unitary matrix obtained by the Gram-Schmidt ortogonalization of the system

H(A, B) = [bl, A b l , . . . , A k' -1 b l , . . . , b,~, A b m , . . . , A k''-I b,,~]. T h a t means

X = H(A, B) C

(1.4.1)

where C is the upper triangular matrix with positive real numbers on diagonal. From (1.4.1) we have

X O -1 = [bl, A b l , . . . , A k1-1 b l , . . . , bin, Ab,~,..., A k''-lbm] where C -1 has the form like C. We consider the following cases: (i) If kj > 0 (1 < j < m), then by (1.4.2), bj is of the form

:J

aj 1

*-- row kl + k2 + . . . + kj-1 + 1

0

6 where ajl is the positive real number and the entries * are d e m e n t s of F .

(1.4.2)

240

(ii) If kj = 0, then bj E (bl, A b l , . . . , A k ' - l b l , . . . , bj-1, A b j _ l , . . . , A ki-' - l b j - 1 ) -- ( X ° , .

.,Xkl ~.-1

yki-z--l\

X 0

where the notation ( v l , . . . , v t ) indicates the vector space generated by vectors v l , . . . ,yr. Hence bj is of the following form

bj = X

~- row kl A- k2 A- ... "4- kj-1

where the entries • are elements of F. Thus, it follows from (i) and (ii) that B has the form B = XB~ (1.4.3) Also, it follows readily from (1.4.1) that A X = lab1, A2bl,.. . , A k' bl,. .. , Ab,,~, A 2 b i n , . . . , A k'' bm]C.

(1.4.4)

We will show that the right side of (1.4.4) has the form X A k . Since [bl, A b l , . . . , A j''-I b l , . . . , b,~, A b , ~ , . . . , A k ' ' - I b,~] = X C -1, it follows that [ A b l , A 2 b l , . . . , A k ~ b l , . . . ,Ab,~,A2b,~,,... ,Ak.,,b,~] has the form

{'Y12 ¥1~ "'" Y1,-,-,'~ =: xY 0

(1.4.5)

"-

where Yti (1 < t , j < m) are excluded if either kt = 0 o r kj = 0 and Ytj is the kt x kj-matrix if both kt > 0 and kj > 0. Furthermore, if kj > 0 then l~j has the form

f,

.......

, zjki

where z j 2 , . . . ,zjk~ are positive real numbers. Thus, it follows from (1.4.4) and (1.4.5) that AX = XYC.

241

Since C is the upper triangular matrix with positive real numbers at diagonal entries, we conclude that Y C is of the form like Y. Let

Axa

""

AI,,~

(1.4.6) where Aii has the following form

(,

....... ,

A"=ta 0"" *: i) "'.

(1.4.7)

ajkj

where a j 2 , . . . , aik i are positive real numbers. Since A* = A and X is the unitary matrix, it follows that (re)" = Yc.

Combining (1.4.6), (1.4.7) and the fact (YC)* = Y C , we see that Y C has the form Ak. Thus AX = XAk (1.4.8) From (1.4.3) and (1.4.8) we get that (A, B) is similar to (Ak, Bk). The proof of Theorem 1.4 is complete. • It follows from the Theorem 1.4 that H ( k ) # 0 for every k E K,,,,=. Hence the family {H(k), k e g,,,,,} is a partition of the space .~,,,,,,; S,~,,~ = U keK..= _~(k)By this Theorem we get the following:

COROLLARY 1.5. For every k E Kn,m, H(k) is an analytic submanifold of the analytic manifold S,,,.~ and the map h : if(k) ~ V(n) × g ( k ) ,

(A, ~) ~ (X, a ~ , . . . , a~,,,..., a m , , . . . , ar,,~,,, b L . . . , b~) is an analytic homeomovphism. Here H(k) := R~_ x R n x F g(k) ~ R2'~+da(k),

g(k) := k, + (kx + k2) + . . . + (kx + k2 + . . . + k,,_,), d:=dimaF=l,2,

R+ := {z E R , z > 0},

k0 := 0.

Furthermore, the map (A, B) ---r (Ak, Bk) is the analytic canonical form on ~r(k) and the orbit space ffI(k)/U(n) is analytic isomorphic to 82"+ag(k). Particularly, we have: The case m = 1 =*. S,~,1 ~ U(n) x R 2'~ and S,~,I ~ R2'L

242 The case n = 1 =~ S~,m ~ F × ( F "~ \ {0}) and $1,,~ ~ F x ( F 'n \

{0})Iv(i).

Hence S~,,~(F) is homotopy equivalent to the projective space P,~_~ (F). 2. T h e principal U(n)-bundle structure on S,~,~. In this section we shall prove the following: THEOREM 2.1. The space of reachable symmetric linear systems Sn,m iS an analytic manifold. The canonical projection p : S,~,,,~ ~ S,,,,~ is a principal U(n)-bundle. It is trivial if and only if m = 1. PROOF. Let the Lie group acts analytic and freely on an analytic manifold M so that the graph of the action (i.e. the set Q := {(z,# .z), z E M and g 6 G}) is a closed analytic submanifold in M x M. Then the orbit space M / G is a principal G-bundle (see e.g. Dieudonn6 [6], 16.10, 16.14). Clearly that U(n) acts analytically and freely on ,~,~,,~,. So it surf[ties to prove the following two lemmas: • LEMMA 2.2. The set Q := {((A, B), S . (A, B)), S E U(n) and (A, B) E g,,,,,}

is closed analytic submanifold of S,~,,~ x S,~,m. PROOF. Since U(n) acts analytically and freely on Sn,,n hence the map

q: vcn) × g=,. -- g.,m × (s, (a, B))

((a, B), S. (a, B))

is an analytic embedding and im q _~ U(n) × S,~,,~. Hence Q is an analytic submanifold in ,q,,,~, × g,~,,~ (see Dieudonn6 [6]). Now suppose ((At, B,), (At, B,)) is a convergent sequence in Q. Since U(n) acts analytically and freely, this sequence is of the form ((At, Bt), St.(At, Bt)); limt-..oo(At, Bt) = (A, B) and limt--.oo S , . (At, B,) = (A, B). We have to prove that ((A, B), (A, B)) E Q. We shall prove that the sequence St has a limit in V(n). Denote R(A,,B,) := [Bt,A,Bt,...,A'~-IB,]. Since rank R(At,Bt) = n hence R(At, Bt)R(A~,B,) ° is an n × n-nondegenerate matrix. So we have

St = SiR(At, B,)R(At, B,)* (R(At, Bt)R(A,, B,) °) -1 = RCSt, AtS~ -1, St, Bt)R(At, Bt)* (RCA,, Bt)R(At, Bt)*)-1. Hence llmt-..co St = R(A, B)R(A, B)* (R(A, B), R(A, B ) ' ) - I .

Since U(n) is closed in

CL(,,), we have ( mt-.oo st) e V(n). The proof of the Lemma is completed. • LEMMA 2.3. P : Sn,m ~ S,~,m is trivial principal U(n)-bundle if and only i f m = 1. PROOF. The "if" part follows from the comments after Corollary 1.5.

243

The "only if": If n = 1, we have $1,.~ -~ R x ( F "~ \ {0}) and $1,.~ -~ P,~-a (F) (Sa,m is homotopy equivalent to P,,~-I(F)). If P is trivial then $1,.~ ~ Sa,,~ x U(1). Hence F '~ \ {0} ~- U(1) x P,,~-I(F). This is a contradiction if m > 1. I f n > 1 and m > 1: Let 1 < r < min{n,rn} and let V,.,,,~(F) be the Stiefel manifold consisting of all orthogonal vector r-frames in the vector space F m. Let G,.,,~(F) be the Grassmann roarer.old consisting of all r-.elanes in Fm. For any (A, B) E S,~-r,,,~ with det A # 0 we consider the following monomorphism of principal bundles

0

0

'

Since (V,.,,,~(F), P, G,.,,,~(F)) is a nontrivial principal V(~)-bundle for m > 1, the principal U(n)-bundle (S,,.,~, P, S~,,~) is also nontrlvial for m > 1. The proof of the Lemma is completed. • 3. A n a n a l y t i c cell d e c o m p o s i t i o n o f t h e s p a c e S,~,,~,. 3.1. We equip

K,~,,,, with

the order given by J

J

k kl + k2 + . . . + ki. Hence k __. l. Now in order to show (iii)~(i) we need two following Lemmata. LV.MMA 3.3 Let (At,Bt) be a sequence S,~,,,, which converges to ( A , B ) E S,~,,~. Then for every S E U(n), the sequence ( S A t S -1, SBt) converges to ( S A S -1, SB). PROOF. It is well-known that for S E U(n), then the map ( A , B ) ~ ( S A S - ~ , S B ) is an analytic map from .~,~,r, into ,9,~,r,. Hence the Lemma follows. • LEMMA 3.4. For 1 < j < rn, the eigenvalues of the kì × kì-matriz N i are pairwise different, where N i is indicated as in Theorem 1.4. PROOF. Let z = ( 1 , 0 , . . . , 0 ) be a kì-vector. It is clear that the couple ( N i , z ) is reachable. Hence

rank [~,NÌ~,...,N~'-I~I

= ki.

Since Arj is symmetric, there exists a unitary matrix Sì E U(ki) such that

SìNiS-f 1 = diag (Yl,... ,~/ki ) =: Dj. We put Sjz = ( z l , . . . , zk; ) =: ~'. Since the couple (SiNÌS~q, SiN.i)is also reachable, it follows that kj-1 r~kj

0 # det [~',Dj~',...,zJ i

-1

.--I

z I = det

0

"'.

0 ) det zki

1 I

.

~/ki

.

k;-1 ...

~/ki

The second factor on he right side is a Vandermonde determinant. Hence z t ~ 0 for l 2. Suppose that (A, B) is of the Hermite canonical form:

AI, = d i a g [Na,...,Ni,Ni+a,...,Nm],

Bk = [ba,...,bi,bi+a,...,bm]

and suppose that the n x 2-matrix [bi, bi+a] is of the following form

[bi,bi+l] =

( Bi_~ Bi ~ ~'- rowka + - . - + k i - i +1 ] ~- r o w / c a + . + k l - a + k i + l / 0

'

where the (ki + ki+a) x 2-matrix Bi has the form all

*

0

:

i

•

!

hi+l,1

i

o

o

o

*-- row ki + 1

)

BY an argument analogous to the previous one for the couple ( ( N/

Ni+a

) Bi), '

we can construct a sequence (At, Bt) E ~r(1) such that (A,, Bt) converges to (A, B). The proof of Theorem is completed. • COROLLARY 3.5. (i) For k E Kn,m, ~ ( k ) = U {if(l), l ~_ k} and'-H(k) is an analytic

,ub,et ol g.,.., i.e. fo. = e -~(k), the.e e=i,t, a neighbo=.hooa W in g.,.. of= and the.e ezist analutic functions f l , . . . , ~ : W ~ R such that H ( k ) n W = fl~,=a f~-a(O). In particular, f f ( ( n , O , . . . , O ) ) is open and dense in •.,,, and 9 ( ( O , . . . , O , n ) ) is cto,ea

analytic subrnanifold in S,~,,~,. (fi) The boundary in L,,,, o] each if(k) is the union o.f some if(t) with dim H(l) < d~mH(k). PROOF. (i) By Theorem 3.4 we have ~ ( k ) = U{K(I);/_--< k}. It foUows that

(A,B) ~ H(k) ~, rank [ h , A b l , . . . , A " ' h , . . . , b , A b , . . . , A " - % ]

1.

PROOF. If m = I then the set Kn,a = {(n)} (the set K,~,a has one element) hence the canonical form in Theorem 1.4 is the continuous canonical form. Now we consider case m > 1. Assume the contrary that c : Sn,,n --~ S,,,,, is a continuous canonical form. We shall prove that Sn,,~ is homeomorphic to the product v c n ) x g,,.,Iv(,~). Denote by Q := {((A, B), S.(A, B)), S 6 U(n), (A, B) e g.,.~} the graph of action of U(n) and denote by R(A, B) := [B, A B , . . . , A~'-aB]. Since U(n) acts analytically and freely on .~,~,.~, the map f : Q ---, U(n);

((A, B), (A', B')) ---* R(A', B')R(A, B)* (R(A, B)R(A, B)*)-I

is the continuous map. Hence the map

g: U(n) x S.,,,~ ~ g.,.~,(S, [A,B]) --* S . c(A,B) is the continuous bijection. The inverse is

(A, B) --~(/(c(A,B), (A, B)), [A,B]).

250

Therefore, we have H.(U(n) × S,,,~) ~- H.(S,=,,~). But this is incompatible with H.(Sn,m; G) ~ H.(Gn,n+m-a; G) (G = Z2 or Z) by Theorem 3.10 and 7ri(S,=,m) = 0 for i < dm - 2 by N.H.Phan and L.C.Dung [17], Thm. 2.1. The proof of the Corollary is complete. • 4. A b o u t t h e c o n j e c t u r e o f H u y n h M u i . In this section we study the homology and homotopy relation between the space of reachable symmetric linear systems Sn,m(F) and the space of reachable linear systems ~,=,m(F) and prove that the conjecture of Huynh Mui is true in case F = C. Recall that ~ , , ~ ( F ) = {(A,B) e F ~×~ × F " x ~ ,

( A , B ) is reachable}.

The general linear group GL(n) acts on ~,,,m(F) by

T(A, B) = ( T A T -1 , TB). The orbit space ~,=,,~ = ~n,,~,(F)/GL(n ) is called the space of reachable linear systems. Suppose that inc : S,=,m -* ~ , m is the canonical embedding and J : Sn,,= ~ ~,=,~ is the map which makes the following diagram commutative

(4.0) d

The following Theorem is main result of this section: THEOREM 4.1. If F = C then J : S,~,,~ ~ ~,~,,,~ is a homotopy equivalence. First we consider some topological properties of the space ~,,,~(F). The following Theorem has been proved by Helmke [7] in the complex case (see Helmke [7], Thm 2., p. 68 and Thm. 4, p. 135). THEOREM 4.2. If F = C then: (i) ~,~,,,,(F) is an analytic manifold of dimension 2nl"n.

(fi) For k E K,~,,~, let tier(k) := {(A,B) e }],,,,=(F), k ( A , B ) = k} and let Her(k) = I I e r ( k ) / G L ( n ). Then the family { H e r ( k ) , k E K,,,,,~} is a finite analytic

cell decomposition of ~,~,,~,(F) and direr Her(k) = 2n + 2g(k), where g(k) is defined as in 1.5. i.e. g(k) = ka + (ka + k~) + . . . + (k, + k, + . . . + k,,,_a). PROOF. See Helmke [7]. • We now prove Theorem 4.1. PROOF of Theorem 4.1. Since S,,,,~(C) and ~,~,,~(C) are separable manifolds hence orbit spaces S,~,,~(C) and ~,,,,~(C) are separable manifolds, too (see Dieudonn6 [61, 12.10.9). It follows that they have the homotopy type of a (countable) CW-complex

251

(see McCleary [14]). Hence by Whitehead Theorem (see McCleary [14] or Spanier [191), we need only to prove that J induces art isomorphism on integral homology. Let H q (I3,,,,~(C); Z) and H~(Sr~,m(C); Z) be respectively the q-th Alexander-Spanier cohomology groups with compact support of ~=,,~(C) and S~,,~(C), where the coefficients are taken in Z (see Massey [13]). Denote by L' = U {H(k), d i m . H(k) g q}

and

Qq = [.J {Her(k), d i m . Her(k) < q} the q-th skeletons of S.,.,(C) and I3,~,.,(C) respectively. Let Cq(S,~,,~,(C)) := Hqc(Lq \ Lq-~; %) C' (S,.,,~(C)) := H~(Q q \ Oq-l; Z).

By Massey [13] Thin. 3.3, H~(S=,.~(C); Z) and H~ (I].,.~(C);Z) are respectively the q-th cohomology groups of cochain complex with trivial boundary operations

c'(s~,..(c)) = {c~(s.,.~(c)), o}

and

C'(S~,~(C))

=

{c~(r~,~(c)),o}

(note that all cells of Sr,,m(C) and of ~,~,,~(C) have even dimensions). Hence we have

H~, (s.,..(c); z) = o~ (s.,..(c); z) = @ {H~ (HCk); Z), dim. H(k) = q} k

= ~ {Z, dim. H(k) = q}, k

and

k

= ~ {Z, d i m . HerCk) = q}. k

For every k 6 K,,,m, we have J(H(k)) C Her(k) and d i m . H(k) = d i m . Her(k) = 2~ + 2g(k) = = 2~ + 2(kl + (kl + k2) + . . . + (kl + k, + . . . + k ~ _ l ) )

by Corollary 1.5 and by Theorem 4.2. Hence we get Jk :=

Jill(k): H(k) -~ Her(k)

induces an isomorphism on cohomology groups. It follows that J" : H~(~,m(C); Z) -~ H~(S~,~(C); Z) is an isomorphism for every q.

252

Since ~,,m(C) and Sn,m(C) are simply-connected (see Helmke [7] and N.H.Phan and L.C.Dung [17], Thm. 2.1 and Thm. A') hence they are orientable manifolds (see Dold [5], VIII, 2.12 or Spanier [19]). Furthermore, dimR S,~,,~(C) = dimR ~,~,m(C) = 2rim. Hence applying the Poincare duality Theorem (see Massey [13]), Whm. 11.2) we have Hq(~n,m(C); Z) ~-- H2er'm-q(~n,m(C); Z) and

z)

z)

Thus

J,:

z)

Hq(z,,,.(c); z)

is an isomorphism for every q. The proof of Theorem 4.1 is completed. • REMARK 4.4. By the diagram (4.0) and by Theorem 4.1, the inclusion map inc : -~n,m(C) --* ]~n,m(C) induces the isomorphism of homotopy groups. Further, since Sr,,ra(C) and ~n,m(C) have the homotopy type of (countable) CW-complexes (see McCleary [14]) hence the map inc is a homotopy equivalence by Whitehead Theorem. REFERENCES

[1] M. Aigner: Combinatorial theory. Grundlehren der Math. Wissenschaften 234, Springer 1979. [2] G. Birkhoff and Maclane : A survey of modern algebra. Macmillan, New York, 4th edition 1977. [3] A. Borel and A. Haefliger: La classe d'homologie fondamentale d'un espace analy. tique. Bull. Soc. Math. France, 89, 461-513 1961. [4] R. Brockett: Some geometric questions in theory of linear systems. IEEE Trans. Autom. Control AC-21,449-455, 1976. [5] A. Dold: Lectures on algebraic topology. Springer-Vertage, Berlin Heidelberg, New York, 1972. [6] J. Dieudonn6: Foundations of modern analysis. Vol. 3, Academic Press, New York, 1972. [7] U. Helmke: Zur topologie des Raumes linearer Kontrollsysteme. Ph. D. Thesis, Report 100, Forschungsschwerpunkt Dynamische Systeme, University of Bremen, West Germany, 1982. [8] U. Helmke and D. Hinrichsen: Canonical forms and orbit spaces of linear systems. IMA Journal of Mathematical Control and Information, 3, 167-184, 1986. [9] D. Hinrichsen: Metrical and topological aspects of linear control theory. Syst. Anal. Model. Simul. 4, 13-36, 1987. [10] D. Hinrichsen and D. Prgtzel-Wolters: Generalized Hermite matrices and complete invariants of (strict) system equivalence. SIAM J. Control and Optimazation 21, 289-305, 1983. [11] D. Hinrichsen, D. Salamon, A. J. Pritchard, P. E. Crouch and e.a.: Introduction to mathematical system theory. Lecture Notes for a Join Cource at the Univerities of Warwick and Bremen, 1980.

253

[12] R. E. Kalman, P. L. Falb and M. A. Arbib: Topics in mathematical system theory. McGraw-Hill, New York, 1969. [13] W. S. Massey: Homology and cohomology theory. Marcel Dekker, New York, 1978. [14] J. McCleary: User's guide to spectral sequences. Publish or Perish, Inc. Wihning, Delaware (U.S.A.), 1985. [15] J. Milnor and J. Stasheff : Characteristic classes. Princeton University Press, 1974. [16] N. H. Phan: Topo c¢~a kh6ng gian edc h~ thSng tuy~n t(nh d6i z~ng. TAP CHI TOAN HOC, Vol XV, No 1, 26-31, 1987, (in Vietnamese). [17] N. H. Phan and L. C. Dung: On the topology of the space of reachable observable symmetric linear systems. To appear in the Report Series of Forschungssehwerpunkt Dynamische Systeme, University of Bremen, West Germany. [18] V. M. Popov: Invariant description of linear time-invariant controllable systems. SIAM J. Control, 10, 252-264, 1972. [19] E. H. Spanier: Algebraic topology. McGraw-Hill, New York, 1966.

Homotopy Ring Spaces and Their Matrix Rings R. Schw£nzl and R. M. Vogt 0. I n t r o d u c t i o n

This paper is an account of methods and (elementary) results we have used without giving details since 1984 (e.g. see [FSSV], [FSV1], [FSV2]). Utilizing a result of R.. Steiner [St2; Lemma 1.7] we in [SV1] altered P. May's definitions of Aoo and Eoo ring spaces [M2], [M3] to a homotopy invariant one, and could show that a ring space in this new sense can be converted up to coherent homotopy in one of May's sense without changing its homotopy type. This new definition is flexible enough to allow to carry out a large number of classical constructions of basic ring theory in the Aoo and Eoo setting in a fairly straight-forward way~ a point which May discussed rather pessimistically [M3; Introduction, Remarks 10.3 and 12.4]. One aim of this paper is to demonstrate this transfer from classical ring theory to the Aoo world for constructions involving matrices. In Section 1 we give definitions of Aoo and Eoo monoids and rings which are most suited for our purpose and compare them with previous definitions in the literature. Homomorphisms between homotopy ring spaces are too rigid for our theory; in particular, the notion of homomorphism is not homotopy invariant. There are two possible substitutes introduced and shortly discussed in Section 2. In Section 3 we construct the Aoo rings M,,X of n-squared matrices over an Aoo ring X. This cannot be done in May's original setting (see [M3; § 4]). Special cases have been treated previously by K. Igusa [I]. For the generalization of triangular or diagonal matrices we have two options which we introduce and compare. Finally we study the multiplicative Aoo submonoid GI,,X of homotopy invertible n-squared matrices over X. The methods of Section 3 are applied to give a plus-construction definition of the algebraic K-theory K X of an Aoo ring X using a suitable stabilization sequence of GI, X. This definition is in the spirit of [M3] but can do without technical arguments. The alternative approach to K X due to Steiner [St2] will be the subject of [SV6] (see also [SV2]). There we among other things prove Moritainvariance for K X and show that K X of an Eoo ring X is itself an Eoo ring, extending a result of May [M3; Prop. 10.12] who proved this to be true for genuine commutative topological rings. Acknowledgement: The work on this paper has partly been supported by the DFG.

255

1. H o m o t o p y m o n o i d s a n d h o m o t o p y rings

As pointed out in the introduction, many of the concepts of classical monoid and ring theory carry over to the homotopy coherent case if they can be expressed in universal terms, i.e. if they are independent of the particular monoid or ring under consideration. It is the purpose of this section to set up this "universal" language. Throughout this paper we work in the category Top of compactly generated spaces in the sense of [V]. Products, subspaces, function spaces etc. are taken in this category. The corresponding homotopy category is denoted by Toph. The language we are going to use is the one of universal algebra, i.e. we are going to codify (topological) algebraic structures universally by exhibiting the category of all operations which can be written down in the given algebraic structure. Such categories are called theories (see definition below). We always have some operations for free, called set operations: Let S be the category of finite sets n = {1,2, 3, ...}, n >_ 0, with 0 = ~ and all maps. For each space X and each a E S(m, n_n_)we have the set operation

(1.1)

x

x

(=1,..., =,)

1.2 Definition: A theory is a category ® with objects 0,1,2,..., topologized morphism sets O(m, n), and products, together with a faithful functor S °p C ® preserving objects and products. Composition is continuous, and the canonical map {3(m, n) --+ {3(m, 1)" is a homeomorphism. A {3-space is a continuous functor X : (3 ~ Top such that S °p C {3 ~ Top is product preserving. The space X(1) is called the underlying space of X. A homomorphism of {3-spaces is a natural transformation of such functors. A theory functor 0i ---* {32 is a continuous functor of theories such that SaP

/ {31

\ ~ @2

commutes.

We will often find it convenient to overlook the distinction between X and X(1) and refer to X when we mean X(1) and vice versa.

1.3 E x a m p l e s : By definition~ S °p is initial in the category of theories, and each topological space is an S°P-space by (1.1). The first non-trivial example is the theory Om of monoids: ®,,(n, 1) is the free monoid on n generators Xl,..., x~, and O,~(n, k) = (Ore(n, 1)) k. Composition is defined by substitution;

256

e.g.

o ( : x : s , x2 3) =

3

2

1

A monoid X determines and is determined by a O~-space 0 ~ ~ Top, given by sending a word w(xl, ..., x,~) in n generators to the operation X" -~ X,

(z~, ..., z.) ~, w(z~, ..., z,)

where the word is evaluated in X. O,~ and the theories 0 ~ of commutative monoids, Or of semirings, and O~ of commutative semirings, which are defined similarly, will play the central role in this paper.

We are now going to define the homotopy coherent analogues of monoids and semirings. In 0,~ a morphism from n to k is a k-tuple of monomials (1.4)

ziar I • ....:ripr p

in n non-commuting variables, in O,,,, it is a k-tuple of monomials

(1.5)

x ? . . . . z;,"

in n commuting variables, in Or it is a k-tuple of finite sums of monomials (1.6)

rp l. ziar 1 • .... zip

of type (1.4) with coefficients l E / N (0 is contained in fie), and in 0 ~ it is a k-tuple of finite sums of monomials

(1.7)

I. x rxl

. ...

•

X~ n

of type (1.5) with coefficients l E / N . Such a morphism is called simple if all coefficients l are 0 or 1 nnd if in the commutative cases all exponents r~ are 0 or 1. In particular, all morphisms of 0,~ are simple. In the following definition 0 . is O,,, O~,,,, Or, 0 ~ , whatever is appropriate, and sO. its subset of simple morphisms.

1.8 D e f i n i t i o n An A~ o r / ~ monoid or ring theory is a theory 0 together with a theory functor F -- Fe : 0 --* O. such that (1) obO C m o r e is a closed cofibration (2) F : raor0 --* m o r 0 . is bijective on path components and a homotopy equivalence over s 0 .. A O-space is called an A~ or E ~ monoid or ring.

1.9 C o m p a r i s o n of definitions: The definitions of A~ or E ~ monoids in [BV1], [BV2] and of A~ a n d / ~ rings in [SV1] require less stringent conditions of the augmentation functor. As theories ®,~ and ® ~ are generated by the simple morphisms

257

(1.10)

#,=zl.x2.....x,,

n>0,

and # 0 = 1

subject to the obvious relations. Let A C @,~ and CA C O n be the subcategories generated under composition and taking products by the #= and, in the commutative case, by the permutation set operations. In [BV1], [BV2] a theory augmented over O. = O,~ or e ~ , F : O ~ O. is called an Aoo resp. E¢0 monoid theory if F is a homotopy equivalence over A resp. CA. This condition is weaker than (1.8) because A and CA consist of simple morphisms only. The subcategory B = F-I(A) resp. F-I(CA) of e is the spine of a split theory O~r over 0,~ resp. Ocm in the sense of [BV2; p. 58 ff.]. The identity on B extends to a theory functor G : ON ~ 0. It is easy to show that Fe o G is a homotopy euqivalence over simple morphisms. Hence any Aoo or Eoo monoid in the sense of [BV1], [BV2] is in a canonical way an A¢o or Eoo monoid in the sense of (1.8) by substituting O by ON via G. The theories Or and 0,~ are generated by the simple multiplicative operations #, of (1.10) and the simple additive relations (1.11)

A,=xl+...+x,,

n>0,

A0=0.

The subcategories generated by the #, and A, under composition and product contain non-simple morphisms, which complicates the picture. Let R C Or and CR C e= be the subsets consisting of/~0 and the morphisms of the subcategories generated under composition and product by the A,,n > 0, the l.tk, k ~ 1, and the permutation set operations. In [SV1] a theory augmented over O. = Or or O~, Fe : O --* O. is called an Aoo resp. E¢0 ring theory if Fe is surjective and a homotopy equivalence over//resp. CR. We could show [SV1; Thm. 5.1] that there is a universal Ao~ resp. Eoo theory L/derived from Steiner's canonical operad pair [Stl] and a functor G :/2 ~ O of Aoo resp. Eoo ring theories. This G is unique up to contractible choice; i.e. there is a canonical contractible space of Aoo resp. Eoo theory functors/d ~ 0. Since L/is an Aoo resp. Eco theory in the sense of (1.8) ([St2; Lemma 1.7] and [SV1; Prop. 2.5]) any Aoo or Eoo ring in the sense of [SV1] is one in the sense of (1.8), canonically up to contractible choice. Moreover, by [SV1; Cot. 5.2], any/d-space X is homotopy equivalent to a space Y on which Steiner's operad pair acts in the sense of May [M2]. Since any Aoo or Eoo operad pair in the sense of [M3] or [M2] gives rise to an Aoo or Eoo ring theory by [St2], any A¢¢ or Boo ring in the sense of May is one in the sense of (1.8), while any Ac~ or E¢¢ ring in our sense is homotopy equivalent to one in May's sense. There are more combinatorial definitions of Aoo or E~ monoids and rings, the A-spaces and F-spaces of Thomason [Th] and Segal [Se] in the monoid cases and Woolfson's hyperF-spaces in the Eoo ring case [Wo]. By various replacement procedures ([M4], [MT], [SV3]) these functors are homotopy equivalent to Aoo resp. Eoo monoids and rings, and vice versa.

1.12 R e m a r k s : One may think that in Definition 1.8 it would be more convenient to postulate that Fo be a global equivalence. Unfortunately this cuts down the range of examples in the commutative monoid case and hence also in both ring cases basicly to

258

products of Eilenberg-MacLane spaces, because by [BV2; Thm. 4.58] a Q-space would then be weakly equivalent to a topological abelian monoid resp. a (commutative) topological semiring, which excludes important examples such as stable homotopy, complex bordism [M2], algebraic K-theory of Boo rings [SV2] etc. For similar reasons we have to augment our theories over commutative monoids and semirings rather than abelian groups and rings: Let Ocg be the theory of abelian groups. Then Oc,,, C 0~g. In [SV4] we showed

(1.13) Let F : O ~ Ocg be a theory functor which is bijective on path components and defines an Eoo monoid theory upon restriction to Oc,~. Then every Q-space is wealdy equivalent to a weak product of Eilenberg-MacLane spaces.

1.14 R e m a r k : Since the standard CW-approximation functor preserves products and contractability, we may assume that our A¢o and Eoo monoid and ring spaces have structures codified by CW-theories which are contractible over simple morphisms. Moreover, up to weak equivalence, we can substitute our Aoo and Eoo monoids and rings by their CW-approximations. The passage to CW-theories takes automatically care of the technical requirement that obO C morO be a closed cofibration.

2. H o m o t o p y h o m o m o r p h i s m s and h a m m o c k s

As map between Ao~. and Eoo monoids and rings we have two options: Homotopy homomorphisms, studied !in detail in [BV2] in the monoid cases and in [SV1] in the ring cases, and hammocks, introduced in [DK]. Our definition of homotopy homomorphisms below is equivalent to the ones in [BV2] and [SV1] but avoid the use of universal constructions. A homotopy homomorphism can be replaced by a hammock (see [SV5]). While homotopy homomorphisms arise naturally when one translates standard linear maps such as the stabilization Gl~R ~ GI,+IR of the general linear group to the Aoo and Eoo world, functoriality of our constructions can be described more easily in terms of hammocks.

H o m o t o p y h o m o m o r p h i s m s : In general we have a family {Xk; k E K} of Aoo or Eoo monoids or rings. Let Si p be the category of set operations on Xk. Since X~ = * for all k E K, the category of set operations of the whole fazr~ily is op

kEK

with the objects O_ E Si p, k E K, all identifiedto a single terminal object O. W e denote the object n_ E Sk by (n, k).

259

2.1 Definition: A K-coloured theory is a category O with obO = obSg together with a faithful functor S ~ C 0 preserving objects and products. The morphism sets of 0 are topologized, composition is continuous and the canonical map 0((m, k), (n, l)) [0((m, k), (1,/))]~ is a homeomorphism. A theory functor from a K-coloured theory 01 to an L-coloured theory 02 consists of a map f : K --* L and a continuous functor G : 01 ~ 02 such that fB

,

f3

f3 F

O1

'

O2

commutes.

The definitions of O-spaces and homomorphisms extend to the K-coloured case in the obvious way.

2.2 E x a m p l e : Let 0 be a monochrome theory and D a small category. A D-diagram of 0-spaces is codified by an obD-coloured theory 0 o D defined as follows:

(O o D)((m, dl), (1, d2)) = O(m, 1) x D(dl, d2)'. An injection a : 1 --* m__in S~ C SobD is mapped to (a*; idd, ..., idd) which specifies the functor S~b °PO ~ O o D. Composition of c = (a; fl,..., f,~) £ (6) o D)((m, dl), (1, d2)) with (Cl, *.., C-~n) 6 (O 0 D)((n, do)(m, dl)) is defined by (a o (bl, ..., b,n); f l o g n , . . . , f ,

o gx,,,

..., f,,, o g,,,1,..., f,,, o g,,,,~)

where c~ = (bl; ga, ..., gl,). This determines 6)D completely. It is easy to check that a ( 0 o D)-space is a D-diagram of O-spaces and homomorphisms and vice versa.

2.3 R e m a r k : In [SV1] we used the ambiguous symbol 0 x D for G o D. Note that 0 o D is the quotient of the usual product O × D of categories, obtained by identifying all objects (0, d) to a single terminal object 0. The results of [SV1] apply to O o D.

In the following definition 0 . is phisms.

0,~, On, Or or 0,~, and s 0 . its subset of simple mor-

2.4 Definition: Let D be a small category. A D-indexed Aoo or Eoo monoid or ring theory is an obD-coloured theory O with an augmentation functor F = Fo : O ~ 0 . o D of obD-coloured theories such that

260

(1) obO C morO is a closed cofibration (2) F preserves objects (3) F is bijective on path components and a homotopy equivalence over sO. o D. A O-space is called a D-indexed Aoo or Eoo monoid or ring. To subsume the four definitions into one, we call 0 a theory over O. D.

Categories D of special interest for us are L , : 0 -~ 1 -+ 2... -+ n. For constructions generalizing the bar construction D = A °p, the simplicial index category, is of importance (see [FSSV]). Since a homomorphism of O-spaces, 0 a monochrome theory, is simply a (3 o Ll-space, we are led to the following definition.

2.5 Definition: Let Oi be theories over O. and Xi a O~-space, i = 0, 1. An h-morphisms (or homotopy homomorphism) from X0 to )(1 consists of theories ¢i over O . , i = 0, 1, a theory ~ over O. L1, theory functors Pl

qi

¢,

,

= F

I(O.

{i))

i = 0,1

over 0 . , and a qi-space G : q2 --* Top such that ql

1,, Oi

Xi

1o

i=0,1

) Top

commutes. Any map H ( a ) : X0 =-+ X1 with a E F~l(((idl; 0 -=+1)) is called an underlying map of the h-morphism. Although the theories ~i, ~i and Oi might have very little to do with each other, the functors Pi and qi are homotopy equivalences over sO,. This definition allows to compare spaces structured by different theories without the need to introduce the universal theory b/. Using the universality of U it is immediate to relate this definition to the ones of [BV2; 4.2] and [SV1; 3.1]. We recall one important fact from [SV1].

2.6 P r o p o s i t i o n : Let 0i, i = 0,1 be theories over 0 . , let Xi be a 01-space and H : X0 --* X1 an h-morphisms. Let h : X0(1) -+ )(1(1) be an underlying map and a homotopy equivalence. Then any homotopy inverse of h is the underlying map of an h-morphism X1 ---) X0. We say that X0 and )(1 axe homotopy equivalent as Aoo or Eoo monoids or rings (whatever applies).

H a m m o c k s : Fix a theory 0 over 0 , . A homomorphism f : X =-+ Y of O-spaces will be called a weak equivalence if f ( 1 ) : X(1) --* Y(1) is a homotopy equivalence.

261

2.7 Definition: A hammock of length n >_ 0 and width k _> 0 is a commutative diagram of O-spaces and homomorphisms

Zoa

Zo,2

. . . . .

Zo,,,

Z1,1

Z1,2

. . . . .

ZI,,, Y

I

I

Zk,~

Zk,~ . . . . .

1

Zk,.

such that (1) all vertical maps are weak equivalences (2) in each column all maps go in the same direction. If they go to the left, they are weak equiwlences (3) the maps in two adjacent columns go in different directions (4) no column contains only identity maps.

After restriction to a sufficiently large set of G-spaces the hammocks form a simplicial category. Its k-simplices are the hammocks of width k. The i-th degeneracy repeats the i-th row, the i-th face omits the i-th row. It can happen that the resulting "hammock" falls to satisfy (4), but we can reduce it to a genuine hammock by omitting a column if it contains only identities and then compose columns to establish property (3), and then carry on if necessary. Hammocks are composed in the obvious way, and the composition is associative and commutative.

3. M a t r i c e s over Aoo ring spaces

Throughout this section let 0 be an Aoo ring theory and X a O-space. Since Fe : e ~ e r is bijective on path components, X defines a semiring in the homotopy category [X] : e ,

~ Toph

262

such that

X

e

Top proj.

[Xl

er

Top/,

corn_mutes.

Let M,~X denote the space of all n-squared matrices with entries in X. functor M~ : e r --* e~

We define a

such that IX] o M~ is the semiring of n-squared matrices over [X] in Toph: The category S of sets n.n_comes equipped with a canonicM sum and a canoni- cai product. We identify m U n with m + n in blocks and m x n with m . n via lexicographicai ordering of pairs. The functor M~ sends 1 E Or to n 2 = n x n. Think of n 2 as the entries of an (n x n)matrix in lexicographical ordering. A set operation a* : q ~ p is sent to the set operation cr* = M~(a*), where o-: ~

= ( a x ~_) u ... u ( a x ~_) ~ (~ x ~_) u . . . u (~ x ~) = qn__~

maps the i-th summand n × n identically to the a(i)-th summand. In matrix terms, a" is given by the set operation a" applied to a q-tuple of (n × n)-matrices. The operation xl + ... + xp is mapped to the p-fold matrix addition pn 2 ~ n 2 whose (i,j)-th component is • ~½+ ... + ~ where the superscript k stands for the k-th summand n x n in pn 2. Similarly xl ..... xp is sent t o the p-th fold matrix multiplication. This determines the functor M~ completely. Now form the pullback theory M , e : M~e

,

e

1 1

e,

M,,

~ ®r

Since M~ maps simple operations to simple operations, F : M . O ~ e r is an Aoo ring theory, and we obtain

3.1 T h e o r e m : Let • be an Aoo ring theory and X : ® --~ Top an Aoo ring. Then M~O is an A¢o ring theory and M,~X = X o M,, : M,,O .--* Top endows the space of (n x n)matrices over X with an Aoo ring structure such that [M,,X] : O, ~ Toph is the usual

263

matrix semiring over [X]. The correspondence X ~-~ M~X is functorial with respect to hammocks.

To treat the Aoo analogues of upper or lower triangular or of diagonal matrices let X, be the space X(F¢I(Ao)) C X(1), i.e. the space of zeroes in X (which is not the full path component of 0 in the semiring ~'0(X)). Let U~X, L*~X and D~X be the subspaces of all matrices in M~X having elements in X, below, above, respectively below and above the diagonal. We have

(3.2) The subspaces U,~X, L~,X, andD~X of M,,X are M~O-subspaces of M,X. In particular, they are Aoo rings.

3.3 R e m a r k s : Although Fgl(A0) is contractible, Xz need not be, so that U,~X,L~,X, and D~X may not have the desired homotopy type of X½"("+I) resp. X". We are going to construct a functor U,, : 0r + Or similar to M,,, such that IX] o U, codifies the upper triangular matrix semiring with the zeroes below the diagonal ignored. By [SV2; Thm. 4.12], any Q-space X is equivalent as Aoo ring to a Q-space Y such that Y : O(O, 1) ~ Top(*, Y(1)) = Y(1) maps F~I(A0) and F~l(#o) homeomorphicaily onto their images. The homotopy equivalence is given by a O-homomorphism Y ~ X and a homotopy homomorphism X -* Y. So up to homotopy, we can always arrange that Xz ~ *. In this case U~X is equivalent as A~o ring to U,X to be constructed.

The functor U. : Or -~ Or maps is given on objects by 1 1 1 p ~-* p- ~ n . (n + 1) = ~n(n + 1) U ... U ~nCn -t- 1). Each summand should be considered as the ln(n -t- 1) interesting entries of an upper triangular (n x n)-matrix. Set operations are mapped in the analogous way to the M,-case. Addition xl +... + xp is mapped to componentwise addition again, but the multiplication xl ".... xp is mapped to the operation whose (ij)-th component, i _< j, is 1 X2 Xlrl r l r 2

iAb

described above.

Let

be a contravariant functor from ~ to the category Ab of

abellan groups.

Then our approach gives a simple proof of the unpublished

result of Jackowski and

Se~ai which asserts that there exists a

spectral

sequence:

E p'q =

lim p aes (~) o

]] Hq(Aut(m),Npg(m)) ~ [m]~EIsAf(a)

limp+q N ,

w h e r e A u t ( m ) i s t h e a u t o m o r p h i s m g r o u p o f t h e o b j e c t m o f A{(a). In

the

case

was c o n s i d e r e d group

Aut(y)

sequence

where

by the acts

described

M is author

freely

an in

on the

orbit [ 11].

category, If,

morphism

for set

a any

similar

two o b j e c t s

M(x,y),

i n 1 7 . 1 8 o f [ 10] h a s t h e s a m e

spectral

then

x,y

the

sequence o f 4,

the

spectral

Ep ' q - g r o u p s .

In Section 2 of this paper we investigate an example of an E-I-category. Let G be a finite group and let Sub-G be the poser of all subgroups of G. Assume that W is a poset and that G acts on W preserving order. d:W

>Sub-G be an equlvariant poser map such that, for each element w of

W, d(w) is a subgroup of the isotropy group Gw= {g~G: form

Let

gw=w }. Pairs of the

(W,d) are considered for example in [3] and are called there G-posers. We associate to each pair (W,d) a certain E-I-category W(d).

that if ~ is an El-category,

then there exists a group G(M),

WM and an equlvariant map dM:WM Section 2 form W(d),

we aiso study properties

a G(M)-poset

>Sub-G(M) such that WM(dM)=s(M). of functors between categories

which are induced by G-posets maps.

We show

In of the

As corollaries we obtain

results, which can be considered as special cases of the results of Section I. We also apply these results to the case of orbit categories.

275

O. P r e l i m i n a r i e s .

Let ~ be a small category and let ~:~ Grothendieck construction p~rs

(c,F),

on ~,

~j~,

>Cat be a functor.

is the category whose objects are the

where c is an object of ~ and F is an object of ~(c).

morpb/sms of ~J~ are given by the pairs (w,f):(c',F') 7:c'

The

>c is a morphlsm of ~ and f:~(~)(F')

>(c,F),

The

where

>F is a morphism of ~(c).

Composition is defined by (7, f) C7', f' )=C77',f~(7) Cf' )). By =~:~J~

>~,

we shall denote the functor such that ~(c,F)=c,

~ ( 7 , f)=w.

Assume now that ~ is a subcategory of Cat and that ~ is a functor from to ~. notation

Let ~ : ~

>Cat denote the naturai inclusion. We shall use the

~J=~IiR~.

In

particular,

we

shall consider

the

subcategories of Cat. Let Gr be the category of groups. group.

Assume that G is a

By ~ we shall denote the category with one object • G whose

endomorphlsm monoid is equal to G. inclusion of categories

LGr:Gr

dlscrete category with object the

following

naturs/

>Cat.

>Cat.

We

This correspondence

shall a/so

use

the

~" can be regarded as a generalization

direct product of groups.

naturai

If G is a group,

of the semi-

then for any functor ~:~

Let c be an object of ~. Let #c--~(c,@(-)):9

¢c--E(¢(-),c):~)°P

>Gr,

is equal to the seml-direct product of G and ~(.G ) .

Assume now that ~ and ~) are small categories and that #:~) functor.

gives us

>Cat, where Poset is the category of all posers.

The construction

the construction ~

gives us the naturai

We can also consider any set T as a

set equal to T.

inclusion LSet:Set

inclusion iPoset: Poser

This correspondence

>~ is a

>Set and

>Set. It is easy to check that the following two

categories c\¢=~I¢c and ¢/c=(9°PI¢C) °p are the usuai "over"

and

"under"

categories. In the case where ~=~ and ~=idlg , c\@=c\~ and ~/c=#/c. For any small category E and two functors Y:~ y,:Eop

>Top,

construction denote

the

Y: E

one can define the topologicai space Yx~Y'.

is described,

for example,

simpllciai nerve

denoted by B~.

>Top and

of

the

in 2.16.

category

We shall consider the functor

~.

Its realization will be

k-~=B(-\~):~ °p

>Top, then h

ollm

=

=

This

of [4]. By N ~ we shall

I,

where symbol ]] denotes the reai~zation of the simpllciai space.

>Top.

If

276

The

homotopy

collmits on

the following two categorles appear

in our

(i) Let S be a slmplicia/ complex and let K(S)

be the

further considerations.

O.l. Examples. polyhedron

obtained

as

the geometrical

realization of

S.

The

category

assoclated to the poset deflned by S will be denoted by the same Let ~s:S space Xs(S) sgs',

letter.

>Top be the functor such that, for every element s of S, the is equal to the closed simplex K(s)

then X(sKs')

the results of

of K(S)

determined by s. If

is equal to the incluslon K(s)gK(s').

[4], that X S is a

and that for any functor X: S °p

It follows from

free, locally contractible

S-CW-complex

>Top, there is a homotopy equivalence hocolim X -= XsxsX. s°P

(Ii} Let G be a group.

We shall use the notatlon EG=B(,G\~}=E~(OG).

is obvious that EG is a free universal G-CW-complex.

A functor T:~

It

>Top

can be considered as a topological G-space and hooolim T = EGXGT. •

The following fact is a genera/izatlon Thomason's

paper

[12].

of the main

Theorem

of

The method of proof is essentially the same as that

of [12].

>Top be a functor.

O. 2. Proposition. Let Y: ~J~J

Then there is a

homotopy equivalence ~:hocollm hocolim Y(c)

c~

> h o c o ~ m Y,

~(c)

where Y(c) Is the restrlction of Y to ~(c). Proof.

Y:

For any functor ¢:~)

>W of small categorles and for any functor

>Top, there is a homotopy equivalence u:

hocollm hocollm Y/c

>hocolim Y.

The fact above follows from the results of [1 ], [5 ], 9.8, and [4]. Let

vc:¢/c

>~D be the functor equal to x¢c, where ¢c ~)op

that vc is a usual forgetful functor.

>Set. I t is clemr

Then Y/~YPc and the homotopy

equivalence p is induced by the maps hocollm p . c Assume now that ~P=~J~ and that #=ii~. If c is an object of ~, then by l(c):~(c)

>#/c we

shall denote

the inclusion of categories such

that

277 iCc)(F)=((c,F),idc]. p(c):¢/c have

>~(c)

This functor

has

a

left adjoint

such that p(c)((c',F'),W)=~(~)F'.

the natural transformatlon

p from ¢/-

to ~.

functor

In fact, in this case,

we

This natural transformation

Induces the map k:hocollm hocolim Y/c c~ ¢/c From the fact that iCe) p(c)l(c):~(c)

>~(c)

>hocollm hocollm Y(c). o~ ~(c)

is a right adjoint to pCc)

is the identity map

and that the composition

it follows that, for every

object

c of ~, the natural map

induced by p(c),

A(¢) :hocollm Y/¢ >hocollm Y(c) ¢/c ~Cc) is a homotopy equlvalence. Thus A is a homotopy

equlvalence, too. ( [ I ] , [ 5 ] , [4]. ) This implies that hocolim hocollm Y(c) -~ hocollm Y. In fact,

uslng the same method as in [12],

one can construct a

canonlcal map @ such that ~A Is homotoplc to p. •

The

following fact, which

proved using arguments

belongs

to homological

llke those in the proof of 0.2.

[7],IX,6. ) Let us consider a functor N: C~j~j)op

O. 3. P r o p o s i t i o n .

algebra,

can

be

(See, for example

>Ab.

There i s a s p e c t r a l s e q u e n c e l l m p l i m q N(c) ~ l t m p+q N, o~ ~Cc) ~J~J

where NCc) Is the restrlction of N to ~(c).a

We shall now descrlbe a category,

which will be used

In the next

Sectlon In the deflrdtlon of the functor A~. Let (Gr,Set)

be the category whose objects are the pairs

G is a group and T Is a G-set, where ~:C map.

>G' Is a group homomorphism

as a functor T:~

where

and whose morphlsms are the palr6 (~,~), and ~:T

>T' Is an equlvaylant

The one point trivial G-set will be denoted by "G"

considered

(G,T),

Any G-set T can be

>Set. By ~(Gr. Set): (Gr, set)

>Cat

we

shall denote the natural tncluslon such that L(Gr, Set) (G, T)=~ST . It Is easy to check that the object set of ~ST Is equal to T(I G) and

278

that, for any two elements t,t" of T(.G), we have: W/T(t, t' )={g~G: gt=t' }. The composition of morphisms of ~/T is defined by mnltip]ication of G. It is obvious

that

L(GP, Set) (G, -G)=~.

O. 4. Corollary. Suppose that G is a group and T is a G-set. =T:~J'r

Let

>~ be the natural projection.

(1) Let X be a G-topological space.

Then

there is a homotopy equlvalence w(G,T,X):EGXG(XXT)

>hocolim X~ T .

(il) Let N be a Z(G)-module. Then lim " N= T

= H" (G, Homz(Z(T), N) ),

where X(T) is the permutation X(G)-module with the basis equal to T. Proof.

(i) By O. I. (ii), we obtain that EGXGCXXT)=hocoHm XxT=hocoHm hocollm

~

(X~T)(mG).

TC* G)

Now, it is sufficient to apply 0.2. (ll) This fact is a consequence of 0.3.m

i. Main r e s u l t s .

In this section we shall assume the additional condition

that ~ is an

that all Isomorphisms

of

M

E-I-category are

satisfying

automorphisms.

We

shall say that ~ is an E-I-A-category .

Let ~ be the set of all objects of 4. This set can be ordered In such a way that, for any two objects x,y of 4, we have x-Set be the functor such that, for each object a of

%cM), T4(a)=4Cao, al)X . . . . x 4 C a n _ l , a n ) , i f n~o,

and TM(ao) i s equal to the one p o i n t s e t {~a } ' i f n=o. For each o morphism a~a' of So(4), TM(a~a') is defined by the appropriate projections

and compositions of morphlsms of M. Let GM:So(4)

>Gr be a functor such that, for each object a of So(M) , GM(a)=Aut(ao)X .... xAut(an),

where Aut(a i) denotes the automorphism group of the object a i ofM, and for each morphism a~a' of So(M), GM(a~a') is the appropriate projection map. We shall also use the notation T(a)=T4(a) , G(a)=G4(a).

The group G(a)

acts on T(a) in such a way that, for any two elements g=(go ..... gn ) of G(a) and

f=(fl ..... fn ) of T(a), we have gf=(glflgo I ..... gnfngn~l) Hence,

(G(a),T(a)),

for any element a of So(M),

we can consider the pair

which is an object of the category (Gr,Set).

One can easily

check that there is a functor ffM :So(M)

>(Gr, Set)

such that, for every a~So(4) , ffMCa)=CG(a),TCa)), PGrff~--G~ and PSetff~=T4 , where PGr: (Gr, Set) .....>Gr and @Set: (Gr, Set)

>Set

are the natural projections. 1.1.Deflnltion. We define

• ~=L(Gr, Set)ffM

and

s(4)=So(M)I~ M

.m

We shall also use the notation A(=/~M , if=flY" It follows from the definition that s(2)=s (M). The o of definitions.

following farts

are

immediate

consequences

280 I. 2. Corollary. We have ~(a)=~Ca)/T(a)

whenever a~s CM).. o 1.3. Proposltlon.

(i) The object set of s(M) is equal to the set

U T(a) a~s

=

CA) o

U

~(ao,al)x

....

xg(an_1,an).

aoh' ~re all

p~(~)h=h'.

be the subcategory of s(~) whose objects are all f such

If f,f' are objects of p~l(x) then pg-1 (x} (f, f, )={ g~s(g) Cf, f" ) : id a =go=Pg(g) }. o

The

classlfying

space

of the category

p~l(x)

is contractible

because

• ET(x) Is a final object of this category. x

We shall prove that there exists a right adjoint ral Incluslon

~

:p~l(x)

space of x\p~ is contractible.

>xkpM

functor to the natu-

. This fact Implles that the classlfylng

,

281

We define

a functor R:x\p~--->p~l(x)

in the foHowlng vray. Let

f=(fl ..... fn ) belong to T(a ° ..... a n ) and let h:x

>pM(f)=a ° be an object

of xkpM. Then we take R(h)=(flh,...,fn). Now, let ~':h ~:f

>h' be the morphlsm of xkpM induced by the morphlsm

>f" of s(M). Then pM(f')=a~=a r. Suppose that ~ is defined by the

element

g =(go ..... gk ) of G(a').

Hence h'=fr...flgoh

, because h'=pM(~)h.

We can now define R(~') to be equal to the morphlsm • (flh ..... fn)---->(f~h', .... fk0 ) given by (id,g I .... gk)~G(x, al• ..... ~ )• .

It is easy to check that R is a right adjoint functor to ~ classifying spaces of the categories

xkpM and p~1(x)

Hence the

are homotopy

equivalent

and this ends the proof, s For every functor X:M

>Top

there exists a functor

XE:So(M)

>(Gr,Top)

such that XECa)=CGCa),EG(a)xT(a)xX(ao)), where T(a) is considered as a discrete topological space. The map XE(aSa•) G(a)

is the product of the map induced by the projection

>G(a') and the map ~:T(a)xX(a o)

>T(a')xX(a r) such that ~(f,x)=(T(~a')f,X(fr...fl)x).

The functor X E induces the functor

Xo:So(M)

>Top such that

Xo(a)=XE(a)/G(a}=EG(a)XG(a)(T(a)xX(ao)). I. 6. Proposition. Let M be an E-I-A-category. X:a

For any functor

>Top , there are homotopy equivalences hocollm EGCa)XG(a) (T(a)xX(a) )-~hocolim hocollm(T(a)xX(a)o )-~hooollm X aes (~) o aes (4) g(a) 0

Proof.

o

0

The general homotopy coIintlt theory ( [ 1 ], [4 ], [S ] ) and

1. S. imply

that there is a homotopy equivalence

From Proposition 0.2,

hocollm Xp~ >hocollm X. sC~) we obtain that, for any functor Y:s(~)

>Top,

there

is a homotopy equivalence ~0:hocoilm hocollm Y(a) aes CA) /~Ca)

>hocollm Y sC~)

0

where Y(a) is the restriction of Y to ~(a). • (a)--g(a)J~(a).

If Y=Xp~,

We shall use the fact that

then Y(a)(f)=X(a o) and Y(a)(g)=X(go).

case, by 0.3, we have homotopy equivalences

In this

282

vM(a):Xo(a)=EC(a)xc(a)(T(a)xX(ao))~ The family {~M(a)}

hocolim hocolim X(a )s h o c o l i m Y(a). ~(a) TCa) o ~(a)

can be constructed in such a way that we obtain a

natural transformation of functors ~rM:X°

>hocolim Y(-),

~C-) which induces a homotopy

equivalence between the homotopy

colimits of the

above functors. This fact ends the proof. • Let us

now

consider the simplicial complex

sd~-s (A) °p

associated

to

O

the poset ~. From 0. i. (i), we obtain the following fact.

I. 7. Corollary. There is a homotopy equivalence Xox s (~)Xsd ~ ~ hocolim X .• o Assume now that the cateEory M has only two objects. example,

of M and that that the set M(al,a o) is empty. empty,

This is the

which was described in [8]. Suppose that a ° and a I are the objects If the set M(ao, a I) is not

then ~ is equal to the poset {ao, a I} ordered by the relation aoSa I.

In this case

Xo(ai)=E(G(ai ) ) X c ( a i ) X ( a i ) ,

f o r i=o, 1

8_nd Xo(a o, a 1 ) =E(G(ao) xC(a I ) ) XC(ao) x C ( a l ) (M(a o, a 1 )xX(a o) ). The poset sd~ is isomorphic to the poset of all non-empty s u b s e t s s e t {o, 1} and Xsd ~ is the one dimensional simplex. homotopy equivalent

Thus hocolim X is

to the homotopy push out of the diaEram consistinE

the two maps Xo(ao, a l ) - - > X o ( a i ) morphisms Cao, a l ) ~ ( a i )

Consider

, f o r i=o, 1,

which a r e induced by the

of SoCk).

now a contravariant

functor N:~ °p

>Ab.

The

foHowing

result is a further consequence of i.S.

1.8.Proposition.

There exists a spectral sequence

lim p HqCGCa),Homz(l(T(a)),N(ao)))

a~s (~) O

of the

~ lim p+q N .

of

283

Proof. It follows from i. 5 that there exists a group isomorphism lim N M

=

lim NpM. sCM)

Now it is sufficient to apply 0.3 and 0.4. (ii), because

s(M)=So(M)I/~

and

•

.4iCa)=~Ca)/T(a).

Let MGr ,be the category with the same objects as M and with the morphlsm sets defined, for any two objects x,y of M, as follows: MGr(X,y)= ~(x,y) for x~y , There

exists

an

MGr(X,X)=M(x,x)=G(x).

isomorphi,sm of categories S(MGr) = So(M)IG M •

The category So(A)

can be considered as a subcategory

of S(MGr).

There

exists the natural extension of the functor T M to the functor from S(MGr) to Set. This extension will be also denoted by T M. It is easy to check that sC MGr) J'TM--sCM).

1. 9. Corollary. Let M and X be the same as in 1.6. Then there exists a homotopy equivalence hoco].im (T(a)xXCa O) )=- hocollm IX . acs (MCr ) M Proof. This result is a consequence of I. S and O. 2. • I. I0. Corollary. There exists a group isomorphism m

lim M whenever

N: M °p

N =

lim Homz(Z(T(a),N(ao)) aes (MGr )

>Ab.

Proof. This result is a consequence of I. 5 and O. 3. •

Assume now that ~:~--->(Gr, Set) is a functor .such that, for every object c of ~, ~(c)=(~l(c),~2(c)).

By ~/~I

we shall denote the canonical

functor from ~ to Set such that, for every object ('. of ~, C~/~ 1 ) Cc) =~2 (c)/~ I (c). The category ~IC~/~ 1 ) will be also denoted by [ ~ ] . following facts is easy and will be left to the reader.

A proof of the

284 1.11.Lemma.

Let ~ be a poset.

(i) For any functor ~:~ that (p,f)s(p',f')

>Set , the category ~2~ is the poset

i f and only i f

(ll) Let ~:~

>(Gr, Set)

such

pmp' and ~(psp')f=f'.

be a functor. Then

~t:[gJ~]

there exists a functor

>(Gr, Set)

such that,

and

~t (p' [ f] )= (~1 (P)' ~1 (P)f) whenever (p, [f] )e?/(~/~l). • Consider now the case where ?=So(M) and ~=~=(~M, FM). Then So(~)J~

= [s(~)]

and s(~)=So(~)l~ =[s(~)]l~ t, where ~t:[s(a)]

>(Gr, Set) is the functor such that ~t(a,[f])=(G(a),G(a)f).

1.12.Corollary.

Let M and X be the same as in 1.6. Then there exlsts a

homotop¥ equiva/ence hocolim Ca, [ f ] ) e [ s ( ~ ) ]

EG(a)xG(a) (G(a)fxX(a o) ) -= hocolim X. ,~

Proof. This result is a consequence of I. 5, O. 2 and O. 4. (i).•

Conslder now a functor N:~ °p 1.13. CoroUary.

>Ab.

There exists a spectral s(~uence

lim p (a,[f])e[s(~)]

Hq(G(a],Homz(Z(G(a)f]

Proof. This result follows from

1.5,

0.3

N(a ))) ~ llm p+q N . ' o and

0.4.(ll).s

The referee of this paper pointed out to the author that the existence of a homotopy equlve/ence ~:hocollm hocollm X(a) aes (~) ~(a) 0

> hocoHm X

285

is a consequence of Theorem 3.4 of [13]. "locally group~ke category =.C

and

reduced"

is equal to 2.

The Dwyer and Kan's condition

is equivalent

the

E-I-A

condition.

~

The category

p

-I

Their

Their category D is the opposite of s (M). In

N

the definition of the functor h, situation.

to

0

the index k can be ignored in this

D is our category ~(a),

to X[a) and the functor h(X):D

m

the functor j X is equal

>S is our functor a-->hocolim X(a) ~{a}

(except that the variance

is reversed).

To deduce

the exlstence of a

homotopy equivalence @ from Theorem 3.4 of [13] one tmes the following facts I. For any category A the functor ~

A hocollm

:Ho(S~)

>Ho(S)

A is the left adjoint of the "constant" functor (see [ i ], XII. 2.4. ) 2. Ho(S) Ho(S)

The functor Ho(S) > K obtained by composing the constant functor C >Ho(S ~) with h# is equai to the composite of the cortsts~nt functor D D D >Ho(S ~) with the functor xH:S ~ >S~/H wk[ch takes an object Y of

D ~

S

to the projection map YxH

>H.

D 3. The left adjoint of xH is the forgetful functor S /H

D

N

>S

2. C a t e g o r i e s a s s o c i a t e d to G-posers.

Let G be a finite group.

By a G-poset we mean a poser with an order

preserving left actlon of G on it. A G-poset W can be regarded g

>Poset.

The poser of the orbits of G-actlon on W will be denoted

A G-set T will be considered The category shall

as a functor

of ail G-sets

consider

the

natural

as a G-poset

ordered

by

(G-posets)

will be denoted

inclusion

~G-Poset: G-Posset

the identity by G-Set >Cat

by W/G. relatlon.

(G-Poser).

such

that

~G_Poset (W)=gJ ~. The

underlying

WC,G)--~(W).

functor G-Poset

>Poset will be denoted

by

~.

The G-poser of all subgroups of G will be denoted by Sub-G.

Thus

We

286

2.1.Deflnltlon.

A G-poset m a p

for every element w of W

d:W

>Sub-G

is called admissible if,

, dw is a subgroup of the isotropy group

Cw=( g~C: gw=w} = ~ / W ( w , If d is an admissible map,

then W(d)

w)

.

is the category whose objects are

the elements of ~(W) and whose morphism sets are defined as follows: W(d) (w, w" ) = ~/W(w, w" )/,iw' = {g~C: gw~-w' }/dw'= =

The

UJ W(d)

will be

which ccFresponds

denoted

by

i(d).

to [g]eG/dw'

We

shall

is determined

by g. An immediate consequence of thls dnfinition Is the following fact.

2.2. Lemma.

Suppose that d: W

>Sub-G

is

an

admissible

map.

Then

dw i s

a normal subgroup of G .• w Next we consider some examples of categories of the form W(d) describe homotopy collmlts of functors X:%I(d)

2.3.Examples.

and

>Top .

(i) Let d be the adnlssible m a p

such that,

for every

element w of W, dw is the trivial subgroup of G. Then W(d)=~J~W and EGx_hocollm Xi(d) -~ hocollm X . u

CCw)

WCd)

(ii) If the action of G on W is trivial, then (xCd):W

W(d)=~4J'~(d),

where

>Cat is a functor such that, for every element w of W,

~(d) w=~/de~&Gr (G/dw) 8/Id hocolim ECG/dw) -= hocolim X . wE~CW) WCd) If dw=G whenever w belongs to W, then W(d)=~(W)=W/G. (iii) Suppose that H is a subgroup of G and that H' group of H. Let d:G/H d(gH)=gH'g -I. we

>Sub-G be the admissible m a p

The category (G/H)(d)

shall denote the classifying space

equal to G/H'

and

whose morphism

such that

will be denoted by(G/H)(H'). of the

category

whose

By E(G/H')

object set is

sets contain exactly one element.

category is isomorphic to HX(G/H)(H' ). The space E(G/H') as a free right H/H' space.

is a normal sub-

This

can be considered

287

There are homotopy equivalences hocolim X s E(G/H')XH/H,X(H) =- E(H/H']XH/H,X(H) G/H) (H')

Let W d:W

o >Sub-G

.•

be a G-subposet of W. The restriction of an admissible to W

map

will be also denoted by d. If w belongs to W, then the

o category (Gw)(d) is isomorphic to (G/Gw)(dw).

2.4. Example.

Let d :Sub-C

>Sub-G be the identity map.

Then

(Sub-G)(d)=O G is the orbit category of the group G.([2],I.3.)

The objects

of 0 G are the G-sets of the form G/H, where H is a subgroup of G. The morphlsms of 0 G are the G-maps.

For a G-subposet F of Sub-G the subcategory

F(d) of 0 c will be denoted by O(F).

•

Let W and V be G-posets and let d:W admissible maps.

Suppose that ~:W

induces the functor ~(d):W(d) ction W(d)

>Sub-G,

d':V

>Sub-G be

>V is a G-poset map such that d-W/G will be denoted by p(d). We need the following notation.

2.4. Definition.

(i) A G-poset W is called normal if, for any two

elements w, w' of W, the condition w~w' implies that G w is a subgroup of Gw,. (ii) A G-poset W is called regular if, for any two elements w,w'

of W,

the conditions w~w" and wsgw" imply that g belongs to Gw,. • It is obvious that every regular G-poset is normal. 2.5. Examples. dG:W

(i) Assume that W is a normal G-poser.

Then the map

>Sub-G such that dGW=G w , whenever w belongs to W, is admissible.

this case,

the category W(d G) will be denoted by W(G).

In

If d is an arbitrary

admissible map defined on W, then there is a functor ~(d):W(d)

>W(G)

such

that =(d) (w)=w. Assume now that K is a G-complex in the sense of [2]. The G-poset of all finite subcomplexes of K of the form K(s), K(s)

is the smallest subcomplex of K containing

where s is a cell of K and s, will be denoted by SK.

This G-poset is normal and the category SK(G) °p is a full subcategory of the category ~( described in the section I-S of [2]. (ii) Let W be an regular G-poset. is a natural equivalence of categories.

Then the projection p(G):W[G) It follows from

this case, p(G) is a full and faithful functor .

>W/G

the fact that, in

288

(lii) Let W be a G-poset. that ~(sdW°P)=sd~(W) °p.

action on each coordinate. G-poset.

By sdW °p we shall denote the G-poset such

The action of G on this poser is defined by the It is easy to check that sdW °p is a regular

There is a G-poset map

qw:sdW °p

>W,

the underlying posers is a right coflnai functor.

which after restriction to

(See[6].)l

Assume now, that P is a poset and that T is a functor P

>G-Set.

Then

by PIT we shall denote the G-poset such that ~C_Poset (PIT )=PinG_Set TThe elements of PJ'T are the pairs (p, t) ,where p~P,t~x(p),

and

(p,t)-Cw' in such a way that

It is easy to check that this deflnltlon Is

correct and that W= (W/G) IT. •

The followinE result is a consequence of 2.7.

A proof is easy and will

be omitted.

2.8.Corollary.

Let W be a r e g u l a r G - p o s e t and l e t d : W ~ > S u b - G

admissible map.

Then there Is a functor T(d):W/C

categories

and

W(d)

TCd) ( [w] )=COw) Cd). •

(W/G)IT(d)

are isomorphic.

be an

>Cat such that the

For every element w of W,

289

2.9. Example. We shall use the notation of Section 1. Let M be an E-I-A-category and let P=So(M). Define G' ( a ) =

~ ]]

C(x)

G(M)= FIN G[x) and, for a~So(4), x~M where {a}={a ° .....

an}.

x~(~ \ { a } } The group G(M) acts trivially on P. The G(4)-map d':So(4) such that d'a=G'(a}

is admlsslble and So(M)(d')=S(~Gr),

>Sub-G(4)

because

G(M)/G' (a)=G(a). For every element a of s (M}, the group G(M) acts on the set T(a) by o Let z:s (4) >G(4)-Set be the functor defined by this o actlon, l.e. for every a~So(M} , let z(a)=T(a). Let WM denote the G(M)-poset the action of G(a).

So(M) Iz. Then ~ (W4) =So(4) J'r~=s(4Set), where MSe t is the subcategory of 4 with the same objects and such that 4Set(X, y)=M(x, y), 4(x, x)=(idx), whenever x,y are objects of M and x~y. It is easy to check that W~/G(~)=[s(4) ]. There is an admlssible map d4:W 4

>Sub-G(M)

such that for every

element f of T(a) dM(a,f)=G' (a). It is easy to check that WM(d 4) =s (4). Propositions 2.8,2.7 and 2.8 imply that W 4 is a regular G(M)-poset and that there is a functor zt: [s(4)]

>Cat such that sC~)=[s(~) ]I~ t

and zt(a, [f] )=(Gf) (d)=(G(a)/G(a)f) (e}=g(a}J'G(a)f . It is obvlous that zt=i(Gr, Set)~t,

where ~(t is the functor defined in the

end of the previous section. •

We shall now study categories associated to functors O(d):W(d)

>V(d')

induced by G-posers maps @:W---->V satlsfylng condition d-W such that ~H [ (g,w)]=gw is a G-poset map.

290

Let ~(O):~(W)

2.10.Proposition. defined

>~(V) be the u n d e r l y i n g

poset map

by ~ and let v be an element of V .

(i) There exists

an isomorphism of categories

Z: (v\~(O))Iv where v:v\~(@)

>v\O(d),

>Cat is the functor such that v(v-~Ow)=(G/d'~w)(dw).

(ii) There exists an isomorphism of categories ~: (GXd, v(~(O)/v))(d ~) where ~:GXd, v(~(O)/v)

>W

>~)(d)/v,

is the C-poset map such that ~[g,(~w~-v)]=gw.

Proof. We shall consider v\~(9)

(vk@(d) be a functor such that Z(w, gd'@w)

element of V(d')(v,g~w) deterntined by g. If w~w" and m:gd'Ow'

is the

>g'd'~w"

is

a morphism of (G/d'Ow')(dw') determined by an e]ement EoOf G, then Z(w~w',m):gw

>g'w' is the morphism of W(d) determined also by go" A proof

that ~ is an isomorphism of categories is easy and will be left to the reader. (ii) Suppose that w' belongs to O(d)/v. equal to the morphism g'~w' m:[g,w]

Then we define ~[g',w']

>v of V(d') determined by g , - I

>[g',w'] is the morphism of (GXd,v(~(~)/v))(d~)

E'E -I, then ~(m):gw

@(d) i s a l s o a r i g h t

If w~-w' and

determined by

>g'w" is the morphism of W(d) determined also by

g'g-l.m

2.11.Corollary.

to be

(i)

If

~(0)

is

a right

cofinal

functor

and

d=d'O,

then

cofinal functor.

(ii) If G acts trivially on V then there is an isomorphism of categories (~CO)/v) (d)

>OCd)/v.

(iii) If W=V and ~ is the identity map, then the natural inclusion ~: (G/d' w) Cdw)

>OCd)/w

is a right cofinal functor. Proof.

The assertion

(i) follows from 2. I0. (i), because,

the classifying space of the category v(w) belonging to v\~(@).

Hence B(v\~(O))

(See [12]l.2, and [I] or [S] 9.2.).

in this case,

is contractible for every w

is homotopy equivalent to B(v/O(d)).

291

The s t a t e m e n t assertion

(iil)

(ii) is an immediate c o n s e q u e n c e

follows

from

(i)

and

2.10.(ii},

of

2 . 1 0 . (il). The

because

the

natural

inclusion

C(G/d' w)

>~(GXd, w(W/w) )

is a right cofinal functor. •

2.12. Corollary.

L e t X be a f u n c t o r

f r o m W(d) t o Top.

Then there

e x i s t s a homotopy e q u i v a l e n c e

hocollm hocollm X~ ~ hocolim X .• V(d') (GXd, C_)~(O)/-)(d ~) WCd) The result above fo~ows immediately from 2.10. (ii) and from the general homotopy theory

([I], [5] 9.8, [4]).

In particular, it yields the

Suppose

trivially

on V a n d

there

i s a homotopy

fo~owlng facts.

2.13. Corollary.

(i)

that

G acts

w h e n e v e r v b e l o n g s t o V. Then V ( d ' ) = ~ ( V )

and

that

d'v=G

equivalence hocolim

hocolim

X m h o c o l t m X.

wCCV) (C(O)/v)Cd)

W(d)

(11) The natural projection to the orbit set O:W

>W/G Induces a

homotopy equivalence hocollm hocolim X ~ hocollm X [w]eW/G (GCW/w))(d) WCd) (111) Suppose that W Is a regular G-poset.

Then there exist homotopy

equivalences hocollm E(G/dw)x G /d "~(w)w s [w]EW/G w

hocollm hocollm X s hocollm X [w]~W/G (Gw)(d) W(d)

Proof. The statement (i) Is a consequence of 2.12. assertion G(W/w) can

(tt) if

and

be obtatned

follows

immediately

only if, from

there (1t},

is a right coftnal functor.

from

exists

because

(t),

because

geG s u c h t h a t in

this

case,

w"

and 2. II. (11). The is

gw'°sw: the

g

an

element

The a s s e r t i o n

inclusion

Gw

of (1tt)

>G(W/w)

This fact is also a consequence of 2.8. •

2.14. Corollary. Suppose that W=V and that 0 is the Identlty map. (i) There are homotopy equivalences hocollm E(G/d(w))xd, ~ hocollm hocollm X ~ hocollm X w~W(d') (w)/d(w)X(w) w~W(d') O(d)/w W(d)

292

(ii) In particular, d: W

if X:WIW

> Top,

then,

for any

admissible

map

>Sub-G, there are homotopy equivalences hocolim EGXd(w)X(w) w~WCd) Proof.

The a s s e r t i o n

The s t a t e m e n t

(ii)

(i)

follows from

s hoco~m X s EGx_hocolim X . ~/W u ~(W)

is

a (i),

c o n s e q u e n c e of because

2.11.(ill)

~J~/=W(do) ,

and

2.3.(lii).

where doW=(e)

for

every wcW.•

2.15. Example.

Consider the case where,

for every w~W,

X(w) is a one-

point space. Then 2.14 implies that there are homotopy equivalences EGxGBW -= hocolim EG/d(w) = EGx_hocolim G/d(w) weW(d) u weW(d) where BW is the classifying space of the underlyin E poset of W with the G-action induced by the action of G on W. In particular, if W is a normal G-poset,

then

EGxGBW -= EGx_hocolim EG/G w , u weW(G) and if W is regular, then EGxGBW s EGx_hocolim EG/G w .m u[ w] ~W/G 2.16. Examples. Let F be a G-subposet of Sub-G. (i) Consider Example 2.4.

Then 2.15

implies that there exists a

homotopy equivalence

where I:O(F)

EGxGBF s EGx_hocolim I u O(F) >Top is the natural inclusion such that I(G/H)=G/H.

(ii)Let d:F °p

>Sub-G

every element H of F, d(H) category

be the admissible

G-poset

map

such that, for

is equal to the centralizer CG(H)

(F°P(d)) °p will be denoted by CG(F).

regarded as group homomorphisms

~:H

of H in G. The

The morphisms of CG(F)

>H" of the form ~(_)=g(_)g-I

can be for

some gEG. In the case where F is the G-poset of elementary abelian p-subgroups of G, we obtain the category A(G,p) considered in [9]. Example 2. IS implies that there is a homotopy equivalence EGxGBF°P ~- hoco iim EGx G (G/C G (- ) ). CG(F)°P

293

From the fact that the G-spaces BF and BF °p are equal, it follows that there is a homotopy equivalence EGx_hocolim I -= EGxGhOColim G/CG(-) .m u O(F) CG(F)O p

The next result can be regarded as a specification of I. 12.

2.17.Proposition. adNtissible map.

Let W be a G-poset and let d:W

For any functor X:W(d)

>Top,

>Sub-C be an

there are homotopy

equivalences

hocolimW(d)X ~ hocolims(w)(dqw)Xqw~ hocolims(w)(G) E(G/dqw(-))XG-/dqw (-)xqW ~

E(G/dWo)X(G

hocolim [w o . . . . .

Wn]~S(W)/C

where s(W)=sd(W) °p and qw:S(W)

)/dw X(Wo)

~...nG w°

wn

o

>W is the natural projection

(from

2.5. (iii)). Proof. The first homotopy equivalence is a consequence of 2. ii. (i) and the fact that qw is a r~Eht cofinal functor.

The existence of the second

one follows from 2.14. (i) and 2. S. (i). The third one is a consequence of 2.5.(Iii) and 2.13.(iii) or 2.5.(ii).|

We end this paper with applying of the result above to the case of orbit categories.

2.18.Corollary.

Let F be a G-subposet of Sub-G and let X:O(F)

There is a homotopy equivalence hocollm X s hocolim E(G/Ho)XNH n...NH X(G/Ho)'S O(F) [H0 ..... Hn]~S(F)/G o n

For X=EGxGI we obtain a homotopy equivalence EGxGBF ~

hocolim EG/(NH n...nNH ). [H° ..... Hn]eS(F)/G o n

>Top.

294 References.

[ I ] A.K. Bousfield, D.M. Ken Lecture Notes In Math. 304, [2]

G.E. Bredon

"Equivariant

"Homotopy

Limits

,Completion

and

Localization"

1972. cohomology

theory"

Lecture

Notes in Math. 34,

1987. [3 ] K.H. Dovermam_n,

M. Rothenberg

"Equlvariant

surgery

end

classification

of finite group action on manifolds" Mem. Am. Math. Soc. 379, [4] E. Dror Farjoun

1988.

"Homotopy and homology of diagrmns of spaces"

in

Lecture Notes in Math. 1286,93-134. [5]

W.G. Dwyer,

D.M. IC~n "A classification theorem

cied sets" Topology, Vol. 23,No. 2,139-155, [6]

W.G. Dwyer,

D.M. Kan

for diagrams

of simpli-

1984 .

"Function complexes

for diagrams

of simplicied

sets" Indigationes Math. Vol. 45, Fasc. 2, 1983. [7]

P.J.Hilton,

U. Stenm~bach

"A course

in Homologic8/

Algebra"

Springer,

1971. [8]

S. Jackowski,

J . E . McClure

"nomotopy approximation

for

classifying

s p a c e s of compact Lie groups" i n Lecture Notes i n Math. 1370, [9]

S. Jackowski,

J . E . McClure

"A homotopy decomposition f o r

1989. classifying

s p a c e s of compact Lie groups" p r e p r t n t . [10]

W. Li~ck " T r a n s f o r m a t i o n groups and a l g e b r a i c K-theory"

to a p p e a r

in

Lecture Notes i n Math. [11]

J. Slond~ska families

" E q u l v a r i a n t Bredon cohomology of

of

subgroups"

classifying spaces

Bull. Ac. Sc. Pol. Sc. Math. Vol. XXIII, No. 9 - 1 0 ,

of 1980,

503-505.

[12]

R.W. Thomason

"Homotopy

colimlts in the category

of small categories"

Proc. Camb. Phil. Soc. 85, 1979, 91-109. [ 13] W.G. Dwyer, theory

D.M. Ken

of fibrations"

"Reducing equivarient homotopy in

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on

of Peter Hilton", Contemporary Mathematlcs,

Algebredc

theory to the Topolology

vol. 37, 1985.

in honor

On b o r d i s m rings with principal torsion ideal Vladimir V. Vershinin Institute of Mathematics Siberian Branch of the Soviet Academy of Science Novosibirsk 630090 USSR

1

Introduction

Bordism spectra have long been known [12] as a rich source of examples of homology comodules over the dual of the Steenrod algebra, and introducing singularities [2] allows us to vary these structures further. Such computations were pioneered for the unitary case MU in [3], and more recently for the symplectic case MSp in [13,14]. In this paper we introduce some new sequences of singularities into MSp, and study the homology and homotopy properties of the resulting spectra. We prove that they are multiplicative, characterise their homology comodules, and show that their homotopy rings have torsion ideals generated by a single element in dimensions of the form 2i - 3. Our spectra axe mainly of interest at the prime 2, where they lie in the poorly mapped territory between MSp and BP. I owe many thanks to Nigel Ray, for much helpful advice on improving this paper.

2

Homotopy

and homology

computations

Nigel Ray's well known family of elements ¢i {11] lie in the symplectic bordism ring MSp.. Each ¢h i _> 1, is indecomposable in MSpsi-3 and has 2¢i = 0. It is also convenient to write ¢0 for the class 0x of the non-trivially framed circle in MSpa. Consider the following sequences of elements

Ao

=

Aim

(¢1,¢~,...,¢5, .... ), ( ¢ 0 , ¢ 1 , . - . , ¢ 2 I-2,¢5',¢2 I+1 . . . . ),

i~

1,

where Ai consists of all ¢i, J a power of 2, except for ¢~,-,. Let A T be the finite subsequence of Ai, i > 0, consisting of the first n elements. Thus

A? = (¢0, ¢1,..., ¢~--,),

i>n>2

~ = (¢0),

i>1

A? = (¢0, Cx,..., ¢~,-2, ¢2,,..., ¢2--,), i < n . A" We propose to consider the theories MSp. ~( ), i > 0, i.e. sympleetic bordism with singularities A~', and their direct limit MSp~.~( ). We write the corresponding spectra

296

as MSpA't and MSp A'. We studied MSpA~ and MSp A° in [13], where we labelled them as MSp :~ and MSp ~ respectively. Our main tools will be the Adams-Novikov spectral sequence [1,9] and the modified algebraic spectral sequence m.a.s.s. [13] which converges to the E2 term of the AdamsNovlkov spectral sequence. We recall [13,16] that the initial term E[ '''t of the m.a.s.s. for the spectrum X is isomorphic to

Ezt.%(Z/p, BP. ® H . ( X ; Z/p)), where .Av is the dual to the factoralgebra An/(Qo) of the Steenrod algebra and Q0 is the Bockstein operator. Also, BP. is the object associated to BP. (the Brown-Peterson homology [1] of a point) by the filtration of the m.a.s.s.,

BP.=Z/p[ho, ht,...,hi,...], (2,0), deghl = In the case X = to

i = 0,

(1,2(2 I - 1), i > 0.

MSp and p = 2 this initial term was computed in [13] and is isomorphic Z / 2 [ c 2 , . . . , ck, • • •, ul, • • •, u j , . . . , h0, hi, • •., hm,...], k ~ 21 - 1, degck = (0,0,4k), d e g u j = (0,1,2(2 j - 1)),

degh0 = (2,0,0), deghm = (1,0,2(2 '~ - 1)), m > 1. Our main result is the following. T h e o r e m 1. In the theory of symplectic bordism with singularities of type A~ there exists a multiplicative structure such that ~baa,_l = 0 in MSp. '. The groups Hk(MSp~"t; Z) are finitely generated and torsion free for all k, whilst H2j_I(MSpaT; R) = 0 for all j , where R = Z or Z/2. The initial term of the m.a.s.s, for MSpa? is isomorphic to the ring Z/2[c2,...,

ck,...,

u,+l, u,+2 ....

, h0, • . . , h , , , . . . ] ,

where the generators are the images of the corresponding elements in the initial term for

MSp. P r o o f . We will prove the theorem by induction by n. I f / = O, then all the statements of the theorem were proved in [13]. So we suppose that i > 0. If n < i, then we obtain the spectra M S p ~n of [13] and the theorem is also proved there. So suppose it is true for some n - 1 such that n - 1 > i. Then MSpA'~-~ is a multiplicative spectrum. From the theorem 3.2 of [2] it follows that there exists an exact triangle. An--I

MSpA.'~-'(X) ~, MSp , (X) MSp."r(X),

(1)

297

where X is any spectrum, the homomorphism fl is multiplication by ¢:.-~ and 6 is the Bockstein homomorphism; also V is defined by considering manifolds with singularities A~'-~ as manifolds with singularities A['. Let X be the spectrum defined by ordinary integral homology, or homology with coefficients in 7/2. Then the above exact triangle can be rewritten as the exact sequence

... ----* Hk_~.+,+a(MSp a~-'," R) ~ ~ tI~(MSp'~-~' ;R) , Hk(MSpa:';R) ~

An--I

Hk_~.+,+~(MSp ' ;R)

} **.,

where R = Z or Z/2. From the induction hypothesis that H2i_~(MSp/'~-~;R) = 0 it follows that H2i_I(MSp'a~; R) = 0 for all j. Hence, the long exact sequence splits into short exact sequences

0

,H2~(MSpa~-~;R) ",H2k(MSpa?;R)

An--I

6,H2k_2.+,+2(MSp'

;R)

,0,

(2)

from which it follows that H2k(MSpg?, Z) is finitely generated and torsion free. Consequently the Adams-Novikov spectral sequence, the algebraic spectral sequence and the m.a.s.s, for the spectrum MSpa? exist and converge. Let R = l / 2 . Then from (2) we obtain the analogous sequence in cohomology

0,

H2k(MSpAT-,;ZI2) ~

H2k(MSpa~;Z/2) : s H2k_2.÷2+2(MSp/,7-,;Z/2) ,

O,

and also the long exact sequence

0

-----. Hom.~,(ZI2,BP.®H2k(MSpA~_,;Z/2) ) ~o ----* Hom.4~(Z/2,BP.®H2k(MSp ,;Z/2))

,

---* Homx~(Z/2,BP. ® H2k_~.+=+2(MSpa?-';Z/2)) ~0 71

----, E x t , ( Z / 2 , BP. ® H2k(MSpa?-'; Z/2)) ----* ...

... ----* Ext~I(Z/2, BP.®H2k_2.+2+2(MSp , ;Z/2)) ----, E x t ~ (Z/2, B P .

,

® H2k(MSpa~-'; Z/2)) 7" ~J

Ext~t; (Z/2, B P . ® H2k(MSpA?; Z/2)) --~ --'4...} r~/':'xt';'z'2, Be. ® H,,_,.,,,,(MSp, ~"-' ;Zp)) - ~'

where the homomorphisms T' and ~' are induced by the morphisms -y and 5, respectively, and fl' is the connecting homomorphism.

298

BP. ®H.(MSp zx'~-', Z/2)). The corresponding element ¢2--, in MSpa. ?-' goes to zero in MSp~. ?. From consideration of the AdamsWe have the element u,+x in Ext.~ ( l / 2 ,

Novikov spectral squence it follows that under the mapping ~ the element u,+l is sent to A--1 A.--I 0. Using the fact that % : M S p . ' ( ) ~ MSp.' ( ) is a module m a p over MSp., we obtain that u , + l " ExtOl' (Z/2, BP. ® H.(MSp; Z/2)) is sent to 0 by 5° and hence u,+~. E x t ~ ' ( Z / 2 ,

An--l

BP. ® H.(MSp ' ;Z/2))

BP. ® H.(MSpa?-~; Z/2)) for fixed q and t (i.e. E~',-~'t) as a vector space over l/2 coincides

is sent to 0 by 5". By our induction hypothesis, the dimension of E x t ~ l ( l / 2 ,

with the dimension of u,+l . E l ',-l't. Consequently, ~ is a monomorphism for s = 0,1, .... Thus, ~° is the zero homomorphism and ~ is an epimorphism, ~o is an isomorphism, and the kernel of ~" for s > 1 is the module An--I

u~+l'Ext~l(Z/2, BP. ® H.(MSp ' ; l / 2 ) ) . This means that the initial term of the m.a.s.s, for the spectrum MSp zx'7, as a module over the initial term of the m.a.s.s, for MSp, has the form indicated in the statement of our theorem. We obtain the formulas for the first differential of the m.a.s.s, for MSp~'? (for t - s < 2 "+s - 3) from the corresponding formulas in the m.a.s.s, for MSp [13,6]: dl(hi+x) = houi+l, dz(~,+2*-l) : hk+lUi+l, k ~ i, k = 0, 1 , . . . . The module structure of the E1 term EI(MSpa?) of the m.a.s.s, for MSpa? generates the multiplicative structure, which in turn generates the ring structure in E2 of the m.a.s.s, for MSp a?. This is compatible with the module structure of E2 of the m.a.s.s. for MSp zx", over E2 for MSp. The ring E2(MSpa?) has the following generators in dimensions t - s < 2 '~+3 - 3: ho; ui+l; Cm, m # 2 t - l , m # 2 k + 2 i - 1 , k,l = 0,1,...; II

//~ = ~ h . , + 1 c 2 - ~ + 2 ~ - 1 " - " j=l

4",+2'-1

"" • c2-~ + 2 ' - 1 ,

i v = ( , ~ , . . . , n , ) , , ~ > . . . > ,,1 >_ o, n j # i, ~ ( N ) = ~ ___ l, ~.1 = h , , + l ;

hi+t2,"

cz~+2,2-1,"

~M =

hoc2.,~+2~-1 • "c2..~+~-1 Jr hi+l~M,

where M satisfies the same conditions as N. There are some evident relations involving the above generators. In particular, we need ui+l~N = 0

for all N . Evidently E~ = E = if t - s < 2 "+3 - 3. As a consequence, we obtain that in these dimensions, multiplication by ui+l is monic on E ~ "'t if s > 1, and hence this is true for E2 of the Adams-Novikov spectral sequence for MSp A'/.

299

The obstruction to the existence of a multiplicative structure [8,4] in the theory An

An

MSp. '( )

£~?--t

is an element 3'[P'] Z MSp2.'+~_5 where [P'] E MSP2"+~_5. So the dimension 2 "+3 - 5 of this element is odd. Consider its projection into the E~ term of the Adams-Novikov spectral sequence. It must be of the form ~2j-1 V 2i_1 ~

(3)

for some j = 1 , 2 , . . . , and Y E Ext°(BPo,BP.(MSpA?)), and Y must be a cycle with respect to the differential da. Now the relation ~ = ~l=o k-1 ~btMl in MSp. has been proved recently [7], for some

Mt E MSp..

If we take k = T -1, then we obtain that in

MSpa. ~, with

k > i,

= 0.

Hence, we must have an element z E ~2 ~'°'3"2'+2-s of the Adams-Novikov spectral sequence for MSpA? such that d3(z) = ~ _ , . It follows that if t - s < 2 ~+s - 3, then d, = 0 for An

r > 3. Then in formula (3), Y must lie in MSp. ', and i f j > 1 then the obstruction is equal to zero (see the table below, which displays s against t - s):

3

I

~2,-I

~2 n

,Z 3 • 2 i+2 - - 8

2 "+3 -- 5

2 "+3 - 3

So the only possibility for the obstruction to be non-zero is that it has the form ~ - i Y, where Y E MSp~,,~,_~,+2_2. Thus the dimension of Y is equal to 2 mod 4. Hence, its projection to the Eo~ term of the m.a.s.s, must be of the form

J

for some ]~N# and YNj. But we have in the m.a.s.s, that ]~N#Ui+1 ----0, so we obtain that ~b2~-IY = 0. Thus the obstruction to multiplicativity is zero. The obstruction to commutativity is in our case an element of order two and liesin dimension 2 "+3 - 4 [8,4], and hence equals zero. For the obstruction [r] to associativity, we have the condition 3[r] = A . 01,

300

and 0~ = 0 in our theory M S p ~ . ? ( ) so long as i > 0. If i = 0 then more detailed considerations in the re.u.s.s, for M S p " ' 3 show that the obstruction also vanishes. Our induction is now complete. T h e o r e m 2. The multiplicative structure in the theory of symplectic bordism with singularities A~' may be chosen such that, for p > 2, its coefficient ring is isomorphic to the polynomial algebra M Sp(p). = Zp[wl, . . . , wl, wi+ 2, . . . , w,+l, x2, . . . , 2:k,

. .

where k # 2i - 1 if 1 < j _< n + 1, j # i + 1, and deg wl = 2(2 t - 1), degxk = 4k. P r o o f . We also prove this theorem by induction on n. As in the proof of theorem 1 we can suppose that i > 0 and n - 1 > i. By the induction hypothesis, the multiplicative structure on M S p ~'~-~ be chosen so An--1

that the ring M S p . ~

is polynomial and isomorphic to Zp[Wl,...,

wi, wi+2,...,

Wn~ X2, • • • , X k , • • . ] ,

where k :fi 2j - - 1 if 1 ~ j _< n, j # i + 1. A--s We have ~ . - , = 0 in MSp(p'). for p > 2. So from the Bockstein exact sequence we obtain short exact sequences 0 ~

~"-~ MSp(~).

, M S p " '(p) ~

M S p ~ -~_ . 2.+2+2

~ 0.

A,l Hence, the module MSp(p). is the direct sum Sp(p).

(., Wn+l,

where w.+t 6 M S p . Z~"' , and deg w.+l = 2 "+2 - - 2 . From the Adarns-Novikov spectral sequence for M S / x ? 2 Wn+ 2 ~

with p = 2, we deduce that

r / $ Z 2 n + t _ 1 -~- a n + 1 "~- Y n + l W n + l ~

where a,+, C= FaMSp~. '~-' and Y,,+I 6 F2MSp~. '~-*. Here F . i M S p ~ '~-' means the mod• he', A n - 1

ule of filtration j in m o p . /2"ors C Hom.~ ( B P . , B P . ( M S / ' ? ) ) corresponding to the re.u.s.s., m is odd and zz-+:-i is defined so that h,+l is associated to it. An It follows from theorem 4.6 in [8] that if in M S p . ' ( ) there exists a multiplicative structure # , , then there also exists a multiplication/~,, such that

where z is an arbitrary element in the bordism group with singularities A~' in the corresponding dimension, and the p~irings #,.,-1 and #,-1.,, are defined by the product of A~'-t-manifolds. In our case we obtain

~(w.+~, w.+,) = ~(W.+l, w.+,) + 4z,

301

An

where z is an arbitrary element of MSP4('~..,_I). Hence, we can vary the multiplicative An

structure in MSp. ' ( ) so as to give Z2n+l-I ~---::[:(W2n+I -- an+l -- Wn÷lyn-[-l).

This means that, after suitable variation, the ring MSp(p). will become polynomial, and the theorem is proved. We shall consider now the homology of the spectrum MSp aT. T h e o r e m 3. There exists an isomorphism of comodules over .A., the dual of the Steenrod algebra, of the form

Ho(MSp ,; 1/2 ) ." 1 / 2. [ ~ , ..

4 .2 ,~i-~,~i-,,~,--

, ~.+~, ~ '. + 2 , . . . ] ® 1/2[c2,... , ck,.. -], -~

where the ~i are generators of A., and ~j _-- ~j rood decomposable in .A.. The elements ck are primitive. P r o o f . Suppose the theorem is true for n - I. The elements ck in H.(MSpa'~; 1/2) are defined as the images of ck E H.(MSpa'~-~; 1/2) by the monomorphism 3' of the exact sequence (2) for R = 1/2. We can also regard the algebra

z/2[~,~, . ..,~L~, . . .'~,_,,~,, ~

,~.,~.+,,...] ~

as the image of the morphism u."-:: H.(MSpAT-'; I/2) ~

H.( Hll2; 1[2),

where v"-1: MSpa? -1 ~ HI/2 corresponds to the generator of H°(MSpA'~-I;I/2). Let us consider now the Atiyah-Hirzebruch spectral sequence

E 2 = H.(BP; r.(MSp"'~)) ==~ MSpa. 7(BP). It is known that in the analogous Atiyah-Hirzebruch spectral sequence for MSp~."(BP)

[:51 d2(2-+l-1)(m.+:) = ¢2--a. We have an evident morphism of spectra 7: MSp a'i'~

' MSp r",

inducing a morphism of spectral sequences, from which it follows that the element cannot be killed in E ~, for r < 2(2 "+1 - 1). In E 2(2"+'-1) we have

¢2--1

d~(~.+,_,)(m.+,) = ¢2.-,, where ~,,+: _ rnn+: modulo decomposable. This means that ~n+a is an infinite cycle in the spectral sequence for MSpa'l(BP). Note that the canonical morphism rr: BP H1/2 also induces a map of Atiyah-Hirzebruch spectral sequences. Let w" be the morphism

w'* : MSp a? inducing the identity on r0(

~ HZ

). Then the following compositions

BPo(MSp a?) ----, BP.(HZ) =" HZ.(BP),

302

HZ/2.(MSp A?) --.--, ~" HZ/2.(IIZ) = tIZ.(HZ/2) are the edge homomorphisms in the corresponding spectral sequences. Here X interchanges HZ and HZ/2. We combine these morphisms in the commutative diagram

BP. (MSp A'~)

,07,

~, =

BP.(HZ)

~.~,

HI.(BP)

,r.~,

HZ/2.(MSp ,)

~r,~,

(4)

, HZ/2.(HZ) ~ HZ.(HZ/2).

The fact that ~.+1 is an infinite cycle means that we have some element P.+l in ~tt

BP.(MSp ~) whose image under the edge homomorphism is m.+l. The image of ~.+1 under the action of It. is ~,,+1, ~2 where ~,+~ = ~,,+~ modulo decomposable elements. We add the following rectangle

HZp..(HZ) ,7.~

x

=~ #Z.(HZ/2) ,~.~ X

HZI2.(HZ/2) ~- HZ/2.(HZI2) to the diagram (4) (y: HZ ~ HZ/2 denotes reduction rood 2) and obtain that the image of ~r.(/2,,+,) e HZ/2.(MSp A?) under v." --- (71 ow"). in HZ/2.(HZ/2) is equal to ~.2+1 = X($.2+,), where v": MSpA? .--* HZ/2 corresponds to the generator of H°(MSp"'t; Z/2), and ~,,+1 - ~n+l modulo decomposable elements. We therefore also denote ~r.(p.+l) by ~2+1.

We see from the exact sequence (2) that H.(MSpA'~; Z/2) is a module over H.(MSpA?; Z/2) on two generators, in dimension 0 and 2(2 "+1 - 1) respectively. The element ~.2+1 does not belong to the image of H.(MSpA'~-~;Z/2) under the action of 7. and so may serve as the generator in dimension 2(2 "+: - 1). Hence, every element of H.(MSpA?; Z/2) can be written in the form P0 + P," ~+I (5) • An--I wherePoandPlmH.(MSp i ;Z/2) arepolynomialsin~,..

¢2

~4

•, M--2~ ~i-l~

.~2

"2 4

t,i ~ " • • ~ ~ n ~ ~ n + l

.... ,c2,...,ck, .... Writing the element (~.2+1)2 in the form of (5) gives "2 2 ff.+~)

Q0+Q,

"2

(6)

Applying v;" to both sides of (6) yields that Q0 = ~,,+t 4 + Q~, where Q~ does not contain ~+1. The polynomial Q1 also does not contain ~+1, and the element ~,,+t4 becomes decomposable in H. ( g s p a ? ; l/2). Now we must prove that the ring Ho(MSpA~; l / 2 ) is polynomial. If this is false, then there must exist a polynomial P in the variables c2,..., ck,. •. , ~ , . . . , ~+1,4~i+2,. -'2 •, ~,+t,~2 ~ + 2 , . . . , which is identically zero. Writing P as a polynomial in ~+1, we have p

"2 k = ak(~,+1) +... + ao,

where ak ~ 0 and aj does not contain%gn+l" "2

303

If k = 21, then we obtain from the relation (6): a

g41

-2

(7)

2t¢;.+1 + qo + qt~,,+l = O,

An-I

where qi E t L ( M S p , ; Z/2) and the polynomial qi can involve ~.+1 4 only in powers less t41 than 1. The left hand side of (7) has the form (5), and so a 2tg,+~ +qo = 0. Hence, a2t = O, contradicting the definition of P. On the other hand, If k = 21 + 1, then using (6) we obtain that 41 "2 (a2t+l~n+l + ql)~.+1 q " qo = O , where qi ~ H . ( M S p " ? - ' ; Z / 2 ) and the polynomials qi can involve ~+~ only in powers less than t. Analogously, we obtain a2t + 1 = O. So our proof is complete. Remark that we have utilised the fact that 3' and 5 of (2) are morphisms of modules over H.(MSpZX'2-t; 7/2). T h e o r e m 4. Symplectic bordism with singularities Ai admits a multiplicative structure, which can be chosen such that, for p > 2,

M SpZ~). = Zv[wl,..., wi, W i + 2 ,

. . . , Z2, . . . , Xk,

. . .],

where k ¢ 2 / - 1 i f j ¢ i + 1, and degwt = 2(2 t - 1), degxk = 4k. The torsion ideal of M S p .a~ is generated by the image of Nigel Ray's element ¢~,-1 and the relation ¢~,-~ = 0 is fulfilled. The groups Hk(MSpZX~; Z) are finitely generated and torsion free for all k, and are zero in odd dimensions; furthermore

H.(MSpA';Z/2) "=" Z / 2 [ ~ ,

..

. ,~_~,

~ ,4_ , , ~ , ,

=2 . . . ~-2j , . . .] ® Z/2[c~, .. .,ck,.. .1.

P r o o f . This follows from the previous theorem by standard direct limit arguments. Note that we may easily dualise our results so as to obtain the cohomology of M S p A? and M S p A' as modules over the Steenrod algebra. Specifically, let :D~, i >_ 0, be the subalgebra of the Steenrod algebra for which a Z/2-basis consists of the elements Sq s (of the Milnor basis), where J = ( j l , . . . ,jk) and the jt are such that jt_ n + 2 act trivially on So. tlence, we obtain an epimorphism

, H'(MSpa?; Z/2), v/

with w as before. Comparing the dimensions of the two vector spaces in this formula we see that it is an isomorphism. We conclude this section with some results we shall require later. P r o p o s i t i o n 1. The Atiyah-Hirzebruch spectral sequence for MSpa.'~(MSp A?) collapses. P r o o f . We again use induction. Let us suppose that the spectral sequence for

MSp~. ~-' (MSp A?-~) collapses. Then maps of the corresponding spectral sequences are induced by 7: MSp AT-' --..~ M S / " / and & MSpA? ~ S2"+~-2MSpa?-' , which on the level of the E 2 terms yield an exact sequence .-,

,

.-1

H,(MSp ~, ; Z ) ® M S p . a' ,

a n

"'j

H.(MSp,aJ;Z)®MSp,

H.(MSpa'2-'; Z) ® 1% i Sp._2.+2+~ a2-'

,

6,

....

First, we prove that the spectral sequence for MSpa. t(MSp ''7-~) collapses. For suppose it does not, and that the first nontrivial differential is dr. This differential is a homomorphism of modules over MSpa. ~, so there exists an element z 6 H"(MSpa'~-';Z) such that dr(x ® 1) # 0, where x ® 1 6 E~,o. The image of E.'0 under the homomorphism (5. is zero because the spectrum S2"+2-2MSpA",-1 is (2 n+2 - 3)-connected. Hence 6.(d,(x ® 1)) = 0 and so there exists an element y 6 E~_r+l.,(n-l,n-l), the E r term of the spectral sequence for MSpa. '~-~(MSp'V;-'), such that %(y) = dr(x ® 1). Let us consider the element z ® 1, belonging to E~,o(n-1,n-1). In this spectral sequence we deduce that dr(z ® 1) = y + y' ~ O, (where y' is such that 7.(Y') = 0), which contradicts the induction hypothesis. Now we prove that the spectral sequence for MSpa.'/(MSp~'~), whose E2 term is isomorphic to H. (MS~*?; Z)®MSp A'I, collapses. Let a be an element in Hu~÷2_2(MSp"r;Z) whose image under 6. is the unit of H.(MSpA'2-~;Z). Every element of H.(MSpt~?; Z) can be written in the form a + ba, where a,b 6 7.(Ho(MSp'7-';Z)). In E~,,(n,n-1) there are no elements of finite order if q < 2"+3 - 3, so dr(a @ 1) = 0 for all r. The morphisms 7 and 6 are morphisms of module spectra over MSpa'~ -~ . We thus obtain the formula d,((a + ha) ® 1) = 0 using the multiplicative properties of the Atiyah-Hirzebruch spectral sequence. C o r o l l a r y 1. The Atiyah-Hirzebruch spectral sequence for (MSpa?)'(MSp a'~) collapses. P r o o f . This follows from the duality between the homology and cohomology spectral sequences. C o r o l l a r y 2. The Atiyah-Hirzebruch spectral sequence for ( M S / " ) ' ( M S p A') collapses.

305

3

Splittings

and

the

BiP

spectra

Now we describe how to obtain more precise information about the ring MSp~. ~. For this purpose we use the spectral sequence of the singularity f2,-, (see [13,5]). If we take the theory MSpa.'( ) and add one more singularity f2'-~ we obtain the theory M S p Z . ( ) , whose coefficient ring is isomorphic to the polynomial algebra

M S p ~. '~ Z[wl,... ,wj,... ,x2,... ,xk,...], where j = 1,2,..., k # 2t - 1 and des wj = 2(2J - 1), degx~ = 4k. The E1 term of the spectral sequence is the polynomial algebra MSp~.[UI+~], where degU,+, = (1,2 i+~ - 3 ) ,

d e g w / = (0,2(2 j - 1)), degxk = (0,4k).

If we filter this term properly we obtain the initial term of the m.a.s.s., and using the following formulas for the first differential

k¢i+l, hoUk, k = i + 1, 0,

dl(hk) =

dl(cl)

f o,

l

t # 2' + 2s-1 _ 1,

hjUi+l,

l = 2 i + 2 j-1 - i, (j ~ i + 1)

we obtain formulas for the first differential of the spectral sequence of the singularity:

d~W,+l) = 0,

dx(z~) =

d~(wk) =

0, wjU,+l,

0, k#i+l, 2U~, k = i + 1,

1~2 i+2 j-I-1, l = 2' + 2~-' - 1, (j # i + 1).

Let P' be a manifold whose cobordism class [P'] in MSp~. '+' , for Y:.i+t = (fo, . . . . ¢2,-t), is the obstruction to the multiplicativity of the theory MSp~.'+'(). Since this theory is multiplicative [13], there exists a manifold Q with singularities of type ~i+1 such that OoQ = P'. There are the following multiplicativity formulas for the first differential of the spectral sequence of the singularity:

dl(ab ) =

d l ( t l ) b -it-

(_l)(dega+degUi+l)degUi+l dl(a)[~(Q)ldl(b) dr_(--1)dega{d¢gUi+l+l}adl(b), Ui+~

where des is the topological (second) degree of an element and 8 is the Bockstein operator. In our case 5(Q) is the manifold with singularities of type ~21 = (fro,..., f2,-2) such that ao(~(Q)) = 2f2,-a. So [d~(Q)] may be taken as w,+~, because the only precondition is that

OWls1 = 2f2,-,, where Wi+l represents the bordism class wi+1. So we obtain the following formula

al(ab) = dx(a)b + (-1) des•+' a, Ca)wi+xd,(b) aax(b). Ui+~ +

306

It follows from this that w~+t and Yv+~-t-1 = x 2v + v - , _ l - Wi+l*dJjX2i+2j-1-1 are cycles of the differential all. Let us denote by L the subring of E1 generated by wi,ui+l, and z2~+v-t-l; then we have an isomorphism

E2~-H(L, dl).[xk],

k # 2'+2i-~-

1,

j#i+l.

The spectrum MSp r" is a free free spectrum, hence from the E2 term the spectral sequence of the singularity Cv-~ coincides with the Adams-Novikov spectral sequence, and d2 = 0. From the fact that ¢~_~ = 0 it follows that there exists an element ~ E E °'3"2'+~-s such that d3(~) = ¢3 . 2 For E °'' in this dimension, we have xa.2'-2 and Y2,+v-~ = xv+v-,_l - wiwi+Ixv+v-~-i as the two generators. We do not know the action of the differential d3 exactly, so we denote by r/the second generator (the first is ~) such that da(rl) = 0. We can choose other generators in E3 so that they will be cycles for d3. If we denote by L the subring of E3 generated by ~, and H(L, dl) except for rl, then we obtain the isomorphism

E ~ = E 4 ~ - ( H ( L , d3)).[zk,rl],

k#2'+2J-~-l,

j¢i+l,

k#3.2'-2.

Obviously there are no extension problems, and if we eliminate the first grading s then we see that E~o ~- MSp~. ~. It is not difficult to deduce that multiplication by yv+2J-l-1 for j ~ i,i + 1, and by r/or ~2, are monomorphisms MSpa. ' ..---* MSp~. '. Let us denote the sequence of all these elements, and also xk for k # 2 / + 2j-1 - 1, k # 3 • 2i - 2, j # i + 1, by E, and order it by dimension. We thus obtain a regular sequence in MSpa. ~. In particular, multiplication by the first element of E induces a map of spectra a: MSp a~ ~ MSp a~, whose mapping cone we denote by MSp 'x"l. After localisation we obtain the sequence

(8)

., ~ MSp(2 ) which generates the commutative diagram

tI.(MSp~;;Z2)

'~', H.CMSp~;;Z2)

II.(MSpg;;Q)

~

I1 .(MSpg;) ® q

H.(MSpg~;Q)

I[ .(MSpg9 ® q

"Y---~ H.(MSp~;'I;z2)

,

".-~ I-I.(MSp~'a;Q)

,

¢',

I1 .(MSpgi") ® q

,.

The induced map

ct.:~r.(MSp~)) ® Q

, r.(MSp~) ® Q

is a monomorphism, as is the canonical morphism H.(MSp~; Z2) --* H.(MSp~; Q). Hence

c~.: H°(MSp~;Z2)

, H,,(MSp~;Z2)

307 is also a monomorphism, and we have the short exact sequence

0 ---* H.(MSp~;Z2) ~

H.(MSp~;;Z2) ----* .~o. H.(MSp(2A,,, ) ; Z2)

The images of the elements of ":- are generators of

tI.(MSp~;

, 0.

Z~) under the Hurewicz

map, hence H.(MSp~";Z2) is free over Z2. It follows from this fact and the universal coefficient formula that c~ induces an epimorphism in cohomology

a*:H*(IVISp~;-12)

a, Z 2). , I~* t (MSp(~);

Therefore a induces an epimorphism of E2 terms of the the Atiyah-Hirzebruch spectral sequences for (MSp~2~)'(MSp~2~). Applying corollary 2 we see that (MSp~2~)'(ot) is also an epimorphism, and so we have a short exact sequence Ai * Ai ~i * Ai Ai * Ai, 1 0 ~.- (MSp(2)) (MSp(2)) #-- (MSp(2)) (MSp(2)) ~-- (MSp(2)) (MSp(~)) , . . O.

Hence, there exists a map

c2:MSp~; ---+ M S p ~ such that ~oa = 1.ea oP(2) ~ . , , and the sequence (S) shows that

MSpg;is equivalent to a sum

MSpg~ ~" MSpg;" V S'MSpg'}. Then there exists a map 1: 1vlop(2)

, MSp~

such that 3'¢1 = 1MSp~,~, and a map ILl:

MSp~ A 1vl "'~op(2) a,,1 ---+ MSp(u)' a, ,

such that the diagram

M S p ~ A MSp~;

MSg

'^~ MSp~; A MSp~¢) 1^'i MSp~; A MSp~; a

°,

MSg;

,o,

MSpg;,1

is commutative. Therefore ~1 = 3,0 o/~ o (1 ^ ¢1), and ~x is unique. Analogously let ~1,1. ~rr, Ai,l . ~-~, Ai,l ,we, Ai,1 ~vl op(~) A lvL~P(2) ) -na'-~P(2) be such that /~1,, = /~1 o (¢1 ^ 1) = 3'0 o/~ o (¢1 ^ ¢1). Then /~': is associative and commutative. Taking the second element in -- and repeating the above procedure, we obtain a lrt', Ai,2 multiplicative spectrum m op(2) , which is a direct summand in M S p ~ 'x (and so in

MSp~)),

and a morphism of spectra

3"1. spg;., -. Mspg;.'.

308

Using induction, we build a chain of morphisms MSp

;

,

,

....

In this chain each spectrum is obtained from the previous one by attaching cells of increasing dimension. The direct limit of this diagram is a spectrum which we denote by B i P . Obviously it is multiplicative, and we have canonical maps , BiP,

7: M S p ~

¢: B i P ~

MSpg;,

where ¢ is defined from the sequence of maps ¢, o . . . o ¢,: M S p ~ " ----* M S p ~ for all n. Evidently, 7¢ = 1sic, and there exists a splitting

M S p ~ "" V S " , B i P , J where the nj are the degrees of the elements from =:. Using theorem 4 and the construction of the spectrum B i P we obtain an isomorphism of comodules over .4.:

H . ( B i P ; Z / 2 ) . .----. Z / 2 [ ( ~ ' " " " ~ x2 ~4 ;.2 %i-2~ %i-1~ %i,

.

.

,~], ..]®E. . .

Here E is a Z/2-module whose elements are primitive over .4. and whose generators we denote by e,, of degree 4n, where

n = 2"~ + ... + 2"" - u . 2i - v,

0<ml 0.

The summand with n = 2 i+a + 2 i - 2 arises from the nontriviality of the differential dz in dimension 4(2 I+~ + 2 i - 2), which kills ~,_,. If we turn to cohomology we obtain H ' ( B i P ; Z/2) ~ ~ ( - 4 d - 4 2 D , ) " S., n

where the elements S~ are dual to en. If i = 0 then our spectrum has the same homology as the spectrum BoP, constructed by David Pengelly in [10]. Using the methods of his paper (p.1116-1120) it is easy to prove that in this case B i P is equivalent to BoP. We may now summarize all the above considerations in the following theorem. Theorem 5. There exists an indecomposable multiplicative spectrum B i P , for each i = 0 , 1 , . . . , such that MSp~ ~ V S" BiP J wad H . ( B i P ; Z / 2 ) "=Z [ 2 [ ~ , . • • ~ ,2 ~4 %~.2 ;2 %1-2~ %i-1, i, " as c o m o d u l e s over -4..

309

4

References

1. J.F. Adams, Stable homotopy and generalized homology, University of Chicago Press, Chicago, 1974. 2. N.-A. Bans. On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279-302. 3. N.-A. Baas and I. Madsen, On the realization of certain modules over the Steenrod algebra, Math. Scand. 31 (1972), 220-224. 4. B.I. Botvinnik, Multiplicativity in the cobordism theories of manifolds with singular-

ities, Proc. of Inst. of Math. Novosibirsk 7 (1987), 44-61 (in Russian). 5. B.I. Botvinnik, V.V. Vershinin, Multiplicative properties of spectral sequence of singularities. Siberian Math. J. 28 (1987), No.4. 569-575. 6. B.I. Botvinnik, V.V. Vershinin and V.G. Gorbunov, Some applications of spectral sequences in cobordism theory, Preprint, Novosibirsk, 1986 (in Russian). 7. V.G. Gorbunov and N. Ray, Orientations of Spin bundles and symplectic cobordism, Preprint, Manchester University (1989). 8. O.K. Mironov, Existence of multiplicative structures in the theories of cobordism with singularities, Math. USSR Izvestiya 9 (1975). 9. S.P. Novikov, The methods of algebraic topology from the viewpoint of cobordism theory, Math. USSR Izvestiya 1 (1967). 10. D. PengeUy, The homotopy type of MSU, Amer. J. Math. 104 (1982), 1101-1123. 11. N. Ray, Indecomposable in Tors MSp., Topology, 10 (1971), 261-270. 12. R.E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J. 1968. 13. V.V. Vershinin, Symptectic cobordism with singularities, Math. USSR Izvestiya, 22 (t984), No.2., 211-226. 14. V.V. Vershinin, On the decomposition of certain spectra, Math. USSR Sbornik, 60 (1988), No.2., 283-290. 15. V.V. Vershinin and V.G. Gorbunov, Ray's elements as obstructions to orientability of symplectic cobordism, Soy. Math. Dokl. 32 (1985), 855-858. 16. V.V. Vershinin and V.G. Gorbunov, On a rectangle of spectral sequences, Proc. of Inst. of Math., Novosibirsk, 9 (1987), 41-59 (in Russian).

"Localization and the Sullivan Fixed Point Conjecture" Amir H. Assadi (1) University of Wisconsin (Madison, WI USA), Max-Planck-Institut ftir Mathematik (Bonn, FRG)

Introduction: Let K be a finite dimensional G-space, where G is a p-elementary abelian group, i.e. G % (~/p~)n.

The

Borel--Qnillen-Hsiang

localization

theorem

states

that

* * G * * HG(K;[]:p) ----* HG(K ;O-p) is an isomorphism modulo HG({point};[l:p)-torsion, where H G is Borel's eqnivariant cohomology ([B] [Q] [Hw] ). The above theorem is not true for infinite

dimensional spaces in general. As we shall see below, the Sullivan conjecture imphes that such a localization holds for infinite dimensional G-spaces Map(EG,K), where dim K < •. Conversely, the main result of Section 2 proves that if the Borel-Qnillen-Hsiang localization holds for Map(EG,X ) , then E G x X is G-homotopy equivalent to E G x K with dim K < m. Here E G is the usual universal contractible free G---space. This provides an answer to a problem posed in [A2]. This question and other problems of this nature arise naturally in the geometric and differential topological aspects of transformation groups of manifolds. In particular, at present most methods of constructing group actions on a given manifold yield only infinite dimensional free G--spaces. See [A1] [AB] [W] and their references. While the localization theorem apphes to p-elementary groups, and the Sullivan conjecture holds only for p-groups, we have formulated our results for all finite groups. The proof of the main topological results, (Theorem 2.4) is reduced to the case of cyclic groups of prime order using an inductive argument. The main tool which provides such a local-to--global passage is the

(1)

Acknowledgement. Much stimulus for the question and the results of the present work was provided during my joint work [AB] and conversations with W. Browder, to whom I would like to express my gratitude. It is a pleasure to thank G. Carlsson, H. Miller, and J. Lannes for conversations on their proofs of the SuUivan's fixed point conjecture. The present version of this work was completed at the Max-Planck-Institut fiir Mathematik (Bonn). I would like to thank Professor F. Hirzebruch for his encouragement and support and the staff of the Institute for their hospitality during my stay at MPI. Research partially supported by an NSF grant and the Max-Planck-Institut fiir Mathematik.

311

algebraic result (Theorem 1.1) of Section 1 which is a projectivity criterion for integral and modular representations occuring as the cohomology of certain G--spaces. The proof of our converse of the localization theorem for G = ~]p~ does not use the proof of the Sullivan conjecture, but merely a statement of this kind. Therefore, it seems appropriate to present the statement and proofs in a sufficiently flexible manner to accommodate the possible improvements. Since the Borel--Quillen Localization theorem is essentially of homological nature, so are the proofs of our theorems. Thus, "the quasicompletion functors" which are modeled homologically after Bousfield-Kan's completion functors will also work in the context of Section 2. This approach emphasizes those homological properties of these functors which are relevant for our purposes and how they are used in the course of the proof. To apply the converse to the localization theorem, one needs to develop computation tools. At present, Lannes' results in [L] are the best available for G = (~/p~)n. Such results in conjunction with our theorems yield more general results for finite groups which are not necessarily p-elementary abelian. In non-technical terms, let us mention one corollary: Corollary: Let

G be a finite group and let X be a free G--space. Then there exists a finite

dimensional G--space K such that E G x K is G-homotopy equivalent to X if and only if for each prime p I G [ , and a representative p---Sylow subgroup Gp ~: G , there exists a finite dimensional Gp---space K(p) such that E G x K(p) is Gp-homotopy equivalent to X . P An interesting feature of the localization theorem as pointed out by Quillen in [Q] is that it is valid for compact G---spaces even if they are infinite dimensional. This motivates the following. Problem: Suppose G = E/p~ conjecture hold for X ?

and

X is a compact G---space. Does the Sullivan fixed point

Section 1. Algebraic Preliminaries Let G be a finite group, and let k be an algebraic closure of 0:p = the field with iv-elements. All modules are assumed to be finitely generated. A classical result of Rim [R] states that a ~G-module M is ~G-projective if and only if its restrictions M ] ~ P are EP-projective for all Sylow subgroups P ~ G . Chouinard has refined this result [Ch] by replacing the p-Sylow subgroups in Rim's theorem by (maximal) p-elementary abelian subgroups. Thus the projectivity of M is detected by all its restrictions to M]~A for all p--elementary abelian A I: G , i.e. A ~ ~p $ ~p • ... • ~p. To decide the projectivity of M I ~A, it suffices to consider the kA-module M ® k . Thus, let A be a Iv--elementary abelian group of rank n and with {el,...,en} a set of generators, and let I be the augmentation ideal. It is possible to choose a k---subspace L C I with

dimkL = r

and such that

I ~ L • 12 as

k-vector spaces. Then L generates kA as a k-algebra and for each A E L , (,~+1) p = 1. The

312

elements ~r E kA of the form ~r = A+I, ~ E L (for such an L ) are called "shifted units" and the cyclic subgroups S = (or) of order p are called "shifted cyclic subgroup". (See [C j] ). In [D] Dade has proved that a given kA-modnle M is kA-projective (hence kA-free since kA is local) if and only if M IkS is kS-projective for all such shifted cyclic subgroups of kA. (Note that almost all shifted cyclic subgroups of kA do not come from cyclic subgroups of A .) W e will fix L for the rest of the following discussion. In [A2], the author proved the following projectivity criterion which will be used in Section 2. 1.1 Theorem. Suppose

X

is a connected G---space such that for each maximal p-elementary $

abelian subgroup A C G, the H (-;k)---spectralsequence X

~E A x A X

.~B A

collapses.

Then

• Hi(X;k) is a projective kG-module if and only if it is projective as a kC-modnle for i>0 every subgroup C C G of order p. Similarly, • Hi(x;~) is a projective ~G-modnle if and i>0 only if it is ~C-projective for all cyclic subgroups C of prime order.

Note that if X

is a Moore space with G-action and

X G ~ 0 , then the conditions of

Theorem 1.1 are satisfied, and we get a projectivity criterion for the cohomology of Moore spaces with G-action.

Section 2. A Converse to the Localization Theorem By a "converse" we mean the following. Given a finite dimensional G-space

X , the

Borel-Quillen-Hsiang theorem tells us how to get information about the cohomology of the fixed point sets for

G =

p--elementary abelian. Now suppose instead of the finite dimensional G--space

X , we are given only the Borel construction ~r : Y infinite dimensional free G-space

~ = ~r ( E G ) )

~BG (or equivalently the corresponding and the localized equivariant cohomology

information of the type in the conclusion of Borel-Quillen-Hsiang theorem. Then we can recover a finite dimensional G-space X whose Borel construction E G x G X ., BG is "the same" as Y ~BG (in the sense of fibre homotopy equivalence). The key to such a construction is a statement of the type of SuUivan's fixed point conjecture. In this section, we will use "completion functors",

homologically modeled after

Bousfield-Kan's completion functors [BK]. For simplicity of exposition, we will assume that our functors are defined for all topological spaces; however, such functors may have smaller domains of definition in the course of applications, in which case, the appropriate modification of the following properties is necessary. Recall that the BousSeld-Kan ~p-Completion functor satisfies the following:

(CO) (C1)

R is a functor from the category of topological spaces to itself. R commutes with arbitrary disjoint unions and finiteproducts.

313

(c2)

Y induces an There is a coefficient ring R associated to R such that if f : X isomorphism F, : H,(X;R) J H,(Y;R), then R(f), : H,(R(X);R) , H,(R(Y);R) is also an isomorphism. There is a full subcategory of topological spaces Top(R) (associated to R ) and there is a natural transformation r : identity ~ R which satisfy: i) If ~rl(X ) = 0 , then X E Top(R). ii) If X E Top(R) then R(X) E Top(R). iii) For all X E Top(R), the map r(X) : X ~ R(X) induces an H,(-;R)-isomorphism.

(c3)

Definition (i) A functor R satisfying (C0)-(C3) above is called a "quasicompletion functor". (ii) Let G be a category of groups and R be a quasicompletion functor. We say that R is adapted to G if the following is satisfied. Here E is a universal contractible G-space. (C4) For all G E G and all finite dimensional G--spaces X such that H,(X;H) and H,(xG;R) are finitely generated, the map of constants X G

~MaPG(E,X )

induces an isomorphism H,(R(xG);R) , H,(MaPG(E,R(X));R ) . When R = [l=p and Rp is the Bousfieid-Kan [BK] [l=p---completion, then Rp is adapted to the category of all finite p--groups by the validity of the Sullivan's conjecture mentioned above. For the Bonsfield-Kan Top(R) consists of Fp---good spaces [BK].

Remarks. (a) The condition on ~rl(X) in (C3) may be weakened to ]~,(~rl(X);R) = 0, or even HI(X;R ) = 0 in applications. (b) The condition (C4) is essentiallythe Sullivan fixed point conjecture which has been proved for p-group independently by G. Carlsson, J. Lannes, and H. Miller. The important specialcase where G acts triviallyon X was done by H. Millerin [M]. See also [C] [L]. (c) Sullivan had stated his conjecture for p-groups. It is worth noticing that the Sullivan fixed point conjecture is not true for G--spaces where G is not a p---group.In [A3] the author has shown that for any finitegroup G which is not a p--group there exists a fixed-point free G-action on in. These easilyprovide counterexamples. However, one may stillask the following: Under which circumstances for a finite G - C W complex X the existence of an equivariant map E G

~X implies that X G ~: ~ ?

W e consider firstthe case G = ~/p~. This case is sufficientfor many applications. 2.1. Theorem. Let G = ~/p~, and let X be a free G---spacesuch that Xl(X ) = 0 and H,(X) is finitelygenerated. Let R be the Bousfield-Kan ~:p---completion(or any quasicompletion functor adapted to G = {~/p~} whose coefficient is g:n )" The conditions (A0)---(A2) together are necessary and sufficient for the existence of a finite dimensional G---complex Y such that: (i) y G E Top(R), (ii) H,(Y G) is finitelygenerated, and (iii) E x y and X are G-homotopy

314

equivalent. (A0) MaPG(E,R(X)) belongs to the image of R up to If=p-homology isomorphism. There exists a finite dimensional complex F E Top(R) with tt.(F) finitely (AI) generated, and a map r / : F ,MaPG(E,X ) such that "the induced map" h

r/: R(F) ~ MaPG(E,R(X)) below.) The map ~ : MaPG(E,R(X))

(A2)

induces

tI.(-;[]=p)-isomorphism. (See the remark

J Map(E,R(X))

induces a Borel-Quillen localized

,

isomorphism in IIG(-;[]:p)-theory. Remarks: 1. "The induced map" ~ is obtained as follows. The map 77 of (A1) has an adjoint map :E x F E x R(F)

J X.

Then

h

r/

is

, R(E) x R(F) ~ R(ExF)

2. Let us observe that for G

the

adjoint

map

of

the

composition

R(~--) ~R(X).

=E/pE and t E H2(G;E/p~) nilpotent for p = o d d

t E HI(R/2E;[]=2) for p = 2 , the Tate cohomology

~*

or

1 (G;[]=p) coincides with tt * (G;[]=p)[~-] .

• When X = point, the localized equivariant cohomology reads: HG(point;[l=p) [ 1 ] ~ fi*(g;~=p) Thus, we may denote the functor HG(-;[]=p) [ ] properties of Tate cohomology as well.

by

for short and suggest the

Proof. Suppose such a Y exists, and let F = y G . Consider "The map of constants" F ,MaPG(E,Y ) which becomes a homology equivalence upon applying R (by virtue of condition (C4)): H.(R(F);[I=p) ~ H,(MaPG(E,R(Y)ilI=p) ) . For simplicity of notation, H, denotes homology with [}=p-COefficientsthroughout this proof. Since MaPG(E,E ) has the homotopy type of a point, one has MaPG(E,X ) ~ MaPG(E,ExY ) _~ MaPG(E,E ) x MaPG(E,y ) on the level of path components. Thus one has the map F ~MaPG(E,X ) such that the composition below induces an H.---equivalence: R(F)

, MaPG(E,R(y))

, MaPG(E,E × R(Y)

, MaPG(E,E ) x MaPG(E,R(y))

., MaPG(E,R(E x Y))

, MaPG(E,R(X)) _

(Note that the homotopy fixed-point set is an invariant of G-maps which are non---equivariant homotopy equivalences.) Hence R(F) ,MaPG(E,R(X)) is also an H,--equivalence and conditions (A0) and (A1) are seen to be necessary. To see the necessity of (A2), consider the diagram:

315

R(yG)

hl

~YG

~Y

b Map(E,E x y )

, Map (E,X)

1

MaPG ( E, R(Y))

, MaPG(E,R(X))

, M a p ( E , R(X) )

MaPG(E,R(E x Y)) (Diagram 1) In the above, all maps which are not labeled induce H,-equivaience, since various spaces involved belong to Top(R) and R = ~: by the hypothesis. ~r is an H,--equivalence since R P satisfies (C4). h 1 induces ~G(-;U:p)-isomorphism by the Borel-Quillen localization theorem ([Q] [Hw] ). It follows that 2 also induces such an isomorphism, and condition (A2) is also necessary. We sketch now the proof that the above conditions are also sufficient. Consider the composition f0 -= ~ " r/ where ~ : F J MaPG(E,X ) , and L : MaPG(E,X ) ~ Map(E,X) are given by (A1) and the inclusion, respectively. The strategy is to add (finitely many) free G-cells to F so that the map f0 is extended to a highly connected map. One obtains the diagram below: F

Y0

r/

, M aPG(E,X )

1

~ Map (E,X) (Diagram 2)

Here, Y'~ = F and we may assume that the cofibre of f is a Moore space with finitely generated homology, since H.(X) is assumed to be finitely generated. At this point, it may be helpful for the reader to consider the special case where r l ( F ) = 0, where (C4) is true without any completion (according to the validity of the Sullivan conjecture in this case.) In this case the proof is much simpler technically since f is readily seen to satisfy the following claim. We claim that f induces an isomorphism for the functor ~[G(-;~:p) so that the reduced homology of the cofibre o f f is cohomologically trivial as a G-module. From this claim, the proof of the theorem is completed as follows. Let Cf be the cofibre of f , and i~ (Cf) be its reduced homology. Then ~[(G;]~*(Cf)) ~ ~tG(Cf,{point}) vanishes ( Fp --coefficients), which is sufficient for the cohomological triviality of H (Cf;~:p) for G = E / p g . Since H (Cf;g) is finitely generated and we may assume it to be g-free as well, it follows that it is EG-projective ( [R] ). By standard arguments (e.g. [A3] Chapter I and II) we may add free G---cells to Y0 and extend f to a homological equivalence, which we continue to call f : Y J Map(E,X) . Since X and Y

316

are 1--connected, this yields a homotopy equivalence. The evaluation map e : E x Map(E,X) ~X, e(f,e) = f(e), e E E , is equivariant and a homotopy equivalence. Hence it is a G-homotopy equivalence since both spaces are G-free. (Note that the action on E x Map(E,X) is the diagonal action and the action on Map(E,X) is by conjugation, i.e. fg(x) = gf(g-lx)-) This finishes the proof and it remains to establish the claim. The proof of the claim is based on studying a number of commutative diagrams:

F = YO G

f

hl

R(Y0)G MaPG ( lg, R(Y0) )

, Map(E ,X)

~ YO

, R(Y0)

~ R Map(E,X)

, Map ( E, R(Y0) ) ~

MaPG(E,R(X))

M a p ( E , R(Map(E,X))

, Map(E,~(X)) (Diagram3)

R ( F ) MaPG(E,R(Y0) )

~o

, MaPG(E,R(X))

(Diagram4)

F

h1 -¢'Y0

(Diagram5) In diagram 3 we have the following l~G-isomorphisms: hl, al, 81 , a2, 82 and 83 ; and in diagram 4, we get a and ~a induce tI,-isomorphisms. In diagram 5, the dotted arrows exist by the functoriality of R and 70 induces an H.--isomorphism. It follows from spectral sequence arguments that ~ induces a Borel-Quillen localized isomorphism. Combining these with a study

3;7

of the diagram: h3

Mapn (E,R(Yn))

~7

J Map(E, R(Y0) )

, Map(E, R(X))

MaPG(E,R(X))

we finally conclude that ~ induces a Borel-Quillen localized isomorphism. This is used in conjunction with a spectral sequence argument to show that in the diagram below f induces a Borel-QuiUen localized isomorphism: Y0

f

Map(E, R (y0))

Here f14 --

fl3"~2"~1" Thus, ~G(Cf,{Point})

!~Map(E,X)

t

1

, Map(E,R(X))

= 0 and the claim is established. []

For the case of finite complexes, we find obstructions in ~0(~G) which are algebraic in nature and may be treated separately from the homotopy-theoretic side of such problems. 2.2. Theorem. Given G,X, and R as in Theorem 2.1, suppose that (A0)---(A2) are satisfied, and in (A1) F is a finite complex with similar properties. Then there is an obstruction w(X) E I ~ ( ~ G ) such that w(X) = 0 if and only if there exists a finite G---complex Y such that E xy

and X are G-homotopy equivalent and y G = F . The obstruction

depend on F as long as F satisfies (A1). ( ~ ( ~ G ) general).

w(X)

does not

is a certain subquotient of I~0(~G ) in

2.3. Remark. It is not always true that w(X) is independent of F for any finite group G . For G = ~ ] p ~ , this is a consequence of the triviality of the Swan homomorphism aG : (~/p~)X ~~0(~G) . Thus, in this case if such a finite Y exists, and if F ' is any finite complex which admits an ~=p-homotopy equivalence F ' .~ F , then there exists a finite G-complex Y~ such that ( y , ) G = F ' (cf. [A3] ).

and E x y '

is G-homotopy equivalent to X as well

2.4. Example. Let R be the Bousfield-Kan f23-~mpletion functor, and let G be the cyclic

318

group of order 23 acting on M ~

~/47~

via the inclusion of G C Aut(///47~) ~///46//. The

calculations of Swan shows that there is no finite G-complex

X

with

H,(X) ~ M

as a

//G-module. However, there are finite dimensional G---complexes Y such that H,(Y) ~ M as //G-modules. For any such Y I~(Y;[]=23) = 0 and y G E Top(R). However, it is not possible to find a finite G-complex K such that E × K and E × Y are G-homotopy equivalent. Next, we briefly outline how we can generalize 2.1 from ~/p~ to general finite groups. Let G be a finite group and p be any prime dividing order of G . We define the following sets of subgroups of G :

Pp(G) = {P ~: G I Pl is a p-power }, P(G) - P[ U]G]Pp(G) , Ap(G) - {(P1,P2)[Pi E Pp(G), i = 1,2; P2 ~ P1 and p 1 / p 2 ~ ( / / / p g ) r for some r >- 0 }, A(G)--- p[ U[G[Ap(G) . The following proposition provides us with the necessary conditions for "finiteness" of G---spaces in the appropriate context. A similar result with appropriate modifications hold for finitely dominated G--spaces in the equivariant sense. As pointed out earher, the recent proofs of the equivariant Sullivan conjecture show that the quasicompletion functors which are used in the following proposition form a nonempty set ! 2.5. Proposition. Suppose that G is a finite group and p is any prime dividing I G I , and let Rp be the Bousfield-Kan completion or any quasi--completion functor whose associated coefficients is I]:p. Assume that Y is a finite dimensional G-space such that

H,(YP;II=p) is

finitely generated for each P 6 Pp(G) and YP belong to Top(Rp). Let X be a free G-space such that E G x y and X are G-homotopy equivalent. Then for each P 6 P(G) and each (P1,P2) 6 Ap(G) the following hold: (B0) All spaces Mapp(E,Rp(X)) are H,(-;ll=p)-eqnivalent to spaces in the image of Rp. (B1) There exist finite dimensional complexes F(P) 6 Top(Rp) with finitely generated H,(F(P);ll=p) and maps r/(P) : F(P) , Mapp(E,X) such that h

r/: Rp(F(P)) ~ (82)

The

map

Mapp(E,Rp(X)) is an H,(-;l[p)---equivalence. I(P1,P2): MaPPl(E,Rp(X)) ~

MaPP2(E,Rp(X))

induces

a

Borel--Quillen localized isomorphism for the group A - P 1 / P 2 . Proof. Let F ( P ) = YP and r/(P) as in Theorem 2.1 (where F(P) and r/(P) are denoted by F

319

and ~/ respectively). Since the first two conditions are consequences of the properties of quasi---completion functors as in Theorem 2.1, we will justify the last condition only. Consider the following commutative diagram. F(P 1)

' MaPPI(E,Y)

F(P2) 1 ~

\

, MaPP2(E'Y)

Rp(F(P1))

~rl

a2

Rp(F(P2)) The

maps

p(y))

F(Pi)

'MaPP2(E,Rp(Y))

~Mapp.(E,Y)

are

given

by

the

maps

of

constants

1

P. y 1

* Mapp.(E,Y) , and the maps r i and a i induce H.---isomorphisms, where t], denotes 1

homology with Borel-Quillen

~:p---coefficients as in 1.1. Moreover, since localized

isomorphism

in

dim F ( P 2 ) < co,

~IA-theory ,

where

a1

induces a

A - P1/P2.

Thus

~[A(F(P2),F(P1)) = 0. Comparison of the Serre spectal sequences of the Borel constructions of various spaces involved show that the map MaPPI(E,Rp(Y))

' MaPP2(E,Rp(Y))

induces also a Borel-Quillen localized isomorphism as well. Since G-homotopy equivalent, the map ,1: MaPPl(E,Rp(X))

E xy

and

X

are

~MaPP2(E,Rp(X))

induces a Borel--QuiUen localized isomorphism, as in Theorem 2.1.

2.6. Theorem. Let G, p , and R be as in Proposition 2.1 above. Let X be a free G---space such P that the conditions B(0)-B(2) of Proposition 2.1 are satisfied. Then there exists a finite dimensional G---space Y such that tt.(YP;~ = ) are finitely generated, YP E Top(Rp) for each P P E P ( G ) , and E x y and X are G-homotopy equivalent. If the complexes F(P) are taken to be finite complexes, then there exists an obstruction w(X) E ~(}(EG) such that w(X) = 0 if and only if Y is G-homotopy equivalent to a finite G---complex.

320

Outline of proof: In order to prove that such a Y exists, we actually proceed to construct U YP for each p ] I G I , in order to obtain maps the p-singular set of Y , i.e. Sp(Y) = ISPEPp(G) hp: Sp(Y)-----,Map(E,X) ~pP: Rp(Sp(Y) P) H, = H,(-;~p) U h : U Sp(Y) p P p

which are equivariant

and

such that

the

induced

maps

, Mapp(E,Rp(X)) induce H.--isomorphisms for each P E Pp(G) , where as before. By adding free G--cells to

U Sp(Y) P J Map(E,X) highly connected and we obtain f : Y0

we make the map DMap(E,X) so that

the cofibre Cf of f is a Moore space, and Sp(Y0) - Sp(Y). Then we try to show that ]~.(Cf;E) is EG-projective. In cases where we deal with finite complexes, the class []~.(Cf)] E i~0(/~G) will represent the finiteness obstruction w(X) which will be only well-defined up to ambiguity arising from different choices of Sp(Y) in the course of this construction. This leads, then, to a well-defined obstruction, denoted again by w(X) (by abuse of notation) in a subquotient of

In order to show that ~.(Cf;/7) is ~G-projective, we use the projectivity criterion Theorem 1.1 to reduce the problem to showing that H.(Cf;~)I~C is /TC-projective for each C ~ G , I C ] = p . But in this case, we are in the situation of Theorem 2.1, since by construction Rp(YoC ) ~ MaPC(E,Rp(X)) induces a homology isomorphism, and other conditions are also satisfied, as one can check from the hypothesis. Hence the proof of Theorem 2.1 shows that H.(Cf) [ ~C is ~C-projeetive for any such C. Fix a K E Pp(G). It remains to show how to construct Sp(Y) K . We proceed by induction on the lattice of p--subgroups Pp(G). Suppose that h P : Sp(Y) P for all subgroups P such that

K~P,

hp P :Rp(Sp(Y) P)

, Map(E,X) is constructed ,Mapp(E,Rp(X))

induces an

H,-isomorphism. Let L denote Sp(Y) for short. We add free W ( K ) - N ( K ) / K cells to LK and extend it to G-orbits (which are added to L in the usual fashion) so that the map a : L0 ~Map(E,X) in this way satisfies the following: the cofibre of a(K): LoK----~ MaPK(E,K), call it

C(e(K))

has homology (i.e.

]~.(-;~:p))

only in one

dimension, i.e. it is a H,(-;[Fp)-Moore space. Now ]~,(C(a(K))) is an rFp(W)-module and we claim that it is IFp(W)-free. Using the modular version of the projectivity criterion (Theorem 1.1), we need to check this for each cyclic subgroup of order p , say C C W , [CI = p • We have the exact sequence: 1

JK JK r ~C ----, 1 where I K ~ I = P " ] K I • Hence, by the K / induction hypothesis Rp(L 0 ) J MaPKr(E,R ) (X)) induces an H,--isomorphism. Translating this into W-actions, we have (LoK)C)P= LoK~ and Rp((LoK)C the diagram

~MaPc(E,Rp(X)K ) is a homology isomorphism. On the other hand, by studying

321

L0 K

a(K)

, MaPK(B , X)

l

1

(LoK)C

t.

, MaPK(E,X) C ~MaPKr ( E , X )

L0

1

K'

Rp(L0 as in Theorem

)

2.1, we conclude that

1

£'

' MaPK, (E,Rp(X)) H,(C(a(K)))

is cohomologically trivial, hence

FpG-projective. This ~:pG-projective module can be killed and the map

more connected so we achieve the inductive step.

a(K)

will be made

D

W e have the following interesting application: 2.7. Theorem. Let G, p, and Rp be as in Proposition 2.5. Let X be a G-space such that X and Mapp(B,X) belong to Top(Rp) for each P 6 Pp(G) . Then there exists a finite dimensional G-complex K such that E x X and B x K are G-homotopy equivalent, if and only if for each cyclic subgroup C i of order Pi there exists a finite dimensional Ci--complex K i such that B x X and E x K i are Ci-homotopy equivalent. []

Section 3. Some Anpli¢~tion~ and Problems To show that the theorems of Section 2 are useful, we need to verify the hypotheses in some geometrically interesting situations. This involves, in particular, cohomology computations of some equivariant function spaces, or equivalently, the space of sections of 5brations over B(~/p~) n arisingfrom Borel constructions. In this respect, J. Lannes'work [L] is quite relevant. Combined with some cohomology calculations of certain classifyingspaces, Lannes' theorem leads to finiteness results, from which we derive the validity of the hypotheses of the main theorem 2.1 for G = ~/p~. Then Theorem 2.7 allows us to derive the finiteness conclusions for a general finitegroup. W e recall below the following theorem of Lannes (conjectured by H. Miller in [ M m ] ). Let x be a p--elementary abelian group, and let K be the category of unstable algebras over the mod p Steenrod algebra. For any space X a homotopy class of maps B x ~X induces a homomorphism

H ( X ; ~ : p ) ~ H (B~r;Fp) in K.

3.1. Theorem (J. Lannes [L] ). Let X be a simply--connected space such that dim Hi(X;fp) < for all i > 0. Then the natural map

O8

322

[Br,X]

, HomK(H (X;~=p) , H (Br;l]=p))

is bijective. The first interesting case that we consider is a classical problem. Let X be a free G---space which is (non---eqnivariantly homotopy equivalent to the n---sphere Sn .

3.2. Problem. When does there exist a G-action on Sn such that E G x Sn is G-homotopic to X? In homotopy theory, this is a problem about spherical fibrations. Let monoid of self-maps of degree one of Sn . Then the spherical fibration X classified by a map homeomorphisms. B Top+(S n)

2 : BG

, B Jg+(S n)

provided that

G

Jg+(S n)

be the , BG is

X/G

acts on

X by degree one Problem 3.2 now translates into a lifting problem for the fibration

~B ~ + ( S n) for the map 2 . A more refined question is the following:

3.3 Problem. When is a spherical fibration over BG fibre homotopy equivalent to an ortthogonal fibration ? This problem involves a similar lifting problem for the fibration B0(n+l) for 2 .

, B Jg+(S n)

According to Theorem 3.1 this is reduced to a tiffing problem on the level of cohomology over the Steenrod algebra (which is not an easy problem in general either !). Now let us recall that according to Theorem 2.7, it suffices to solve the lifting problem of 3.2 for g/piT. (Note that Bousfield-Kan's completion [BK] suffices in this case). The tiffing problem of 3.3 for G = g / p g in fact can be solved on the level of cohomology due to deer, calculations of the structure of H (B o~d+(sn);i]=p) over the Steenrod algebra due to F. Cohen [CLM] and related computations of J. Milgram and Madsen-Milgarm (Cf. [M J ] , [Mj] and [MM] for example). Positive solutions to Problem 3.3 for G = ~/p~ and Theorem 2.7 give a partial answer to Problem 3.2. Namely, let X be a free G---space such that

X '~ Sn . Then there exists a finite

dimensional G---complex K such that E G × K is G-homotopy equivalent to X . In fact K may be taken equivariantly finitely dominated in the appropriate context. This result is the first step towards a complete solution of Problems 3.2 and 3.3 via methods of equivariant surgery, and it suggests that there are interesting relationships between Problem 3.3 and Atiyah's theorem on the K-theory of BG (to the effect that K(BG) is the I-adic compeltion of the representation ring R(G) CF. [ A t ] ) . Another interesting case is to consider G-actions on simply-connected Moore spaces. Let X

323

be a Moore space on which a finite group of square-free order acts freely. Suppose that ]].(X) has the following property with respect to the induced //G-module structure: For each prime order subgroup C C G , H.(X) I / / C - P • Q , where P is //C-projective and Q is indecomposable. ( P and Q depend on C ). Then there exists a finite dimensional G--space K such that E G x K is G-homotopy equivalent to X . The proof of the existence of the G---space K is reduced to the special case G = / / / p / / , thanks to Theorem 2.7 above. In this case,

it* (C;Q)

is isomorphic to either l~i* (C;//) or ~ * (C;I), where I is the augmentation ideal. This allows one to modify the arguments (involving the Sullivan fixed point conjecture and Lannes' Theorem 3.1) for the above special case X = Sn in order to construct the desired K . Finally, the above discussion leads us to the following conjecture which has intersting implications for the topological realizability of homotopy actions and the Steenrod problem, cf. [A2] for related discussions. 3.4. Conjecture. Let M be a finitely generated//-torsion free //G-module, where G is a finite group. Suppose that there exists a Moore space X with G-action such that H.(X) is isomorphic to M as //G-modules. Then there exists a finite dimensional Moore G---space K with the same property.

References [hl] [A2] [A3] [AB] [At]

[S] [BK] [Br]

[c j] [c]

Assadi, A.: "Extensions libres des actions des groupes finis", Proc. Aarhus Top. Conf. 1982, Springer LNM 1051 (1984). Assadi, A.: "Homotopy Actions and Cohomology of Finite Groups", Proc. Conf. Transf. Groups, Poznan, July 1985, Springer-Verlag LNM 1217 (1986) 26-57. Assadi, A.: "Finite Group Actions on Simply-connected Manifolds and CW complexes", Memoirs AMS 257 (1982). Assadi, A. - Browder, W.: "Construction of finite group actions on simply-connected manifolds" (to appear). Atiyah, M.F.: "Characters and Cohomology of Finite Groups", Publ. LH.E.S. Borel, A. et al- "Seminar on Transformation Groups", Annals of Math. Studies, Princeton University Press, Princeton, N.J. Bousfield-Ka.n: "Homotopy Limits, Localization, and Completion", Springer-Verlag LNM no. 304 (1972). Brown, K.: "Cohomology of Groups", Springer-Verlag GTM, no. 87 (1984). Carlson, J.: "The varieties and the cohomology ring of a module", J. Algebra 85 (1983), 104-143. Carlsson, G.: "The Homotopy Limit Problem", (Preprint 1986).

324

[Ch] [cE] [CLM]

[D] [Hw] [L] [M] [Mm] [Mj]

[MJ] [MM] [Q] [R] [Su] [w]

Chouinard, L.: "Projectivity and relative projectivity for group rings", J. Pure Appl. Alg. 7 (1976), 287-302. Caftan, H. - Eilenberg, S.: "Homologieal Algebra", Princeton University Press, Princeton, N.J. Cohen, F.R. - Lada, T.J. - May, P.J.: "The Homology of Iterated Loop Spaces", Springer LNM 533, (1976). Dade, E.: "Endo-permutation modules over p-groups II", Ann. of Math. 108 (1978), 317-346. Hsian#, W.Y.: "Cohomology Theory of Topological Transformation Groups", Springer, Berlin (1975). Lannes, J.: "Sur la Cohomologie Modulo p des p--Groupes Abeliens Elementaires", (Preprint 1986). Miller, H.R.: "The Sullivan Conjecture on Maps from Classifying Spaces", Annals of Math. 120, (1984), 39---87. Miller, H.R.: "Massey-Peterson Towers and Maps from Classifying Spaces'~, Proc. Alg. Top. Aarhus 1982, Springer LNM 1051 (1984). Milgram, J.: "A Survey of the Classifying Spaces Associated to Spherical Fiberings and Surgery", Proc. Syrup. Pure Math. 32 AMS (1978) 79--90. Milgram, J.: "The rood-2 Spherical Characteristic Classes", Ann. Math. 92 (1970) 238--261. Madsen, I. - Milgram, J.: "The Classifying Spaces for Surgery and Cobordism of Manifolds", Ann. Math. Studies, Princeton University Press (1979), Princeton, N.J. Quillen, D.: "The spectrum of an equivariant cohomology ring I", and "II" Ann. of Math. 94 (1971), 549-573 and 573--602. Rim, D.S.: "Modules over finite groups", Ann. Math. 69 (1959), 700-712. Sullivan, D.: "Genetics of Homotopy Theory and the Adams Conjecture", Ann. Math. 100, (1074) 1--79. Weinberger, S.: "Constructions of group actions: A survey of recent developments", Contemporary Math. Vol. 36 A.M.S. (1985).

C h a r a c t e r i s t i c n u m b e r s and g r o u p a c t i o n s

SUNG SOOK KIM

Let G denote the finite cyclic group of order n. The problem of determining necessary and sufficient conditions for F to be the fixed point set of a smooth cyclic group action on some sphere has been solved when n is a prime power. P. A. Smith proved that F must be a Z,-homology sphere. If n is odd it is also known that F is unitary. L. Jones has shown that these conditions are Mso sufficient to realize F as the fixed point set of smooth cyclic group action on some sphere when n is a prime power [J]. In the general case it is known that F is a union of smooth ma~folds, unitary if n is odd. And if Z,, acts on some even dimensional sphere, then the Euler characteristic number of F is 2 if action is orientation-preserving and 0 if action is orientation-reversing. We may ask about possible restrictions on Pontryagin numbers of components of the fixed set F. If n is a prime power, the Pontryagin numbers of the fixed point set all vanish by the P. A. Smith theorem and the Hirzebruch signature theorem. It is natural to ask whether such restrictions hold for other types of smooth cyclic groups acting on spheres. I n case where n is not a prime power, by work of R. Oliver [01] mad its extention of A. Assadi [Asd] and K. Pawalowski [Pa 1-4], we can construct exotic actions on spheres such that Pontryagin classes of fixed point sets do not vanish. But in these examples the Pontryagin numbers all vanish because these actions bound group actions on disks. R. Schultz has shown in [$2] that if G is a cyclic group whose order is not a prime power, then there are smooth actions of G on spheres such that the fixed point sets have nonzero Pontryagin numbers provided the dimension of the fixed point set is greater than 16. Testing to see if the lower bound in dimensions is necessary. There are two possibilities. First, more sophisticated computations might make it possible to remove the restriction on dimensions. Second, there might be some unusual things happening in low dimensions (compare Ewing's result for Zp actions on spheres). Main Theorem is evidence for the first one. M a i n T h e o r e m . Let G be a cyclic group of order p, where p is an odd prime, and let q ~ p be another odd prime. For each r > 0 there is a smooth G-action on some Zq-homology sphere such that the fixed point set is a closed connected 4r-dimensionM manifold with nonzero Pontryagin numbers. In fact, there are subgroups J. w"~X4r t,,q Of t h e oriented bordism group f~4r such that (i) every d e m e n t of Fix~'~q contains a representative that is the fixed point set of some smooth G-action on some Zq-homology sphere, (ii) w'-P'q s o ® Q, for a/1 r / > 1 and p # q. - ~ 4 r ® Q = f~4r R e m a r k : If p = q, then the fixed point set of some smooth G-action on some ZqhomoIogy sphere is a rational homology sphere by the P. A. Smith Theorem. It follows that the fixed point set maps to zero in f2.s ° ® Q.

326

A c k n o w l e d g m e n t s : I wish to express my gratitude to my advisor, professor Reinhard Schultz for his invaluable guidance, encouragement, and for his generous support during my research. I am indebeted to professor Mikiya Masuda for his encouragement and fruitful discussions. 1. P R E L I M I N A R I E S In [$2], lZ. Schultz showed the existence of closed smooth manifolds F that are fixed point sets of smooth Zpq-actions on homotopy spheres and have nontrivial Pontryagin numbers. In fact, sufficient conditions for such F were obtained and one can see that the argument works in our setting. We shall give a brief explanation in this section. We assume F is connected and unitary; i.e., the stable normal bundle v'F of F has a prescribed complex structure. Let G denote the cyclic group of order p, where p is an odd prime, and let ~ be a complex G-vector bundle over F such that

(1.1)

~a = F

and ~ = VF e K ( F ) if we forget the action o n ~. Decompose ~ into eigenbundles of the G-action as follows: p--1

(1.2)

¢k o t k,

¢= k=l

where ~k is a complex vector bundle and the generator g E G acts on tk(= C) as multiplication by (k (( = e2,~i/p). We define p-1 (ke~J + 1 S~(g) = C o n s t a n t ( / : ( F ) H H (ke=.j --1' [F]), k=l

j

where /:(F) denotes the Atiyah-Singer /:-class [AS] of the bundle tangent to F, [F] denotes the fundamental class of F, _ 9) have one fixed point smooth actions of As ( [16] - [21]). Moreover, N. P. Buchdahl, S. Kwasik and R. Schultz [6] showed that all S" (n >_ 6) have one fixed point locally linear Asactions. However, they also proved that none of S n (n _< 5) has a one fixed point locally linear action of any finite group. (Compare with M. Furuta [11], and [17]). We now mention how the present paper is organized. Our first observation will be made on the singular sets. In general, if a group G acts on X , then for any subgroup H of G the H-singular set Xs(H) is defined by

xs(H)= U (xexlgx=x}. g~H\{1} In particular, if H is the entire group G, then we write Xs instead of Xs(G) for convenience and call it the singular set of X. For three dimensional Mmology spheres with one fixed point smooth action of -45, there are at most four As-homeomorphism classes of singular Sets, which we call types. (Proposition 1.9.) These types give a device for studying cobordisms or surgeries, which will be considered in this paper. However, determining the As-homeomorphism classes itself may be of interest in its own right. As an example, we shall determine the type of the Poincar6 sphere E with a standard Asaction. (Theorem 1.13.) We next consider equivariant cobordisms in Section 2, where the main result will give a sufficient condition for two As-manifolds to be coborda~lt (Theorem 2.1.), and, in Section 3, G-normal maps and G-normal cobordisms will be studied following T. Petrie [23]. It seems that the notions and terminologies introduced in Sections 2 and 3 may be useful in tile other situations. Thus, we state the definitions and the assertions more generally than is necessary for the proof of our theorem. The proof of our theorem above will be given in Section 4 assuming a key lemma, whose proof will be found in Section 5. Throughout this paper, a G-action on a smooth manifold is understood to be a smooth G-action and a G-map is assumed to be continuous unless otherwise stated. We denote by Z, t t and C the ring of integers, the real number field and the complex number field, respectively, on which an action of any group is understood to be trivial. For a set X, we denote by IX[ the cardinarity of X.

339

1. T y p e s o f t h e s i n g u l a r sets In this section we first investigate the singular sets of three dimensional homology spheres with one fixed point As-actions. We denote the cyclic group of order m by Cm and the dihedral group of order 2m by D2m. Also, A4 means the alternating group on four letters, which is isomorphic to the tetrahedral group. For elements g l , g 2 , . . - , g n of As, the subgroup of A5 generated by g l , g ~ , . . . , g , is denoted by < g l , g 2 , . . . ,gn >. In the first two lemmas, we summarize some data on A5. Perhaps, it may be supposed that the reader is familiar with them. (cf. [12] Chapter 2, Section 2.4.) LEMMA 1.1. (1) The isomorphism cIasses of nontriviat subgroups of As are C2, C3, Ca, D4, D6, Dlo and A4.

( 2 ) A n y two subgroups of A5 are isomorphic if and onty i f they are conjugate. We now put x ----(1,2)(3,4), y = (3,5,4), z ----( 1 , 2 , 3 , 5 , 4 ) and u -- (1,3)(2,4) in As. LEMMA 1.2. (1) We have z 2 = y3 = z 5 : u 2 ~_ (uz2)3 : 1, x y z : y - l , z z z

= z - l , u z ---- xu 1,y = z - l u z -1 and uzu = zuxz. (2) T h e subgroup < x > is properly contained in the following seven subgroups. <x,u>(~-D4),

<x,y>(~-D6),

< x , z > (~- Dlo),

<x,uyu>(~-D6),

< x, uzu > (~- Dlo),

< x, z2uz > ( ~ A4) and A5

(3) The above < x, z2uz > contains < x, u > . Throughout this paper, unless otherwise stated, the above elements x, y, z and u are fixed and we write the subgroups of A5 as follows. C2 = - < x >,C3 = < y > , C a D4 = < x , u > , D 6 = < x , y >,D10 = < x , z >

=< z > and A4 = < x , z 2 u z >

It might be helpful to keep the following figure in mind.

A5

Figure 1.3. In this section, As-actions in the following family will be considered.

340

Definition 1.4. We denote by ,.q the family of topological As-spaces X satisfying the following conditions (1) - (5): (1) x = x o .

(2) IX a. l= 1. (3) X n = X K whenever H C K C A s , H ~ D4 and K ~- A4. (4) [ X H 1= 2 whenever H C As and H ~- D2m for some rn = 2, 3 or 5. (5) X H is homeomorphic to S 1 whenever H C As and H ~ C,~ for some m = 2, 3 or 5. Moreover, we let .MS be the family of all closed, oriented, three dimensional smooth As-manifolds X whose singular sets Xs lie in S. PKOPOSITION 1.5. Let X be ~ three dirnensionM homology sphere having a smooth As-action with exactly one t]xed point. Then X lies in .MS, that is, the singular set X8 of X belongs to S. Remark. It is well known that the Poincar6 sphere E with standard As-action is a homology sphere with one fixed point action, (cf. the paragraph following Proposition 1.9). Thus, E belongs to .MS. PROOF: Denote by p(Ah) the fixed point of X. Let V be the tangential representation of X at p(Ah). Clearly dim V = 3 and V A5 = O. Since dimensions of nontrivial irreducible real As-representations are at least 3, V is irreducible. Thus, from the character table of As, we can conclude that if H is a noncyclic (resp. nontrivial cyclic) subgroup of As, then dimV H = 0 (resp. 1). Also, d i m X H is 0 or 1 accordingly (if all components have the same dimension). The condition (5) of S follows immediately from Smith's theorem. Also, the conditions (4) and (3) follow subsequently to X c= ~- S 1 and X D• ~- S o by Smith's theorem since D2,n/Crn ~- C2 and A4/D4 ~- C3, respectively. Thus, X lies in .MS. This completes the proof. Let 7"/be the set of all subgroups of A5 isomorphic to A4, D10 or Do. Once we fix a space X in .MS, for a subgroup H in 7-/or H = As, we denote by p(H) the point in X, with isotropy subgroup H. Now note that the numbers of subgroups of A5 isomorphic to C2, C3 and C5 are 15, 10 :rod 6, respectively. Since Xsg = X for all g E As, X, is a union of at most 31 circles. ~-hrthermore, if Hi a n d / / 2 are distinct cyclic subgroups (# {1} ) of As, then they generate a noncyclic subgroup. Hence X H' ~ X H2 is either one point or S °. This means that X~ is a union of exactly 31 circles. These circles intersect at the points p(H) for some H. For example, in order to find the points at which the circle X ~ 2 intersects with the other circles, it suffices to look for the subgroups that properly contains 6'2. By ob:3ervations like this, we get the following. PROPOSITION 1.6. Let X be in .MS. Then ;

(1) The cirele XC~" mter~ects with ~he other circIes a~ e ~ c t I y six points p( A~ ), p( A, ), p(D6), p(uDou) = up(Do), p(D,o) and p(uDlou) = up(Dlo). (2) The circle X c~ intersects with the other circles at exactly four points p( As), p(D6), p( z2 A4 z3 ) and p( z3 A4 z2 ). (3) The elrele X ~ 5 intersects with the other circles at exactly two points p( As) and p(Dlo).

341

Next, we conversely see how many circles in Xs intersect at the points in the above proposition. For instance, to see the intersection at p(De), we must look for nontrivial cyclic subgroup of A5 properly contained in De. In this case, < x >, < y >, < x y > and < x y 2 > satisfy this condition. Therefore, the four circles X~, X ,v, X~ y and X~ ~* intersect at p ( D 6 ) . Similarly, counting the number of cyclic subgroups with the desired property, we get the following. PROPOSITION 1.7. Le* X be in M S . T h e n *he following s h o w s h o w m a n y circles in Xa/n~ersec~ a* *he poin*s in Proposi*ion 1.6. (1) 4 circles in*ersec* a* each p ( D 6 ) and p ( u D e u ) = u p ( D e ) .

(2) in*ersec* at each p(Dlo) and p(uD o ,) = (3) 7 cles intersect a* each p(A,), p(z A,z and (4) 31 circles in*e sec* at

p(Dle).

Now imagine that we walk on the circle X ,c2 starting from and ending at p ( A s ) . Since u x = z u , the action of u gives a diffeomorphism of X ~ 2 fixing p ( A s ) and p ( A , ) and interchanging p ( D 2 m ) and p ( u D 2 m u ) for m = 3 and 5. ( See also Proposition 1.6 (1).) Hence, on X ,c2, we must meet the intersection points p ( H ) ' s in one of the following order. (Note : In each case, we do not specify a direction.) (1) (2) (3) (4)

p(As) - p ( D 6 ) - p ( D l o ) - p ( A , ) - p ( u D l o u ) - p ( u D 6 u ) p(As) - p ( D 6 ) - p ( u D l o u ) - p ( A 4 ) - p ( D l o ) - p ( u D 6 u ) p(As) - p ( D l o ) - p(Ds) - p ( A 4 ) - p ( u D 6 u ) - p ( u D x o u ) p(As) - p(Dlo) - p(uD6u) - p(A,) - p(Ds) - p(uDlou)

-

p(As) p(As) p(As) p(As)

Definition 1.8. According as the above (1) - (4), we say that X , E S (or X E A/,.q) is of type (As - De - D10 - A4), (As - De - u D l o u - A4), (As - D10 - De - .44) or (As - D10 - u D # u - A4), respectively. Remark. In the above definition of types, the subgroups C2, A4, De, Die, u D 6 u and u D x e u are fixed (e.g. De = < x, y >), and Lemma 1.2 (2) implies that they are those subgroups that contain C2 = < x >. Hence by Lemma 1.1 (2), if we choose another involution in A5 instead of x = (1, 2)(3, 4), then we get conjugate ordered sets of subgroups, which means that, in some sense, the definition of type does not depend on the choice of an involution. However, notice that, fixing C2 = < x >, the subgroups De and D10 are choice-free. For example, there is no reason why < x, u y u > is called u D 6 u instead of De. So, once we fix an involution, the subgroups in the description of types should be considered just as themselves not as those isomorphism classes. Also, since there is an automorphism p of A5 such that #(C2) = C2, p ( D e ) = De and p ( D l o ) = uDzou, equivalent manifolds may have different types. Here we say that two As-manifolds X and Y are equivalent if there are an automorphism p of A5 and a diffeomorphism a from X onto Y such that the following diagram commutes. (Here the horizontal arrows mean actions of As.) AsxX

, X

AsxY

, Y

342

Now Lemma 1.1 and Propositions 1.5, 1.6 and 1.7 yield that the equivariant homeomorphism classes of the singular sets in $ are determined by the above types. Namely; PROPOSITION 1.9. Let X and Y be in .ADS. Then X , and Y, in $ are Ah-homeomorphic if and only if they have the same type in the sense of Definition 1.8. Now we concentrate on a standard As-action on the Poincar4 sphere P, and figure out the type of its singular set. As is well known, for a nontrivial representation p: A5 --* S0(3), P, = P,(p) is defined to be a (left) coset space of SO(3) by a subgroup p(Ah). Here SO(3) is the special orthogonal group of degree three over the real number field. The standard As-action on Y](p) is the action A5 x P,(p) ---+ P,(p) ; (g, ap(Ah)) ~-+p(g)ap(Ah). Notice that, by the definition, P,(p) is a three dimensional homology sphere with one fixed point As-action. Thus Proposition 1.5 implies that 5] lies in M S . Moreover, it is easy to see that the tangential As-representation Tp(E(p)) at the fixed point p of £(p) is As-isomorphic to the As-module V(p) associated with p. Using this fact, we can see that P.(p) is Ah-diffeomorphic to 2(p') if and only if the associated characters Xp and Xp, coincide with each other. Note that As has two inequivalent 3-dimensional irreducible real representations. Thus, there are two Ah-diffeomorphism types of the Poincar4 spheres. (ghrthermore, there are two As-homotopy types of the Poincar~ spheres.) Let us begin our computation. We use the special unitary group SU2(C) of degree 2 over the complex number field, SU2(C)= {(2~

b)I

a, b E C , l a 1 2 + l b 1 2 = l } ,

for computational convenience. It is a double cover of S0(3), i.e., SU2(C) has the center {4-1} of order 2 and SU2(C)/{4-1} is isomorphic to S0(3). Moreover, there is an injectire homomorphism/5 from the binary icosahedral group SL(2, 5) to SU2(C) such that the image of/5 contains the center of SU~(C). Also, the factor group/3(SL(2, 5))/{4-1} is isomorphic to As. Hence, not only as topological spaces, but also as As-spaces, P~ is diffeomorphic to SU2(C)/~(SL(2, 5)). (For these and related facts, we refer the reader to [28] §4.4.) We now construct fi concretely. Put

=

(4 + ~ - , ) 2

2rr

= 2 c o s 2 --5-'

7 = _~a = - ( c o s - ~ + x/-Z-Tsin _~_),6rr (5 -- ~ + ~- 1 _ ~ 2 _ ~-=

~

27r

cos 7- and sin -~

~2 _ ~ - i - - 2 s i n ~ , where ( = cos ~ + vrz"f sin ~ . Note that ~ = a, ~ = - $ and ~7 = _~,. Also, we define six matrices as follows:

343

L=-~-

1

a

1

'

It is easy to show that the above six matrices lie in SU2(C). Also, write E and B to mean the identity matrix in SU2(C) and the matrix C - 1 D C -1, respectively. Then, we have the following. LEMMA 1.10. (1) A 2 = D 2 = C s = B 3 = ( D C 2 ) 3 = - E ,

A C A -1 = C -1, A D A -1 = - D , A B A -1 = B -1 and D C D = C D A C . (2) A, C and D g e n e r a t e a subgroup o f SU2( C ) isomorphic to the binary icosahedral group S L ( 2 , 5). (3) N - 1 C N = D - 1 C D , M - 1 B M = B -1 and L - 1 D L = D A .

PROOF: (1). For the equations not involving B, see p.93 of [28]. Also, noticing D -1 = - D , we have B 3 = (C-1DC-1) 3 = -C(DC2)-3C

-1 = - C E C -1 = - E

A B A -1 = ( A C - I A - ~ ) ( A D A - 1 ) ( A C - 1 A

-1) = -CDC

and

= B -~.

Also, (2) is found loc.cit., and (3) follows by easy computations. By Lemma 1.10 (2), we henceforth identify SL(2,5) with the subgroup of SU2(C) generated by the above A, C and D, that is, we regard S L ( 2 , 5 ) = < A , C , D > and define ~ to be the inclusion map from S L ( 2 , 5) to SU2(C). We now write A', B ' , C' and D' to denote the elements of S L ( 2 , 5)/{=t=1}(-~ As) obtained as the images of A, B, C and D, respectively, under the natural epimorphism from SU2(C) onto SO(3). Then, since Lemma 1.2(1) gives a generating set and relations of As, considering ~ modulo the center {=t=l}, Lemma 1.10 (1) yields the following. (See also p.93 of [28].) LEMMA 1.11. There exists an injective h o m o m o r p h i s m p from A5 to SU2( C ) / { + I }('~ S0(3)) sending x, y, z and u into A', B', C' and D ' , respectively. Remark. By direct calculation, we have Xp(Z) = ¢ + ~-1 + 1 = (1 + yrb)/2, where Xp is the character associated with p. If we put ¢ = cos ~ + ~/'Z-fsin -~, 4,~ then we similarly have an injective homomorphism p' from A5 to SO(3). However, this gives the Poincar~ sphere whose Ab-diffeomorphism type is different from that of E(p) since X,,(z) = (1 V~)/2. -

In our computation the Poincar~ sphere E = E(p) given by the above homomorphism p win be used. For any matrix Q in SU (C), we denote by Q the left coset of (SL(2, 5)) in SU2(C) containing Q. Thus, we may consider Q as a point in E. Now the points on the intersections of the circles in the singular set E~ are obtained as follows.

344 m

m

m

LEMMA 1.12. It follows that p(A4) = L, p(D6) = M, p(Dlo) = N, p(uD6u) = D M and p(uDlou) = DN. PROOF: Let Q be a matrix in SU2(C). Then Q lies in ~c2 = ~x if and only if AQ and Q lie in the same coset of SL(2, 5), that is,

Q-1AQ lies in SL(2,5). Since any two elements in SL(2, 5) of order 4 are SL(2, 5)-conjugate, the above is equivalent to

Q-1AQ = Q,AQ ,-1 for some Q' in SL(2, 5). Since -Q = QQi, we can conclude that if Q lies in ~=, then we can take Q from the centralizer Csu2(c)(A) of A in SU2(C) :

Csu,(c)(A) = {Q 6 SU2(C) I QA = AQ}. Conversely, for all Q in Csu~(c)(A) the point Q clearly lies in E ~. On the other hand, by an easy computation, it follows that Q lies in Csu2(c)(A) if and only if it has real entries. Thus, we get

Csu=(c)(A)=

(cose

sine'~

\-sin0

cos0J

I e~R}.

However, since CSL(2,s)(A) = {+E, +A}, we may write

< 0 _< ~r/2}.

cos e J ' 0 _

Namely, the above gives the circle ~ , on which the points are parameterized by 0 in 0 < 0 < ~r/2 with 0 = 0 and 0 = 7r/2 being identified. Now since L, M D and N have real entries, it follows that L, M D ( = -M) and N lie in E =. Next, notice that Lemma 1.10 (3) implies that

M - 1 B M , N - 1 C N and L - 1 D L lie in SL(2, 5). Thus, for example, B M and M lie in the same coset of SL(2, 5), which implies that M lies in EY. Likewise, N and L lie in Ez and E u, respectively. Thus, "M E ~z ["l ~]Y(----~D6), "~ e ~= 1"7~ z ( = ~Dt0), and L E E = A Eu(= E D' = ~A,), and we obtain the first three equalities. Finally, since p(u) = D', the last two equalities follow dearly. The type of standard actions can be determined as follows. THEOREM 1.13. Let p: A5 ---+SO(3) be a nontrivial representation. Then the singular set P,(p), of the Poincar4 sphere P,(p) with standard As-action is of type (As - De -

345 uDlou - A , ) (resp. (As - De - D~o - A , ) ) if xp(z) = (1 + v~)12 (resp. (1 - v/5)12), where Xp is the character associated with p.

PROOF: First let p: A5 ~ SO(3) be the homomorphism in Lemma 1.11. Then, we have Xp(Z) = (1 -t- V~)/2. Recall the six matrices in Lemma 1.10. We look at the points D M , N and L. For a matrix Q in SU2(C), let us denote its ( i , j ) entry by (Q)id- Then we have; cos~ -cos 2 ( D M ) ~ j = ~ l - ~ a 2 (a6' - ~) =

1 + 4 cos 3

(DM)I,2 = ~'-l--~a,2 (a~ + 8') = COS

(lv)1,1 = 4 = - / 6

=

"~

(N)~,2 = vrZT6 ' = ~ 1 2 sin

= 0.3568...,

sin ~g~~/1 + 4 cos4 "~ = 0.9342...,

2 s i n ' ~ ~/1 + 4 cos4 ~

= 0.5257...,

= 0.8507.. . and

(L)~,~ = (Lh,~ = 0.7071 . . . . Note that they are all real and positive. Also, we have 0 < (DM)I,1 < ( N ) I j < (L),.~. Hence, by the argument in the proof of Lemma 1.12, we can determine the order of the points on ~(p)z as in the statement of the theorem, which shows that the type of ~(p)s is (As - De - uDlou - A4). Let p be an automorphism of A5 which is not an inner automorphism. (Note : Those p are actually given by conjugation by some elements that lie in the symmetric group on five letters but not in As.) Then, we have Xp,(z) = (1 - V~)/2. And there is such an automorphism p that satisfies p(C2) = C2, #(D6) = D6, #(Dlo) = uD~ou and p(A4) = -44. (e.g. The conjugation by the transposition (1,2).) Hence it follows that the type of ~ ( p p ) , is (As - D6 - D~0 - A4). Since A~-diffcomorphism type of a Poincar4 sphere is determined by the character, this completes the proof of Theorem 1.13. Remark. For any nontrivial real As-representation p: A5 --~ SO(3) and any type 7 of the singular set, there exists a three dimensional homology sphere with one fixed point As-action whose tangential representation at the unique fixed point is isomorphic to V(p), whose type is 7 and which is Ah-cobordant to ~(p), where V(p) is the As-module associated with p. This will be proved in [4]. 2. E x i s t e n c e o f e q u i v a r i a n t c o b o r d i s m s Let G be a finite group. In this section, we suppose that a G-manifold, a real G-module and a real G-vector bundle possess a G-invariant riemannian metric, a G-invariant inner product, and a G-invariant metric, respectively, and that all are oriented. We explain notations and terminologies which will be used in the rest of this paper. Let X be a topological G-space. For a real G-module M, let c x ( M ) be a real G-vector bundle whose total space is X x M with diagonal G-action, base space is X and fiber

346

is M . If the base space is clear from the context, we write e ( M ) instead of ex(M). Let and 7/be real G-vector bundles over X. If there is a G-vector bundle isomorphism a from ~ ¢ x ( M ) to ~Oex(M) for some real G-module M , we say that ~ and 77 are stably G-isomorphic. We usually write this isomorphism simply by a: ~ --* q instead of the precise description such as a: ~ @ ¢x(M) -~ q G ¢x(M) and call it a stable isomorphism from ~ to 77. This notation will be used even if a is actually an isomorphism from ~ to r/ (not from ~ @ e x ( M ) to q @ ex(M)). However, if this is the case, we call a an unstable isomorphism from ~ to q. Stable or unstable isomorphisms of a particular type will have the following special names. A stable (resp. an unstable) G-trivialization is a stable (resp. an unstable) G-vector bundle isomorphism a: ~ ---+ex(V) for some real G-module V. Note that a stable G-trivialization a is actually an isomorphism from ~ @ e x ( M ) to e x ( V @ M) for some real G-module M. For a real G-vector bundle ~ over X and H C G, let ~Hdenote the H-fixed bundle of over X H, and let ~H be its orthogonal complement in ~ [ x ' . So, in particular, we may write ¢x(M) H = e x ~ ( M H) and cA-,, (]~r)H = e x n (MH), where ]~/H is the submodule of H-fixed elements in M and Mii is its orthogonal complement in M. Suppose that we have an unstable G-isomorphism a x : ~ @ ex(M) --+ ~' • ex(M). Then, for each subgroup H of G we have unstable N c ( H ) - i s o m o r p h i s m s

aH:{H ®eX,(k/IH) ~ {,H O e x , ( M H) and OtXH: ~H (~ e xH( MH ) ---+~I (~ e xu( MH )" Finally, we give the following remark, which is used freely thereafter. Let ax: ~ -+ ¢x(V) and ay: ~' --+ ey(V) be stable G-trivializations. Then they are precisely unstable isomorphisms

ax:~ G g x ( M x ) --* ¢x(V O Mx) and a t : ~' ~ ~ v ( M r ) --, e v ( V • Mr), for some real G-modules Mx and My, which m a y be different. However taking stabilizations of t h e m if necessary, we can regard them as

a x : { @ex(M)---+ ex(V O M ) and av:{' @ev(M) ---*e v ( V @ M ) , for the same real G-module M (e.g. M = M x ® M y ) . In the rest of this section, we assume that our group G is As. As in the previous section, denote by T / t h e family of all subgroups of As isomorphic to A4, D10 or D6. A n d for a manifold X in M S and H in 7g, we let p(H) = p x ( H ) be the point in X with isotropy subgroup H . Also, let T(X) be the tangent bundle of X , and let Tp(X) be the fiber of T(X) over p in X . Now we can state our main theorem in this section. THEOREM 2.1. Let X and Y be manifolds in M S . Suppose that they have stable Astrividizations ax: T(X) --+ ex(V) and a v : T(Y) --~ ev(V) for the same read As-module

V. Then X and Y are As-cobordant. Remark. This V must be an irreducible real As-module of dimension three.

347

Let a x : T ( X ) ( 3 e x ( M ) ~ e x ( V @ M ) be an unstable As-trivialization such that the map between the fibers over the point p(As),

otD4 Ip(A~):Tp(A~)(X) D4 ~3 M D4 --* V D'i O M D" (i.e., M D" --~ MD4), is orientation preserving. The set of those trivializations can be written as a disjoint union of two subsets. The elements in one subset are called type plus and those in the other subset are called type minus. We now explain this fact. First, notice that we can adopt the orientation of V so that the restriction OtX lp(As);Tp(Ar,)(2) (~ J~[ --'+ V (~ M

preserves the orientation. This orientation on V is regarded as the orientation of VD~, since VD4 = V as sets. Consequently, the restrictions otD4 [p(At,).. Tp(As)(Xr ) D4 @ 1~fD4 .__4. v D 4 (~ M D 4

and

• . a X D 4 ]p(As).Tp(A~)(X)D4 • MD4 -~ VD4 (3 MD4

preserve the orientation. This implies that the restrictions OtD'l Ip(A,t):Tp(A4)(.X) D4 (~ M D4 --* V D'l @ M D4 and

OtXD4 Ip(AD:Tp(A4)(X)D4 G MD4 ~ VD4 • MD4 both preserve the orientation or both reverse the orientation. In the former case, we say that a x is of type plus (or +), while in the latter case we say that a x is of type minus (or - ) . Here we give an example. Example 2.2 (cf. [8] or [9] Section 2). Let V be an irreducible real As-module of dimension three, and let p be the homomorphisln from As to S 0 ( 3 ) associated with V. Then, p determines the Poincar6 sphere E with standard action. The tangential representation of E at the unique fixed point is isomorphic to V. Moreover there exists an unstable As-trivialization a~:+: T E --* e s ( V ) , which is of type plus. Remark. Let X E .ADS, and let a x : T ( X ) ~ e x ( V ) be a stable As-trivialization. We write it as an unstable isomorphism ~ x : T ( X ) @ ¢(M) --, e(V (3 M). In the case where d i m M A5 k 1 (i.e. M __DtZ as As-modules), we can find an unstable As-isomorphism a~x:T(X) ~ e ( M ) ~ e ( V @ M ) such that a~-° ' [p(A~):M 04 ~ M D4 is orientation preserving, as follows. If a D4 [v(A~) is orientation preserving, then obviously we can set a~x = a x . Clearly, the map - 1 : l:t --~ Ft is orientation reversing. If a D4 [p(A~) is orientation reversing, then we can take a~. as the composition of a x and the stabilization of e x ( - 1 ) : s x ( R ) ~ e x ( I { ) . Thus, for the proof of Theorem 2.1, we may restrict stable As-trivializations a x to ones such that o~D4 ]p(A~):M D4 ~ MD4 are orientation preserving (i.e. ones having a type plus or minus). The rest of this section is devoted to proving the above theorem. The proof consists of several lemmas (Lemmas 2.3 - 2.9), in which we state the assertions in general context. In the first lemma, we shah show that, given a stable As-trivialization of any type,

348

we can construct another stable Ah-trivialization of the other type. So, to prove the theorem, we may assume that X and Y both have the trivializations of type minus. LEMMA 2.3. Let X lie in .MS. Suppose that there is a stable As-trivialization axe: T ( X ) --, ~ x ( V ) of type ~ = q- or - , then there also exists a stable Ah-triviaiization a x _ Q : T ( X ) ~ x ( 1 % ) ~ e x ( V ~ B.) of type -Q. PROOF: Note that V is an irreducible As-module of dimension three. So, we have a homomorphism p from As to S0(3) as in Example 2.2. Then, we can take a covering ~: SL(2,5) ~ SU2(C) of p, and identify SU2(C) with S(H), the unit sphere of the quaternion field H. Thus we obtain a four dimensional real SL(2, 5)-module H~d/ by sending (g, a) e SL(2, 5) x H to ~(g)a~(g) -1 e H. This H~dj can be regarded as an As-module, and moreover is isomorphic to V • 1%. Hence, we identify them, namely, H~dj = V $1%. Let f : X ---* S(H~dj) be the As-map obtained by pinching the outside of the open As-disk neighborhood of the fixed point p(Ah) of X. We may assume that f(p(Ah)) = 1 e H,di. Then, we get f ( p ( g ) ) = - 1 for H in 7-(. For an integer k, define an As-trivialization Twistk(ax~): T ( X ) • ~x(1%) ---* e x ( V @ R) by Twistk(o~xa)(a) = ax~(a)f(p(a)) k for a E T ( X ) ~ ¢ x ( R ) , where p: T ( X ) ~ e x ( R ) --~ X is the bundle projection, and the multiplication is taken in the fiber H~dj = V ~ 1%. Since Twistk(axQ) is of type (--1)kp, taking Twists(axe), the lemma is proved. Later "Twist" is used again. Note also that if k is even, then the restriction of Twistk(ax~) to the fiber over p(H) (H E 7"[ or H = G) is the same as that of axe. Concerning the type of the singaalar sets, we have the following, by which, in the proof of the theorem, we may assume that X~ and Y, have the same type. (Note : As-surgery does not change cobordism classes.) LEMMA 2.4. Let X be a manifold in .MS, and let a x : T ( X ) --~ e x ( V ) be a stable As-trivialization of type ~. Choose an arbitrary type 7 of the singular set. Then, one can perform As-surgery on X of isotropy type (C2) to obtain a manifold X ~ in .MS of type 7 and a stable As-trivialization ax, : T ( X ' ) ~ e x , ( V ) of type ~. PROOF: Let X~ be of type (As - H~ - H2 - An). Take a point p of X c2 between the points p(H~) and p(H2) which are points in X with isotropy subgroups H~ and H2, respectively. Further take embeddings ¢i: S o --* X c2, i = 1, 2, such that (1) (2) (3) (4)

¢1(1) lies between p(Ah) and p(H,), ¢1(-1) lles between p ( g l ) and p, ¢2(1) lies between p and p(H2), and ¢2(-1) lies between p(g2) and p(A4).

Consider D4-surgery on X C2 along indcD~¢i: D4/C2 x S O --* X C~, i = 1,2. Then X C2 is changed to a D4-space Y ( X ) consisting of five circles as in Figure 2.5. We note that the D4-diffeomorphism type of Y ( X ) is independent of any initially given X in .MS. From this observation it holds that by As-surgery of isotropy type (C2) on X, we can obtain a manifold X ~ in .MS of type 7. Since As-surgery employed here is of isotropy type (C2) and of dimension 0, there are no obstructions to obtaining a stable As-trivialization of type # after the surgery. This proves the lemma.

349

.f.--..~ u p (Hl)

p(H~)

p

(H2)

p (A4)

~

up (H:z)

Figure 2.5. We denote the points in X and Y with isotropy subgroup H in 7-/by p(H) = px(H) and q(H) = pv(H), respectively, for notational convenience. Since we may assume that Tp(AD(X) and Tq(As)(Y) are isomorphic in the proof of the theorem, the next lemma shows that we can choose closed As-regular neighborhoods RN(As, Xa) of X , in X and RN(As, Ys) of Yo, between which there is an orientation preserving As-diffeomorphism. LEMMA 2.6. Let X and Y be manifolds in MS. Then, we can choose closed Asregu/ar neighborhoods R N ( As , X8 ) and RN (As, Y~) such that there exists an orientation

preserving As-dit~eomorphism fl:RN(As,X,) --~ RN(As,Y,) if and only if X and Y have singtdar sets of the same type and the tangential As-representations Tp(AD(X) and Tq(As)(Y) are isomorphic to each other. PROOF: Since the 'only if' part is obvious, we prove the 'if' part. Note that the two As-modules Tp(As)(X) and Tq(A=)(Y) have orientations since X and Y are oriented. Multiplying the real number - 1 if necessary, we may assume that the isomorphism from Tp(A,)(X) to Tq(As)(Y) preserves the orientation. By the equivariant tubular neighborhood theorem, Tp(A~)(X) (resp. Tq(A~)(Y)) can be regarded as an open Asdisk neighborhood of p(As) (resp. q(As)). Taldng the restriction to the unit disk, we obtain an orientation preserving As-diffeomorphism from a closed As-disk neighborhood RN(As,p(As)) of p(As) to RN(As,q(As)) of q(As). Let H be a subgroup in 7-/. If C is a nontrivial cyclic subgroup of H, then X c and y V are connected, and hence res

H) (X) ~ = rest

~ res (A~)(X) =

(As) ( Y ) ~= res

(H) (Y) •

350

This implies that Tp(H)(X ) is H-isomorphic to Tq(H)(Y). Similarly to the above, we can obtain an orientation prescrving H-diffeomorphism from a closed H-disk neighborhood RN(H,p(H)) of p ( H ) to RN(H,q(H)) of q(H). We put X~=

U

XH"

HET/

A dosed As-tubular neighborhood RN(As, X~) of XT~ is obtained as the disjoint union of RN(As,p(A~)) and RN(H,p(H))'s, where H runs over 7/. From the above argument, there exists an orientation preserving As-diffeomorphism from RN(As, X~) to RN(As, Y~). Let C be the set of all nontrivial cyclic subgroups of As. Since Proposition 1.9 implies that Xs is A~-homeomorphic to Y~, for C E C we can choose a closed, thin, Nns(C)-tubular neighborhood RN(NA~(C),X c) of X C. Then a closed equivariant regular neighborhood RN(As, X~) can be obtained as follows.

RN(As,X~) = RN(As,X~)U U RN(NA'(C)'XC)" CEC Since dimXa = 1 and NA,(C)/C ~ C2 for all C C C, the above orientation preserving As-diffeomorphism from RN(As, X~) to RN(A~, Y~) can be easily extended to one from RN(As, Xs) to RN(As, Ys). This proves Lemma 2.6. Now suppose that X and Y lie in fl45 and that there are stable A5-trivializations

ax:T(X) ----*ex(V) and ay:T(Y) --~ ey(V) of type minus. Moreover, assume that there is an orientation preserving As-diffeomorphism fl from a closed A5-regular neighborhood RN(As,XT~" of X~ in X to RN(As, Y~) of Y,. Since the general 'fiber' of RN(As,X~) over X t'2 is a two dimensional disk, for each integer k we obtain an Asselfdiffeomorphism of RN(As,X~) by equivariantly twisting the 'fiber' k-times along X c2. Denote this selfdiffeomorphism by 7k, and set ~k = ~'Yk. For any subgroup H of As, we can choose a closed equivariant regular neighborhood RN(NA~(H), X~(H)) of Xa(H) in X so that

RN(NA,(L),X~(L)) C RN(NA~(If),X~(K)) whenever {1} ¢ L C It" C As, and we can regard RN(NAs(H), Y~(H)) = ~(RN(NA~(H),Xs(H))). Now consider the stable, real A4-vector bundle map

¢~,i

= ( a y ])(d~j. [)(Twist~i(ax)[)-x:

RN(A4,X~(o~)) × V --. RN(A4,Y~(D4)) × V.

For "Twist", see the proof of Lemma 2.3. Notice that ¢

=

¢i,j

is actually a map from

RN(A4,X,(DD) × (V ~ M) to RN(A4,Y,(o~)) × (Y @M) for some real As-module M. Moreover, ¢ [p(As): V ~ M ---* V @ M (the restriction of ¢ to the fiber over the As-fixed point) is an As-isomorphism and the map ¢ Ip(A4): V @ M --~ V @ M is an A~-isomorphism, which are independent of i and j . Let R, U and W be irreducible real A4-modules of dimension 1,2 and 3, respectively. Then, resAA:V -~ W, and for adequate integers £, m and n, we have

res :(Y

M) Z m • mU •

351

In particular, dim(V @ M ) = e + 2m + 3n. In this situation we have the following. LEMMA 2.7. Suppose that the map ¢0,0 [p(Ab) is regularly As-homotopic to the identity map. (Note: By the remark above, this becomes true for nil ¢i,j.) If g >_ 3 and n > 3, then there exist integers i and j such that the A4-vector bundle map ¢ = ¢i,j is regularly A4-hornotopic to the product map of the base map with the identity map on the fiber V (gM.

PROOF: We use the notation: G = As, H = An, D = D4 and C = C2. First consider ¢ -- ¢i,j for arbitrarily fixed i and j. It may be assumed without loss of generality that

¢ [RN(O,p(G)):RN(G,X a) x (V • M) ~ R N ( G , Y a) x (V @ M) is the product map of the base map with the identity map on the fiber. We note that the space Aut(H, V ® M) of H-automorphisms of V @ M is homeomorphic to GLe(R) x G L , , ( C ) x GL,,(R.) by Schur's lemma. And ¢ Ip(H):~?R.@ mU ® n W --* gR G mU ~ n W is a direct sum of isomorphisms eL:L ~ L, where L -- ~R.,mU or nW. Siince Aut(H, mU) ~- G L m ( C ) is connected, emv is reg~alarly H-homotopic to the identity map on mU. Without loss of generality, we may assume that emu is the identity map. Since Twist21(ax) and a y are of type minus, etp~ and enw are orientation preserving. This fact implies that etP. (resp. e n w ) is regularly H-homotopic to the identity map on gR (resp. nW). Hence we may assume that ¢~p~ and ¢,,w are the identity maps, and consequently that ¢ Ip(g) is the identity map. Thus we can assume that

¢

[RN(H,XIt):

R N ( H , X H) x (V @ M) ~ R N ( H , yH) × (V ~ M)

is the product map of the base map with the identity map on the fiber. The H-space Xs(D) \ IntRN(H, X D) consists of six line segments, among which there is no H-fixed one. Let [a, b] be one of the two line segments lying in X C. If ¢ ][a,b] is regularly C-homotopic to the product map t of the base map with the identity map on the fiber relatively to the boundary {a,b}, then we can conclude that the map ¢ is regularly H-homotopic to tile product map of the base map with the identity map on the fiber. The obstruction a to constructing a :regular C-homotopy between ¢ [[a,bl and ~ ( relatively to the boundary) lies in ~rl(Aut(C,V ~ M)). We note that Aut(C, V@M) = Aut(C, (V@M) C) x Aut(C, ( V ~ M ) c ) . Thus the obstruction a can be written as (al, ~2), where aa E 7rl(GLe+2,,+,(R)) -~ Z / 2 and a2 E 7rl(GL2,,(R)) ~ Z/2. Now consider the effect of changing the choice of i and j . If we replace ¢i,j by ~)i-bk,j, then al does (resp. does not) change if k is odd (resp. even). The change o f j has similar effect on a2. However, ¢i,j and ¢i,j+k give the same al. Thus, we can find integers i and j such that the obstruction a vanishes. Therefore, we have proved Lemma 2.7. Return to the proof of Theorem 2.1. By the argument given so far, we may assume that we are in the situation in the paragraph preceding Lemma 2.7. Now take M sufficiently large so that M includes at least three isomorphic copies of each irreducible real As-modules. Then, the conditions e > 3 and n _> 3 are satisfied. If ¢ Ip(A~) (for aX

352

and ay) is not regularly As-homotopic to the identity map, then, in the following way, we can construct a~,: T(Y) ~ ey (M') --~ ey (V ~BM') such that ¢' Iv(As) for o~X and o~, is regularly As-homotopic to the identity map. Let o~ be the map oty @ey(¢ [v(A,)): (T(Y) @ey(M)) @ey(V @M) --4 ey((V @M) @(V ~ M)). Then we can obtain a stable As-trivialization a~, of type minus by the method used in the proof of Lemma 2.3 (for M' : M ~9V @M ~BR) and it is easy to see that ¢' [p(A,) for aX and a~, is regularly As-homotopic to the identity map. Thus we may assume that the conclusion of Lemma 2.7 holds. Hence, changing ax and/3 suitably, we may assume that ¢ is regularly A4-homotopic to the product map of the base map with the identity map on the fiber. For a subgroup H of As, we define an NA~(H)-manifold Z(H,X) by

Z(H, X) = X \ IntRN(NAs(H), X~(g)). We define Z(H, Y) similarly. Glue Z(H, Y) and Z(H,X) along the boundary by the restriction of the As-difl'eomorphism/3 from RN(As, X~) to RN(As, Ya) and get a closed Na(H)-maxfifold Z(H) with free H-action, that is,

Z(H) = Z(H, r ) uzl Z(H,X). Then, by Lemma 2.7 we may assume that the A4-manifold Z(D4) has the tangent bundle stably A4-isomorphic to SZ(D4)(V). Since the D4-action on Z(D4) is free, we have the classifying map fD4:Z(D4)/D4 ---+BD4 of the principal D4-bundle Z(D4) --~ Z(D4)/D4, where BD4 is the classifying space of principal D4-bundles. LEMMA 2.8. The map (Z(D4)/D4, fo4) is null cobordant. PROOF: We abbreviate Z(D4) to Z. Let wi = wy(Z/D4 ) • HY(Z/D4; Z/2) be the j-th Stiefel-Whitney class of the manifold Z/D4. For each partition k + kl + ... + kr = 3 and each element c • Hk(BD4; Z/2), the element

< Wk, ...wk, f~(c), [Z/D4] > in Z/2 is called a bordism Stiefel-Whitney number of (Z/D4, f]),), where [Z/D4] is the orientation class in H3(Z/D4; Z/2). From bordism theory, it follows that the bordism class [Z/D4, fD4] is null if all the bordism Stiefel-Whitney numbers are zero, (see P. E. Conner and E. E. Floyd [7] Chapter II Theorem 17.2 or F. Uchida [30] Theorems 2.15 and 2.18). Since Z/D4 is orientable, closed, three dimensional manifold, its tangent bundle is stably trivial. Thus, all the Stiefel-Whitney classes vanish. It follows that [Z/D4,fo4] is null if fb4(c) = 0 for all elements c e H3(BD4; Z/2). Let D4 = C2 x C~, where C~ C D4. Then BD4 = BC2 x BC~, and we can regard H*(BC2; Z/2) as t h e polynomial ring Z/2[ a ] of indeterminate a mad H*(BC~; Z/2) as Z/2[ b ] of indeterminate b. Let Ir be the projection from BD4 to BC2. We observe the cohomology element (~rfo,)*(a 2) in H2(Z/D4; Z/2). The restriction of the homomorphism p: A5 SO(3) associated with V to the subgroup C,.~ is conjugate to the homomorphism given

by

353

x~-*

(10 ) 0 0

-1 0

,

where x is the generator of C2. Thus, (TrfD,)*(a ~) coincides with the second StiefelWhitney class w~(ez(V)/D4) of the vector bundle ¢z(V)/D4 over Z/D,. Since the tangent bundle T(Z/D4) (which is stably isomorphic to ¢z(V)/D4) is stably trivial, w2(T(Z/D4)) = 0 = w2(¢z(V)/D4). Thus, (~rfD~)*(a2) = 0 and f~),(a 2) = 0. This implies f~),(a a) = f~,(a2)f~,(a) = 0. Similarly we obtain f~,(b 2) = 0 and f~,(b a) = 0. Since H (BD,;Z/2) has a basis ab ,b over Z/2, we that/5,( ) = 0 for all elements c E Ha(BD4;Z/2). Consequently, (Z/D4,J:D,) is null cobordant. This completes the proof. Let fA5 :Z(As)/A5 ~ BA5 be the classifying map of the principal As-bundle Z(As) Z(As)/As. The following lemma completes the proof of Theorem 2.1. LEMMA 2.9. The map (Z(As)/As,fA~) is null cobordant. Consequently, X and Y axe As-cobordan$ (relatively to the singular set) to each other. PROOF: If all the cobordism Stiefel-Whitney numbers of (Z(A5)/As, fAs) are zero, then (Z(As)/As,fA,) is null cobordaa~t. First note that all the Stiefel-Whitney classes of Z(A5)/A5 vanish. Thus, [Z(As)/As,fA~] is null if f.~s(a) = 0 for all elements a 6 Ha(BAs; Z/2). Since D4 is a Sylow 2-subgroup of As,

7r~dD4: Ha(Z(As)/As; Z/2) --* Ha(Z(As)/D4; Z/2) is injective (see G. E. Bredon [5] p.121). It follows that (Z(As)/As,fA~) is null cobordant if (Z(As)/D,, fD4) is null cobordant, where f~)4:Z(As)/D4 --~ BD4 is the classifying map. It is easy to see that (Z(As)/D4,ffD4) is cobordant to (Z(Da)/D4,fD,) which is null cobordant by Lemma 2.8. Thus, (Z(As),fAs) is null cobordant. This implies that Z(A5) is null As-cobordant, and also that X and Y are A5-cobordant (relatively to the singular set) to each other. This completes the proof. 3. G - n o r m a l m a p s Let G be a finite group. In this section, we introduce the notion of G-normal maps and G-normal cobordisms defined by T. Petrie [23] and prove Proposition 3.5 below, in which we construct a G-normal map and a G-normal cobordism from a real G-module. These will be used in the next section for the proof of Theorem 0.1. Given a finite G-CW-complex X, the G-poset II(X) associated with X is defined by

n(x) = ]_I n0(x"); HC__a see [19] and [23]. For a E II(X), we set Go = {g E G I ga = a). A G-vector ~oundle (with G-invariant metric) over X gives a Go-vector bundle ~r~ over Xa (the underlying space of a) by

354

where H is the subgroup such that a E ~ro(XH). The collection ~r~ = {~r~( I a E II(X)} is called a II(X)-vector bundle over X. If ~ happens to be the tangent bundle T(X) of a compact G-manifold X (with G-invariant riemannian metric), then 7rT(X) is simply denoted by uX. Let 0( denote either ~ or 7r~. The stabilization s(o~) of #~ is defined

by where M is a real G-module with G-invariant inner product. The stabilization of a G-(or II(X)-)vector bundle isomorphism b: Q( --+ Q(' is defined by

s(b) = b @ 0(ida(M)): s(~)~) ~ s(o~'). Let X and Y be compact, oriented G-manifolds, and let ~ be a G-vector bundle over Y with fiber-dim ~ = dim Y. By a term G-normal map we mean a triple of the following maps. First, f is a G-map from (X, 0X) to (1I, OY) of degree one. Secondly, b is a stable G-vector bundle isomorphism from T(X) to f*~. And finally, c is a II(X)vector bundle isomorphism from uX to 7rf*~ such that ~r(b) = s(c) (cf. [19] and [23]), where ,(c) is a stabilization of c. Note that in the current paper, we use the term 'a Gnormal map' in the sense of [19] and [231 not in the sense of [18] nor [20]. A G-normal map is denoted by, for example, w = (f; b; c): (X, OX; TX; vX) --t (Y, OY; f*~; 7rf*~). However, if the boundaries OX and OY of X mad Y, respectively, are empty, then we write it by w = (f; b; c): (X; TX; uX) --+ (Y;/*~; rrI*~). Given two G-normal maps w = (/; b; c): (X, OX; TX; uX) --+ (Y, OY; f*~; ~rf*~) and w' = (I'; b'; c'): (X', OX'; TX'; ~,X') - , (Y, 0Y;/'*~; ~f'*~), the notion of a G-normal cobordism W = (F; B; C): (IV, OW; TW; uW) -, (I x Y, O(I x Y); F*(ez(R) x ~); ~rF*(eI(R) x ~)) between them can be given generalizing naturally the corresponding concept in ordinary surgery theory. Here I = [0, 1]. Let f: X --* Y be a G-map. For a prime p, f is called a {p }-equivalence if fP: X P YP is a rood p homology equivalence for every nontrivial p-subgroup P of G. If f is a {p}-equivalence for any prime p, then we call f a ;O-equivalence. A G-map f is called a singularity equivalence if it satisfies one of the conditions (1) and (2) below : (1) The restriction f,:Xs ~ Y, of f to the singular set gives an equivalence of homology with integral coefficients. (2) The reduced projective class group K0(Z[G]) of the integral group ring ZIG] is trivial and f is a 7:'-equivalence. Let f : (X, OX) --+ (IF,OY) be a G-map. It is called a boundary equivalence if its restriction Of: OX --+ OY to the boundaries is a homology equivalence. Let w be the orientation homomorphism w: G ~ {1,-1} given by w(g) = 1 (resp. - 1 ) if g in G preserves (resp. reverses) the orientation of Y. Using this, we can define an involutive anti-antomorphism - of the integral group ring Z[G] by ~ = w(g)g -~ for all g in G. Let G(X) be the subset of G consisting of all elements g of order two such that d i m X ' = [ ( n - 1)/2], where n = d i m X . The form parameter FG(Y) on Z[G] for A = (_1)[-/21 is defined to be the smallest form parameter containing all elements of G(Y). The following two results may be fundamental in G-surgery theory.

355

LEMMA 3.1 ( [18] Theorem A). Let Y be a compact, connected, simply connected and oriented G-manifold of dimension n > 5, and let w = (f; b; c): (X, OX; T X ; v X ) (r, OY; f*~; zrf*() be a G-normal map. Suppose that the following conditions hold. (1) 2 dim Y, < n. (~) dim y L < [(n - 1)/2] whenever L ~ {1,g} for some g e G ( Y ) . (3) f i s a boundary and singularity equivMence. T h e n w d e t e r m i n e s an e l e m e n t a(w) in the Bal~ g r o u p W~(Z[G],rG(Y); ~), and if a(w) = O, then one can perform G-surgery keeping the boundary and the singular set fixed to convert w so that f: X -* Y is a homotopy equivedcence.

Remark. In [20] Theorems A and B, it is proved that, under the same conditions as in Lemma 3.1, one can pe14orm G-surgery as above if and only if a(w) = 0. Namely, a(w) gives the G-surgery obstruction. LEMMA 3.2. Let Y satisfy t,he same assumptions a.s in Lemma 3.1, and further, suppose that it is without boundary and satisfies the strong gap hypothesis, i.e., 2(dim ]I8 + 1) < dim Y. Then the following hold. (1) Let w~ = (f; b; c): (X; T X v X ) ---* ( r ; f*(; zrf*~) be a G-normal map and assume that f is a singularity equivalence. Then a(w) lies in the Wall group Lh(G) of homotopy equivalence and it gives the G-surgery obstruction. (2) /f w' = (f'; b'; d): (X'; TX'; v X ' ) ---, (Y; f'*(; 7rf'*~) is another G-normal map and if there exists a G-normal cobordism W = (F; B; C): (W, OW; T W ; z/W) ~ (I x Y, c3(I x Y); F*(eI(R) x (); ~'F*(eI(R) x ~)) between w and w ~ such that F: W ~ I a(w) = a(w').

x

Y is

a

TO-equivalence, then one has

PROOF: This lemma may be well known. ~re refer the reader to [20] Theorem D for the details. In the rest of this section, we fix a real G-module V with G-invariant inner product, and construct several G-manifolds. We denote by S ( V ) (resp. D ( V ) ) the unit sphere (resp. closed unit disk) of I/. Consider R as the real 1-dimensional trivial G-module with standard inner product. The tangent bundle T ( V ) of V is identified with ¢v(V). The G-vector bundle es(v)(R.) @ T ( S ( V ) ) can be regarded as the restriction of T ( V ) to S ( V ) by the standard G-isomorphism. Here e s ( v ) ( R ) should be understood to be the normal bundle ~,(S(V), V) of S ( V ) in V by the above identification. For an integer k with k "> 1, let R k be the k-fold direct sum of R. The G-vector bundle ¢ s ( v ) ( R k) (~ T( S ( V ) ) can be identified with e s ( v ) ( R k-1 @ V) by the standard isomorphism

es(v)(rtk) e T( S(V) ) = ~s(v)(rtk-~ ) ( ~ s(v)(R ) (~ T( S(V) ) = ~s(v)(n,k-~ ) ( ~ s(v)(V). Here the restriction of the isomorphism to s s ( v ) ( R k-l) should be understood as the identity map.

356

For a positive integer j, we define V(j) here to be the j-fold direct sum of V, and we put Y(j) = S(R@ V(j)) and X ( j + 1)' = Y(j) × Y(1). Denote by p(j)+ and p(j)_ the points (1,0) and ( - 1 , 0 ) , respectively, of Y(j), where 1 e R and 0 e V. The tangential representation at (p(j)+,p(1)+) in Z ( j + 1)' is isomorphic to V(j + 1). By pinching the outside of the equivariant open disk neighborhood of (p(j)+,p(1)+) in X ( j + 1)', we get a degree one G-map h(j + 1)': X ( j + 1)' ~ Y(j + 1). We note that e(R) @ T ( Y ( j + 1)) = ~(R @ V(j + 1)), and e ( R ~) @ h(j + 1)'*T(Y(j + 1)) = ~(R 2 $ V(j + 1)). An unstable Gtvector bundle isomo12~hism

b(j + 1)': e ( R 2) @ T ( X ( j + 1)') ~ ~(l:t ~ @ V(j + 1)) is defined to be the standard isomorphism

( e ( R ) ~ T ( S ( R ~ V ( j ) ) ) } × (v(R)q~T(S(R~V(1)))) = { ~ ( R ~ V ( j ) ) } × ( e ( R $ V ( 1 ) ) } . Thus we obtain a G-normal map v(j + 1)' = (h(j + 1)'; b(j + 1)'; rb(j + 1)'). On the other hand, let l a ( j + 1) = (id; id; rrid) be the identity G-normal map on Y ( j + 1). Here the second id is the identity map on T ( Y ( j q- 1)). By our specified identification, we obtain s(id): e ( R 2) q~ T ( Y ( j + 1)) ---, c ( R 2 @ Y(j + 1)). LEMMA 3.3. The above G-normed map v(j + 1)' is G-normally cobordant $o 1G(j + 1). PROOF: Let 5'1 be the sphere of radius 3 with center being the origin in R ~ V(j), $2 the sphere of radius 3 with center being the origin in R @ V(1). We identify $1 x $2 with X ( j + 1). Define $3 by

$3 : {(x,u,y,v) e R ~ V ( j ) @ R @ V ( 1 ) I x

= 5,< u,u > ~- < y,y > + < v,v >= 1}.

We identify $3 as Y(j q- 1). It is e ~ y to find a compact, orientable, codimension one submanifold W o f R $ V(j) ~ R @ Y(1) and a G-map F: W ---, I x Y ( j q- 1), where I : [0, 1], such that (1) the bounday OW of W is $1 ×: $2 U $3, (2) the G-collar neighborhood of OW in W is CN12 U CN3, where

cgl:

= {(x,u,y,,)

ReV(j) mRmV(1)

I<x,x>+=9, and 4 < < y , y > - F < v , v > CN3 = {(x,u,y,v) E R ~ V(j) @ R e V(1) I 4<x_ 2, the standard sphere S 3k has a smooth one tixed point As-action which is As-cobordant to E(k), the k-fold cartesian product of E with the diagonal As-action. Let V be the tangential representation at the unique G-fixed point of E. Then as a real G-module, V is irreducible, and for a subgroup H, dim Var = 0, 1, or 3 if H is noncyclic, nontrivial cyclic, or trivial, respectively, (cf. Definition 1.4). Using this V we can define several manifolds as in the previous section. (e.g. Y(1) = S ( R G V)) So, we keep the same notation as there, but remember that our V here is a special one given above. We first obtain the following by Petrie's transverality construction (see [19] Section 3, or [23]). LEMMA 4.1. There are a G-manifold X(1)" with exactly one fixed point and a G-normal map u(1) = (g(1);b(1)";c(1)"), where g(1):X(1)" ~ Y(1) and b(1)":T(X(1)") ---* g(1)*T(Y(1)), such that the following hold: (1) X ( 1 ) ''a = g ( 1 ) - 1 ( p ( 1 ) + ) c (and of course, I X ( 1 ) ''a I = 1). (The G-flxed point in

x(1)" is ~ o denoted by p(t)+.) (2) If H is a proper, noncyclic subgroup of G, then

x(1) ''H = {p0)+} I_Ig(1)-~(p(1)-) u ~ d

I g(m)-'(p(1)_) u I= 1

(3) For every maximal subgroup H of G, there exists an H-normal cobordism UH(1)

= ( v . ( 1 ) ; B.(1)"; CH(1)") between res~u(1) ~ a res~ 1c(1), where 1G(1) is the identity G-normal map on Y(1). For an integer k with k > 2, let X ( k ) (resp. E(k)) be the k-fold cartesian product of X(1)" (resp. E). Then we have ; LEMMA 4.2. The G-manifold X ( k ) has exactly one G-flxed point and is G-cobordant

to ~,(k). PROOF: It suffices to prove that X(1)" has one G-fixed point and is G-cobordant to E. By Lemma 4.1 (1), the former is clear. On the other hand, from Lemma 4.1 (2), the restriction g ( 1 ) H : x ( 1 ) "H -+ Y(1) H of g(1) to X(1) ''g is a homotopy equivalence if H is isomorphic to A4 or D., where n = 4, 6 or 10. Now let C be an arbitrary nontrivial cyclic subgroup of G. Then X(1) "c is a disjoint union of circles. Among the circles, those which do not contain the G-fixed point p(1)+ consist of points with isotropy subgroup C. Kill the circles not containing the G-fixed point p(1)+ by G-surgery on X(1)" of isotropy types (C2), (C3)and (C5) simultaneously. Then we can modify u(1) so that g(1)c: X(1) ''c --~ Y(1) c is a homotopy equivalence. In particular, the singular set

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X(1)~ of X(1)" belongs to S. Furthermore, the stable isomorphism b(1)" gives a stable Ah-trivialization (of type minus) for the module V. Since E has an Ah-trivialization for V (cf. Example 2.2), Theorem 2.1 implies that X(1)" is G-cobordant to E, which completes the proof. Now we are going to perform G-surgery on X ( k ) , keeping the G-fixed point set fixed, to obtain a homotopy sphere of dimension 3k. Let w(k) = (f(k);b(k);c(k)) be the G-normal map obtained by composing the k-fold cartesian product u(1) x ... x u(1) of u(1) = (g(1); b(1)"; c(1)") with v(k) = (h(k); b(k); 7rb(k)), which are defined in Section 3. Here notice that the target manifold of f ( k ) is the 3k-dimensional sphere Y ( k ) = S ( R @ V (~ ... ~ V). Then by Proposition 3.5 and Lemma 4.1 (3), we have ; LEMMA 4.3. For every maxima] subgroup H of G, there exists an H-normM cobordism W H ( k ) = (FH(k); B H ( k ) , C H ( k ) ) be,veen resT~w(k ) and r e s ~ l a ( k ) , where l o ( k ) the identity G-norma] map on Y ( k ) . Hereafter we abbreviate the notation by omitting (k) if there seems to be no confusions. For example, we use w instead of w(k). Then we have ; LEMMA 4.4. One can perform G-surgery of the above G-norma] map w = (f; b; c) keeping the G-Axed point set Axed and one can perform H-surgery of the above Hnonna] cobordisms W H = ( FH; BH, CH ) between resZw and r e s ~ l a so that (1) the new f: X ---* Y is a 7P-equiva]ence, and (2) if k >_ 4, ~hen a]l FH: WH -~ I x res~Y, for maximal subgroups H of G, are P-equivalences, where I = [0, 1]. The above will be proved in the next Section by using Lemma 4.3 and a case by case arguments. Here we assume it to go on. Note that Lemma 4.4 (1) implies that f is a singularity equivalence because it is shown in I. Reiner and S. Ullom [26] or [25] that, for all subgroups H of As, the reduced projective class groups K0(Z[H]) of the integral group rings Z[H] are trivial. Since f is trivially a boundary equivalence, we may apply Lemmas 3.1 and 3.2. In the case where k = 2 (resp. _> 3), by Lemma 3.1 (resp. Lemma 3.2), f gives a ( w ) which lies in the Bak group W2(Z[G], F; triv.) (resp. the Wall group Lhk(G)), where F is the smallest form parameter on Z[G] containing all elements of G of order two. LEMMA 4.5. The element a(w) is zero. PROOF: In the case where k = 2, the result follows from W2(Z[G],F;triv.) = 0 ([21] Proposition 1.1), and when k = 3, it follows from L~'(G) = 0 (A. Sak and M. Kolster [3] Corollary 4.4). We now show a ( w ) = 0 in the case where k > 4. It suffices to show that r e s a a ( w ) = 0 for H --- D4, D6 and 910 by the Dress induction theorem [2] Section 12, [10] or [29] since every maximal 2-hyperelementary subgroup of G is conjugate to one of D4,D6 and D10. But for H ~ D6, D10 or A4(> D4), we have H-normal cobordisms W H between r e s e T and res~ l c satisfying the condition (2) of Lemma 4.4. Thus Lemma 3.2 (2) implies that resaHa(W) = a ( 1 g ) = 0 for all H = 94, 96 and O10. This proves the lemma.

360

Hence by the above lemma, we can perform G-surgery, keeping the singular set fixed, to converting w = (f; b; c) so that f : X --* Y is a homotopy equivalence. In particular, since Y is a homotopy sphere, so is X. Also, recall that this resulting X is G-cobordant to P.(k) = ~ × ... x P.. (the k-fold cartesian product) by Lemma 4.2. Summarizing the above, we get the following. PROPOSITION 4.6. Let k be an /nteger with k >_ 2. Then, t.here exists a homotopy sphere X of dimension 3k with G-action such ~hat

(1) X has exactly one G-fixed point, (2) X is G-cobordant to ~(k) = P. x ... x P, and (3) for a/1 rnax/ma/subgroups H of G, refiHX is g-cobordant to reSGHY(k), where V ( k ) = S ( R • v • ... • v ) . COMPLETION OF PROOF OF TIIEOREM 0.1: It suffices to modify (the underlying manifold of) the above X in Proposition 4.6 to the standard sphere of dimension 3k with the same properties. Consider the equivariant connected sum X ' of X with G XH resZX at the points of isotropy type (H) for H = A4 and D10. (For details, see [19] Section 3, p. 248.) Then X' also satisfies all the properties (1) - (3) with X being replaced by X'. In fact, the property (2) of X' follows from the properties (2) and (3) of X, and the property (3) of X ' follows from the property (3) of X. On the other hand, let e(3k) denote the group of 3k-dimensional homotopy spheres, which is known to be a finite group by [17] Theorem 1.2. For a 3k-dimensional homotopy sphere Z with G-action, we denote by [Z] the element in 6)(3k) which corresponds to the underlying space of Z. Then for our X and X' we have

[X'] = (1 + e)[X] in O(3k), where if H = A4 then e = 5 and if H = D10 then ~ = 6. Now since 5 and 6 are relatively prime, we can obtain the standard sphere, which corresponds to the identity element of O(3k), as the underlying space of an equivariant connected sum X " of X with several (G XA, resA4X ) s and (G x D,o resD,o-& ) s. Then the resulting X " also satisfies all the properties (1) - (3) with X being replaced by X " . This proves the theorem. 5. A p r o o f o f L e m m a 4.4 In this section G still means As. We restate Lemma 4.4. LEMMA 4.4. One can perform G-surgery of ~he G-normal map w = (f; b; c) keeping ~he G-f*xed point set fixed and one can perform H-surgery of the H-normal cobordisms W t l = (FH; BH, CH) between res~w and resGH1G sO that (1) the new f: X --* Y is a ?P-equivalence, and (2) /t" k > 4, then all FH: WH --* I x r e s ~ Y , for maximal subgroups H of G, are P-equivalences, where I ---- [0, 1].

361

This is proved essentially in [19] Section 5. However, we give a proof also here for reader's convenience. For two subgroups H and K of G with N a ( K ) C H ~ G, we use the following notations. x > K = {x e X

] G~ ~ K} and W > K = {w E W g I H,,, ~ K }

We first give the following lemma. LEMMA 5.1. Let H and K be subgroups of G with N a ( K ) c= H # G. Suppose that there exist a dosed H-regular neighborhood U>K of H W H>K in WH and an Hdiffeomorphism ¢>K: U>K --+ I x ( X N U>K) relative to X N U>K. Suppose further that there exist a dosed H-neighborhood N>K of H X >K U H W > K in WH and an H-diffeomorphism ~>K: N>K --~ I x ( X N N>K) relative to X V1N>K such that N>K D= U>K, ~2>K [ U>K = ¢>I(, N>K is orthogonal to both X and Y, and ff'>Kl({1} X (X VI N>K)) = Y N N>~c. Them one can perform equivariant surgery of isotropy type ( K ) to modify w = (f; b; c) and W H = (FH; BH; Ctt) so that (1) f K : X K --~ y K is a homotopy equivalence, (2) F•¢: W s K ---, I x y K is also a homotopy equivalence, and (3) there exists a dosed H-regular neighborhood UK D__U>K of H W f f in WH and an H-diffeomorphism ¢Ic: UK ~ I x ( X N UK) relative to X N UK such that

CK ] U>K = ¢>K. PROOF: See [19] Theorem 4.2. Let H and L be subgroups of G = As such that L C__H. We say that w and WH are good for (H, L) ~ the fonowing conditions (1) and (2) are satisfied for all subgroups K

that L C=K c= H and K # (1) f K : X K ~ y K are homotopy equivalences. (2) FHK: W ~ ~ --~ I x y K are also homotopy equivalences. In the following, we modify w and WH step by step using Lemma 5.1 in order to obtain the goodness of them for many pairs (H, L) of subgroups of As satisfying N a ( L ) c= H # G PROPOSITION 5.2.

(1) / / n = 3k _> 6, then one can modify w and W H so that they are good for (A4,A4), (D6,D~), (D~o,D~o), (A4,D4), (D~o,Cs), (D6,Ca) and (A4, C2). /n particular, for any nontrivial subgroup K, the resulting fK: X K ~ y K is a homotopy equivalence. (2) H'n = 3k _> 12, ~hem one can further modify them so that they are good also for (-44, {I}),(D6, {I}) and (D,o, {I}).

362

Proposition 5.2 (1) gxtarantees the property (1) in Lemma 4.4 and (2) guarantees the property (2) there. Thus Proposition 5.2 proves Lemma 4.4. PROOF: We shall modify w and W H step by step to obtain good ones.

Step 1: ( H , L ) = (A4,A4). In this step, H = I ( = L = A4 = NG(A4). Thus H W > K = and H X >K = X a = one point. ( X H consists of 2k points.) Consider U>K as the empty set. It is easy to see that 14zH includes a path connecting X a with a point p+ in y a . Take a closed H-tubular neighborhood N>H of this path in WHH. Then N>K is clearly H-diffeomorphic to I x ( X N N>H). Then, by Lemma 5.1 for (H, K ) = (A4, A4), we Call perform equivariant surgery of isotropy type (A4) to obtain a new G-normal map w = (f; b; c) and an H-normal cobordism W H = (FH; BH, C n ) between res~w and res/~lG which are good for (H, L) = (A4, A4). Step 2: (H, L) = (D6, D6) and (D10, D10). The argument is quite similar to one in Step 1 above, and we omit it. Step 3: ( H , L ) = (A4,D4). Since we have done already for K = A4 in Step 1, it suffices to modify w and W H for K = D4. So we put K = D4. From the construction, X g = X H and f K : x K ~ yl¢ is already a homotopy equivalence. Thus, it suffices to modify FHK: Wff ~ ~ I x y l ¢ to be a homotopy equivalence. Since dim WHK = 1, WHg is a union of W/_tH and several circles liing in IntWgg. Perform H-surgery on W H of isotropy type (D4) to kill these circles. Then the resulting FHI~ is a homotopy equivalence as is required, and thus w and W H are good for (H, L) = (A4, D4). Step 4: ( H , L ) = (Dao,Cs). W e have already done for K = D10 and are required to modify w and W H for K = C5. So, we put K = C5. Note that H X g = X K, H X >g = X H, H W H K = WHK and H W H >K = W~f =~ I x X H.

In particular, H X >K C=H W > K . Take a closed H-tubular neighborhood U of WHH in WH. Regard U>K and N>K in Lemma 5.1 as this U, and apply Lemma 5.1. Then, by equivariant surgery of isotropy type (C5) we can obtain good w and W H for (H, L) = (Dx0,Ch). Step 5: (H, L) = (Ds, C3). We have already done for K" = Ds and are required to modify w and W H for K = C3. So, we put K = C3. The proper subgroups B of G which properly include C3 are isomorphic to D6 or A4. Further, such a subgroup B isomorphic to D6 is unique, i.e. B = D6, and the number of such B's isomorphic to A4 is two which we denote by A(1) and A(2). Then, H X >g = X H 13 X A(1) U X A(2) and H W H >g = WHH = I X X H. Then W H has a closed H-tubular neighborhood U>K in W H which is H-diffeomorphic to I x ( X ~ U>K). Recall that d i m X > g = 0 and dim WHK = k >_ 2. We can embedd I x ( H X >I¢ \ H W ~ g ) to H W H g \ U>K Hequivariantly, and obtain a closed H-neighborhood N>K of H X >K tO H W > g , which is H-diffeomorphic to I x (X N N>K) and contains U>K. Now apply Lemma 5.1 and obtain good w and W H for (H, L) = (D6, C3). Step 6: ( H , L ) = (A4,C2). We have already done for g = A4 and D4. Thus, we are required to do for K = C2. So put K = C2. We list the subgroups of G which properly include C2: Those isomorphic to A4 and D4 are respectively unique. The number of

363

those isomorphic to D6 is two and we denote them by D(3) and D(4). The number of those isomorphic to D~0 is two and we denote them by D(5) and D(6). Then, we have

X >K = X H U X D(a) U Z D(4) [..J Z D(5) [.J X D(6) and HW,.~ 1( = WHu. First take a closed H-tubular neighborhood U>K of WHH in WH. Then, U>K is H-diffeomorphic to I x (X Cl U>K). Next note that d i m X >K = 0. Thus we car, embedd I x ( H X >g \ HWH >u) to HWH K \ U>K H-equivariantly and obtain a closed H-neighborhood N>I( of H X >I( U H W > K , which is H-diffeomorphic to I x (X n N>I¢) and contains U>I(. Now apply Lemma 5.1 and obtain good w and W H for (H, L) = (A4, C2). Step 7: (H,L) = (A4, {1}). Here we suppose that n = 3k > 12. We have already done for K = A4,D4 amd C2. Fhrthermore, w satisfies the condition (1) of goodness for all K , (even for K ~ Ca). So put K = Ca, which is remaining, and we will modify W H to satisfy the condition (2) of goodness. First note that H W >I~ = WHH "~ I x X H. The maps FI_I>g = FH: WHH ~ I x yH and fl~: XI( ...., y K are homotopy equivalences. Let U(H, H) be a closed H-tubular neighborhood of WHH. However, we can not use Lemma 5.1 since N a ( K ) ~ H. On the other hand, since k > 4, we have

dimWl./¢ = k + 1 > 5 and 2(dim W/4>1~ + 1) = 4 < k + 1 = dimWHK. Now try to modify W H by H-surgery keeping U(H, H) U OWH fixed so that FHK is a homotopy equivalence. The obst ruction a (FHI¢) lies in the Wall group Lh+ 1(N), where N = N H ( K ) / K , which is of course trivial. If k + 1 is odd, then a(F K ) = 0 because L~+I(N ) = 0, and thus we can perform required H-surgery. In the general case, we need an add hoe argument. Let I(G) be the identity G-normai map on the closed G-disk D(Ft (3 V(k)), where V(k) is the k-fold direct sum of V. Let I(G)_ and FD6- be the reversed copies of I(G) and FDs, respectively. Now glue these

res

l(V)_, res K o0 FD~_,resKFH . and resT l(a).

We denote the resulting K-normal map by W = (F; B; C), where F: W --* r e s ~ S ( R 2 ~3 V(k)). Then it is clear that a(F I() = a(F~i1~) in Lh+I(N). Let W _ = ( F _ ; B _ ; C _ ) be the reversed copy of W . Then, a ( F _ 1') = - - g ( F K ) . Take an H-connected sum of W H with i n d g w _ at the points in IntWH with isotropy type (K), and denote by ! ! W~H = (F'H,BH,C~t), F~H:W'H --* I x Y the resulting H-normal cobordism. Then we get a(F~HK) = 0. Hence we can perform H-surgery to modify W ~ so that F~/ is a homotopy equivalence satisfying F'H>K = FI~ K. Replacing the initial W H by the obtained W~/, the condition (2) of goodness is satisfied. Step 8: (H,L) = (D6, {1}) aaad (D10, {1}). The aa-gument is quite similar to one in Step 7, and we omit it. This completes the proof of Proposition 5.2 and thus of Lemma 4.4.

364

References 1. Bak, A., The computation of surgery groups of finite groups with abelian 2-hyperelementary subgroups, Lecture Notes in Math. (ed. by M. 1t. Stein) 551 (1976), 384-407, Springer, Berlin Heidelberg - New York - Tokyo. 2. Bak, A., "/f-Theory of Forms," Princeton University Press, Princeton, 1981. 3, Bak, A. and Kolster, M., The computation of odd dimensional projective surgery groups of finite groups, Topology 21 (1982), 35-63. 4. Bak. A. and Morimoto M., Equivariant surgery theory and its applications, in preparation. 5. G. E. Bredon, "Introduction to Compact Transformation Groups," Academic Press, New York London, 1972. 6. Buchdahl, N. P., Kwasik, S. and Schultz, R., One fixed point actions on low-dimensional spheres, preprint, Tulane University and Purdue University. 7. Conner P. E. and Floyd E. E., "Differentiable Periodic Maps," Ergebniss der Mathematik und Ihere Grenzgebiete Neue Folge Band 33, Springer, Berlin - Heidelbrg - New York - Tokyo, 1964. 8. Dovermann, K. H., Masuda, M. and Petrie, T., Fixed point ]ree algebraic actions on varieties diffeomorphic to R ~, preprint. 9. Dovermann, K. H. and Masuda, M., Fixed point free low dimensional real algebraic actions of As on contractible varieties, preprint. 10. Dress, A., Induction and structure theorems for orthogonal representations of finite groups, Ann. Math. 102 (1975), 291-325. 11. Furuta, F., A remark on a fixed point of finite group action on S 4, Topology 28 (1989), 35-38. 12. Grove, L.C. and Benson, C. T., "Finite Reflection Groups," (Second Edition) G r a d u a t e Texts in Mathematics 99, Springer, Berlin - Heidelberg - New York - Tokyo, 1985. 13. Kervaire, M. A. and Milnor, J. W., Groups ofhomotopy spheres I, Ann. Math. 77 (1963), 504-537. 14. Kolster, M., Even dimensional projective surgery groups of finite groups, Lecture Notes in Math (ed. by R. K. Dennis) 967 (1982), 239-279, Springer, Berlin - Heidelberg - New York - Tokyo. 15. Laitinen, E.and Traczyk, P., Pseudofree representations and 2-pseudofree actions or* spheres, Proc. Amer. Math. Soc. 97 (1986), 151-157. 16. Morimoto, M., On one fixed point actions on spheres, Proc. J a p a n Acad. 63 Ser. A (1987), 95-97. 17. Morimoto, M., S 4 does not have one fixed point actions, Osaka J. Math. 25 (1988), 575-580. 18. Morimoto, M., Bak groups and equivariant surgery, K - T h e o r y 2 no. 4 (1989), 465-483. 19. Morimoto, M., Most of the standard spheres have one fixed point actions of As, Lecture Notes in Math (ed. by K. Kawakubo) 1375 (1989), 240-258, Springer, Berlin - Heidelberg - New York Tokyo. 20. Morimoto, M., Bak groups and equivariant surgery II, to appear in K-Theory. 21. Morimoto, M., Most of the standard spheres have one fixed point actions of A~. II, preprint. 22. Oliver, l:t., "Whitehead Groups of Finite Groups," London Math. Soc. Lecture Note Series 132, Cambridge University Press, Cambridge, 1988. 23. Petrie, T., One fixed point actions on spheres, I, Adv. Math. 46 (1982), 3-14. 24. Petrie, T., One fixed point actions on spheres, II, Adv. Math. 46 (1982), 15-70. 25. Reiner, I., Class groups and Picard groups of group rings and order, Regional Conf. Set. in Math. 26 (1976), A.M.S. 26. Reiner, I. and Ullom, S., Remarks on class groups of integral group rings, Symposia M a t h e m a t i c a (ed. by A. Dress) 13 (1974), 501-516, Academic Press, London. 27. Shaneson, J., Wall's surgery obstruction groups for G x Z, Ann. Math. 90 (1969), 296-334. 28. Springer, T.A., "Invariant Theory," Lecture Notes in Mathematics 585, Springer, Berlin - Heidelberg - New York - Tokyo, 1977. 29. Stein, E., Surgery on products with finite fundamental group, Topology 16 (1977), 473-493. 30. Uchida, F., "Transformation Groups and Cobordism Theory," (Japanese) Kinokuniya Suugakusousho 2, Kinokuniya, Tokyo, 1974. 31. Wall, C. T. C., Classification of hermitian forms. VI Group rings, Ann. Math. 103 (1976), 1-80.

A NOTE ON THE MOD 2 COHOMOLOGY OF

SL(][)

Dominique Arlettaz Institut de math~matiques Universit~ de Lausanne CH-1015 Lausanne, Switzerland

1. I n t r o d u c t i o n

Let SL(][) be the infinite special linear group of the ring of integers ][, and for n >_ 2, w,~ E H"(SL(][); ][/2) the n - t h Stiefel-Whitney class of the inclusion S L ( £ ) ~-~ G L ( R ) (wl = 0). The rood 2 cohomology of S L ( Z ) satisfies [3, Proposition

1.21 H*(SL(][); ][/2) ~ Z/2[w:, wa, w4,...] ® A ,

where A is an (unknown) commutative graded algebra. The corresponding result for the infinite general linear group GL(][) is H*(GL(][); Z/2) ~ Z/2[wl, w2, w3,. . .] ® A .

It is not hard to deduce from the knowledge of the integral homology of SL(][) in dimensions 1, 2 and 3 [1] that A contains no element of degree 1 and 2, and exactly one element of degree 3. Dwyer and Friedlander have formulated in [6] a version of the Quillen-Lichtenbaum conjecture concerning the map between the algebraic K-theory spectrum and the 6tale K-theory spectrum, and they have shown that if their version of the 2-adic QuiUenLichtenbaum conjecture is true for the ring Z, i.e., if the map BGL(Z[½]) + I K~-6t (£[~]) induces an isomorphism on mod 2 cohomology, then the following conjecture holds [6, Corollary 4.3]. Conjecture. A=A[u3,us,...,u2n+l,...],

where d e g u 2 n + l = 2 n + l .

A part of this work was done during a stay at McMaster University : it is a pleasure to express my gratitude for its hospitality.

366

Notice that if this is true, then the 2 -torsion subgroup of H4(SL(Z); ][) is cyclic and the isomorphism H4(SL(][); ][) ~ I(4][ @ ][/2 [2, Theorems 1.1 and 1.3] implies that the 2-torsion subgroup of K4][ is trivial. Let us mention the following simple fact. The rational cohomology of SL(][) is known [5]: H*(SL(][); Q) = A[x~,xg,... , z 4 , + 1 , . . . ] , where degx4~+l = 4n + 1. This produces elements us, u g , . . . , u4n+l in H*(SL(][); ][/2) and it is possible to check that they ave actually in A , but it is not clear whether or not they ave exterior. The purpose of this paper is to make the above conjecture more plausible by detecting exterior classes of odd degree in H*(SL(][); ][/2). Consider the finite field Fp (p an odd prime), and denote by fv the reduction mod p : SL(][) -~ SL(Fv) and by f ; the induced homomorphism H*(SL(Fv); ][/2) ~ H*(SL(][); ][/2). By [7], the ring H*(SL(Fv); ][/2) is generated by the Chern classes Ck e H2k(SL(Fv); ][/2) and by the classes ek E H2k-I(SL(Fv); ][/2), k > 2. For any odd prime p and any integer k > 2, we have [3, Lemma 1.4]

f~(ck) = W~. , and we introduce the notation

:= f ; ( e k ) . If p -- 1 (mod 4), then e~ = 0 and consequently, ¢~=0. This is wrong if p -- 3 (rood 4), but we have proved [3, Lemma 1.5] that in this case

¢k = Wk + Tk , where Wk = w2k-l + ~

wjw2k-l-j and

2l.

H*(SL(Z); Z/2) contains an exterior clement of degree 2n + 1 for all

The corresponding results hold also for GL(7) ( k > 1 in this case, but ¢1 = 0, respectively 71 = 0 ). Observe finally that the theorem is wrong if p =-- 1 or 7 (mod 8).

367 2. P r o o f o f t h e t h e o r e m

P r o p o s i t i o n 1. The theorem is true for any positive integer k -- 2 (mod 4). P r o o f . According to Quillen's terminology for the computation of H*(SL(Fp); 7/2) for an odd prime number p [7], the class ek E H2k-I(SL(Vp); Z/2) is defined for any integer k _> 2 as the image, under the reduction rood 2 : H2k-I(SL(Fp); Z/(pk--1)) H2k-Z(SL(Fp); 7 / 2 ) , of the cohomology class "ek E H2k-I(SL(Fp); Z/(p k - 1)) introduced in [7, p.559]; this element ~k has the property that the Bockstein homomorphism associated with the short exact sequence 7 ~-~ Z -* Z/(p k - 1) maps ~'k onto "~k E H2k(SL(Fp); Y), where ~k is the k -th integral Chern class of SL(Fp) [7, Lemma 5]. Therefore, if fl denotes the Bockstein homomorphism associated with the short exact sequence Y }-4 Z -* Y/2, then fl(ek) = ~(p 1 k -- 1)~k for any odd prime p and any integer k > 2. Now, look at the commutative diagram

H~k-I( SL(Fp); 7/2)

H2k(SL(Fp); Z)

f; }

f; )

H2k-I(,..,c'L(Z); 7/2)

H2k(SL(Z); 7 ) .

H*(BU; 7) --~ H*(SL(Fp); Z) --* H*(SL(Z); Z) (the first homomorphism being induced by the Brauer lifting, the second by fp ) coincides, away from the p-torsion, with the homomorphism H*(BU; Z) --* H*(SL(Z); Z) induced by the inclusion SL(Z) ~-~ G L ( C ) , and consequently that

It follows from [4, p.36] that the composition

is, up to p-torsion, tt e k-th integr Che class ck(sn(z)) of the inclusion SL(Z) ~-~ G L ( C ) . By commutativlty of the diagram, /3(~k) = /3(fp(ek)) : f~(/3(ek)) is then an odd multiple of ~ (pk - 1)ck(SL(Z)). The order of ck(SL(Z)) in H2k(SL(Z); Z) is known for k _-- 2 (rood 4) [1] and it turns out that its 2-primary part is exactly the 2-primary part of (pk _ 1) , if p -- 3 or 5 (rood 8). It is then easy to conclude that /3(~k) does not vanish in this case and to deduce assertion (a) of the theorem. If p = 3 (rood 8), remember that ek = Wk q-Tk and observe that

2_<j 2k - 1. The kernel of p* is the ideal generated by the elements cj and ej for j > m : thus, p* is injective in dimensions < 4k - 2. If p - - I ( r o o d 4 ) , then H*(BC"~;Z/2) ~- Z / 2 [ x x , x 2 , . . . , x , n ] ® A [ y l , y 2 , . . . , y m ] with deg z j = 2 and deg yj = i for 1 < j < m . Consider the differential d of degree - 1 defined by d ( z j ) = y j and d(yj) = 0 for l < j < m : d commutes with the Steenrod squares. According to [7, p. 564], p*(ck) = ak and p*(ek) = d(ak), where ak denotes the k - t h elementary symmetric function of xl, x 2 , . . . , x m . Therefore, we obtain

l<j

369

by Wu's formula, and deduce assertion (a) from d ( a ~ + i ) = p * ( e ~ + i ) and d(ajo'k+i-j) = ajd(crk+i-j) + d(crj)crk+i-j = p*(cjek+i-j + Ck+i--jej) . Similarly, (b) follows from p * ( S q 2 i - l ek ) = S q 2 i - l d( cyk ) = d( Sq2i-X o'~ ) = O .

If p -- 3 (mod 4), then H * ( B C m ; Z / 2 ) ~- Z / 2 [ y l , y 2 , . . . , y m ] and p*(ck) = 7-~, p ' ( e k ) = Sqk-17-k, where 7-k is the k - t h elementary symmetric function of Yl, Y2,...,Ym. We must investigate p*(SqZiek) = S q 2 i S q k - l r k and p*(Sq2i-le~) = Sq2i-lSqk-17-k. The Adem relations and the Wu's formula produce the equalities

p*(Sq2iek) = sqkWi-lsqiT- k = sqk-I'i--l((kT1)7-k+ i .-]- ~

(k~j--~l)7-JT-k+i--J) ,

l<j

p*(Sq2i-lek) = (k - i - 1)Sqk+i-lSqi-17-k = (k - i - 1)(Sqi-17-k)2

-(k ---

Thus,

Sqk+i-17-k+ i =

Sqj--17-jSqk+i--JT-k+i--j

p*(ek+i) =

--

I) F,

Li--j--l)7"3 7-k-'ki--j--1 "

O<j 5.

370

P r o o f . If p =- 5 (mod 8), consider Sq4ek = g ( S q 4 e k ) = g(c~ek + cke2) = w~ek + w2¢2 for k -- 1 (mod 4 ) , according to L e m m a 4 : the vanishing of ek would imply w~e2 = O, but this is not the case. If p = 3 ( m o d 8 ) and 7k = O, then ck is a polynomiai in Stiefel-Whitney classes and the calculation of Sq4¢~ produces the contradiction Wk72 2 = O. R e m a r k 7. It is actually possible to prove Proposition 1 for k _> 6 by using the same argument. Proposition

8. The theorem is true for any positive integer k - 3 ( m o d 4).

P r o o f i Let p be a prime = 5 (mod 8) and compute again Sq4~k = ek+2+w2¢k+wk¢22 2 . 2 = 0. Apply now Sq 2 : S q 2 ¢ k + 2 = f;(Sq~ek+2) = 0 by If ¢k = 0, then ¢k+2 + WkF~2 L e m m a 4 and consequently Sq2(w2e2) = w2kSq2¢2 = 0 ; this is impossible (cf. proof of Proposition 5). Similarly, if p - 3 (mod 8) and 7k = O, then 7k+2 + w~72 = 0 and Sq27k+2 + Sq2(w~.72) = 0 ; but this is a contradiction since Sq27k+2 = Sq2ek+2 + Sq2Wk+2 = Sq2Wk+2 e Z/2[w2,Wa,W4,...] and Sq2(w~72) = w~Sq272 q~ 7/2[w2, w3, w 4 , . . . ] .

References

[1] D. Arlettaz : Chern-Klassen yon ganzzahligen und rationalen Darstellungen diskreter Gruppen, Math. Z. 187 (1984), 49-60. [2] D. Arlettaz : On the algebraic K-theory of Z, J. P u r e Appl. Algebra 51 (1988), 53-64. [3] D. Arlettaz : Torsion classes in the cohomology of congruence subgroups, Math. Proc. Cambridge Philos. Soc. 105 (1989), 241-248. [4] M. BSkstedt : The rational homotopy type of ~2whDiff(*), in Algebraic Topology Aarhus 1982, Lecture Notes in Math. 1051 (Springer 1984), 25-37. [5] A. Borel : Cohomologie r~elle stable de troupes S-arithm~tiques classiques, C.R. Acad. Sci. Paris S6r. A 274 (1972), 1700-1702. [6] W.G. Dwyer and E.M. Friedlander : Conjectural calculations of general linear group homology, in Applications of Algebraic K -theory to Algebraic G e o m e t r y and Number Theory, Contemp. Math. 55 Part I (1986), 135-147. [7] D. Quillen : On the cohomoIogy and K -theory of the general linear groups over a finite field, Ann. of Math. 96 (1972), 552-586.

Characteristic classes and 2-modular representations for some sporadic simple groups - II Ch. B. Thomas Max-Planck-Institut fiir Mathematik Gott fried---Claren-Stral~e 26 D---5300 Bonn 3

Dedicated to the memory of J. Frank Adams, teacher, colleague and friend.

0. Introduction As in an earlier paper [Th] we are concerned with calculating the cohomology ring . H (G,Z/) of a sporadic simple group G away form the prime 2. This is easiest when the prime ~. $

concerned divides I GI to the first power, for H (G,~)(£) is then periodic and all one has to do is identify a maximal generator. We complete this part of our programme is section two below. . However our main purpose is at least to begin the determination of H (G,~)(~.) when an £-Sylow subgroup G [ is elementary abehan, and the ~-torsion is detected by the subgroup of $

H (G~,Z/) left invariant by the action of the normaliser N(G~) of G~ in G . We do this for several of the Mathieu groups M k and for Janko's group J1 ' postponing possible consideration of the general case to a future paper. As elsewhere in the theory of simple groups M24 provides an excellent test for the general method, since M24,3 is an elementary non-abelian group of order 27, and the complete description of the stable elements in its cohomology is not easy. A further motive for writing this paper is the wish to understand the relation between H (G,K.~=2t)

and the modular representations of G over the finite field U=2t . In most of the

cases we consider the Chern subring in ordinary cohomology localised away from 2 is generated by the classes of one or two representations of low degree. This suggests a simple structure for Pd]:2t(G ) as a ,~-ring with conjugation, particularly when t = 1 and one exploits the prime factorisation of K2k_l(g=2) ~ ~ / 2 k - 1 . However with the exception of J1 ' which behaves much like a group with periodic cohomology, our results only suggest ways of studying modular representations, since we are faced with the familiar convergence problems of the Atiyah-Hirzebruch spectral sequence. Indeed the generic situation for groups of composite order seems to be that there are universal cycles, which cannot be detected by Chern classes in either the characteristic zero or the modular case. However cohomology does at least make plain which representations are important for the ,~-ring structure: as an elementary example consider M l l , which has irreducible 2-modular representations of degrees 1, 10, 44 and 16. Using eigenvalues it

372

is easy to see that P44 = A2(Pl0)--(1 ) , but because of their characters when restricted to MII,II P16 and its conjugate cannot be obtained in this way. However P16 + P16 -~ A2(Pl0) - Pl0 - (3), showing that this situation is simpler over the prime field F 2 . This is reflected in cohomology by the fact that

Hg(M11'~/45-1)(11) -~ ~ / 1 1 ' but H9(M11,~/25_1)(11 ) is trivial. As a harder example the reader may like to consider M23 in the same way.

The final section of this paper is devoted to 2-torsion in the cohomology of J1 " We include it as a supplement to the partial calculations already in the literature, see [Ch], and also because it represents one of the last contributions to mathematics by J. Frank Adams.

1. Mathieu groups

We recall that the five simple Mathieu groups were originally constructed as examples of multiply transitive groups; the two quintuply transitive groups M12 " ~ S12 and M24 " ~ $24 contain the other three examples as stabilising subgroups. For a description of the various ways in which the groups M k have been described we refer the reader to the "Atlas" - we shall be mainly concerned with the second series: PSL(3,[F4) ,"

,

M22 ~

t M23 "

j M24 •

Since the projective special linear group arises as a stabilising subgroup in this series, we denote it by M21. The importance of the projective special linear group M21 is that it carries much of the structure of H (Mk,/0(3), indeed for the first three groups Mk, 3 is an elementary abelian group of rank 2. Furthermore, with N and Z as usual denoting normaliser and centraliser, we have (i) (ii)

(k = 21, 22 and 23), and (quaternion group, k = 21, 22) and

Z(Mk,3) = Mk, 3 N(Mk,3)/Z(Mk,3) ~ Q8

(semi--~dihedralgroup, k = 23) .

= SDI6

The group SDI6 has presentation {s,t : s8 = t2 = 1, t-lst = s3} . The isomorphisms are not immediately apparent from the tables in the Atlas, but an alternative source is the paper of Z. Janko, [J]. Since the centraliseris as small as possiblethe action of the quotient group on Mk, 3 is faithful.W h e n k = 21 we write G for the normaliser, it is a splitextension of the form

C~xC

b

>

~G

~s,t

"~'~8

373

We shall pick a convenient basis for the normal subgroup as a vector space over an extension field of g:3 below. From now on we use the following notation: Let K be a finite abelian group generated as a direct product by a,b,... The one-dimensional representation a of K is faithful on the subgroup < a > , maps the remaining generators b,... to 1, and a = cl(a ) E H2(K,E). The group M24 has a representation r , the Todd representation, in GL(ll,O=2) described in [Td], which when lifted to characteristic zero has the partial character:

class

1

36

54

Xr

11

2

1

7~

7~

@

~

11

231

232

38

0 - ~ - ~ - 1

Here 73 denotes a conjugacy class consisting of three disjoint 7-cycles with three 1-cycles omitted from the notation, etc. W e shall also denote by r its restrictionto any of the smaller Mathieu groups contained in M24. Away from the primes 2 and 3 we have

THEOREM 1 (i) H*(M24,~[~]) is generated by the classes c3, c4, Cl0 and Cll of the 11-dimensional reuresentation r (suitably restricted to a representative family of Sylow subgrouus). (ii) If k = 11, 12, 22 or 23 H (Mk,~ [ ] ) from the prime 11. In all four cases

has the same generators away

c10(rlMk,ll ) = c~(PklMk,11) , where Pk can be identified from the table

k

Ii

12

22

23

deg(pk}..

16

16

280

896

Remark. The anomalous behaviouz at 11 is explained by the splitting of a single conjugacy class of permutations on passing from a symmetric to a Mathieu group. For a proof see [Th]. Let C h ( ) ( f . )

denote the Chern subring of the even--dimensional cohomology localised at the

374

prime ~ .

THEOREM 2 The subring Ch(Mk)(3 ) of H (Mk,E)(3) is generated by Ci(r [Mk,3) , i = 3,4. *

At least when k = 22 or 23 Ch(Mk)(3 ) is properly contained in H (Mk,~)(3) . Proof. We calculate the 3-primary part of H (G,~), where G is the normaliser of a representative 3-Sylow subgroup in PSL(3,~:4). The spectral sequence for the defining short exact sequence is trivial, so H • (G,~)(3) = H • (C 3 x C3,E)Q8 = E~'0. The odd dimensional contribution is an exterior algebra on a 3---dimensional generator, compare [Le]. In even dimensions proceed as follows: Let V be a 2-dimensional vector space over []=3 and consider the induced , action of Q8 on the symmetric algebra S(V ). Take coefficients in I]=9 rather than [1=3'so as to diagonalise the action of an element s of order 4 in Q8 ' Here we use the usual presentation of Q8 as { s , t : s 4 = 1, s 2 = t 2 , t - l s t = s -1} , and represent Q8 in SL(2,[]:9) by

s-~

.~

°1

-i

, t-~

Having extended the scalars choose a basis of eigenvectors {A,B} for [1=9® S(V ) = Q:q[aft] U=3 with sA = iA and sB = - i B . Formally one first chooses A and then takes B to be the image of A under the Frobenius map ¢. As an automorphism ¢ fixes a and fl, and on the coeffidents ~,~) = ~3. We may further suppose that over the extension field ~:q the bases {A,B} and {aft} are related by the equations A=ia+fl,

B= a+ifl

.

The remark in the preamble about the choice of basis is now clear - G 3 is to be generated by a and b dual to the classes

a and ft. Now I]=9[A,B] <s>

has an []:g-basis consisting of all

monomials AJBk with j + 3 k - 0 m o d 4 , which is equivalent to ( k - j ) - 0 m o d 4 . Since t induces the automorphism A ,. ~- B , B -~ • A , one type of invariant polynomial is "evenly symmetric" in A and B , i.e. one considers symmetric polynomials of the form

AJB k + AkB j = a + k , where

j and k are both even, and (k-j) -= 0 mod 4. The second type must satisfy AJB k - AkBJ= Cjk ' where j and k are both odd and (k-j)-= 0(4). The first few invariant polynomials are A2B 2 = __(a2 + ~ ) 2 ,

A4+ B4= _(a4 + f14), ASB _ ABS= (a2 + f12)(a3 fl + a/~3), . .... One

375

sees immediately that S(V*) Q8 has two generators of degree 4, one of degree 6, . .... On the other hand by counting dimensions we see that all but one of the irreducible representations of G factor through the quotient group Q8 ' and the exception, obtained by induction form the trivial representation, restricts to the regular representation minus a trivial summand on C 3 x C 3 . An easy calculation now shows that Ch(G)(3) is generated by c 6 and c 8 of this restriction, and hence is properly contained in Heven(G,~)(3). This argument applies immediately to the Mathieu groups M21 and M22 since the stable elements in the cohomology of Mk, 3 coincide with those invariant under the normaliser, see [Sw]. Inspection of the character table again shows that Ch(M22)(3) is generated by the Chern classes of the regular representation of C 3 x C 3 . For M23 the argument follows the same pattern, except that one replaces Q8 by SD16, represented over U=q by

SD

(3

' t -~

;

where ( = 1-i

is a primitive 8th-root of unity. A basis of eigenvectors is given by {A,B} , where sA = ( A , sB = (3B, and because t has order 2 rather than 4 the invariant polynomials are AJB k + AkB j with j + 3k = 0 rood 8. As one would except this subalgebra is smaller than for M22, but A2B 2 still provides a generator in degree 4, which is not describable as a Chern class. The situation for the largest Mathieu group M24 is more complicated, since M24,3 is a non-abelian group of order 27 and exponent 3. This cohomology of this group has been worked out by G. Lewis, see [Le], and using this multiplicative relations one can give a surprisingly simple description of the Chern subring. However the determination of the 3-primary part of H (M24,E) is harder, since we can no longer apply Swan's normaliser theorem. 2. Other sporadic simple groups In this section we consider the twelve sporadic simple groups omitted from our previous paper [Th]. Loosely speaking these fall into three classes - the Fischer groups, those closely related to the Monster, and the oddments J3' Ru, O'N, Ly and J4 " Because the last five groups are best described by means of faithful modular representations, our method works particularly well for them. However we start by sumarising the information for primes ~ >_ 5 dividing the order to the first power only, i.e. for which H (G,E)(~ is periodic. A blank space means that the x

1

prime concerned does not divide the order; a space containing a dash (-) means that the Sylow subgroup G£ is not cyclic. An asterisk (*) against an entry means that a generator for the periodic cohomology may be taken to be the appropriate Chern class of a non-trivial irreducible representation of smallest degree

-

-

-

Th B He

Fi2Z Fi23 F~ 4

8

-

8

-

4

}IN

J4

Ly

O'N

Ru

J3

5

-

12

12

-

-

20

20*

10

20

20

12 -

-

10

12 6

22

11

-

12

7

26

12

12

26

24

26

13

32

32

16

32

16

17

36

36

18

12

18"

19

22

22

22

44

23

28

56

28

29

30

30

20

12

30

31

24

36

37

28

43

46

47

44

67

o~

377

For the first groups we can summarise the information from our table in the following result: THEOREM 3 Let the pair (G,q) be as shown

J3

Ru

O'N

Ly

J4

2

5

?

5

11

and let R be the coefficient ring /Z[ if G = J3 and E[ otherwise. Then t~ (G,R) is a sum of polynomial algebras, each of which is generated bv a Chern class of a restricted irreducible represent ation. Proof. This follows the lines of the argument in [Th], and depends on an examination of (a) the character tables and (b) the listed maximal subgroups of G in the Atlas. Remarks on the individual groups. Ru: Perhaps the most revealing representation is that of the related group 2.G in the orthogonal group O28(E [i] ) reduced modulo 5. So far as odd torsion in cohomology is concerned 2.G behaves like G , and the Chern classes of this g:5-representation pick up maximal generators for 7 and 13, and the square of a maximal generator for 29. Ly:

This is perhaps the most interesting group among the oddments, since the period for the

prime 31 (equal to ~ (31-1)) is so low. This is explained by Ly containing G2(FS) as a maximal subgroup (this group of Lie type has periodic cohomology for the primes 7 and 31, the period for both the latter being 12). The remaining maximal subgroups of interest to us are the cyclic by cyclic extensions 67 : 22 and 37 : 18, and the semi-direct product 35 : (2 x Mll ) , which detect 67-, 37- and ll-torsion respectively. However in order to realise a maximal generator for 31 as a Chern class one must go in the Atlas to X39 taking the value 43 110 144 at the identity. J4: This is usually thought of as a subgroup of GLll2(g:2). However comparison with other groups in this class suggests that one look for a more geometrically motivated representation over the Galois field l : l l . Further calculations along the lines of those carried out for the Mathieu groups in the previous section seem possible, although not very rewarding. With the exceptions of HN, Ly and B, the orders of which are divisible by 56 , all the groups on our list have the property that, if ~. >_5 , then £ divides the order to at most the third power. Thus, if ~2 is the highest power

378

occuring, calculation of both Ch(G)(£) and H (G,~)(£) as in Theorem 2 seems to be straightforward. A Sylow subgroup G£ is necessarily abelian, and the image of the restriction map coincides with the subgroup invariant under the action of the normaliser N(G£). For £3 dividing the order one is again forced to use Lewis' calculationsfor the non-abelian group of order £3 and exponent £. The situation is straightforward enough in principle, although certainly numerically complicated. The groups most accessibleto this attack would seem to be Fi22 and Fi23 •

3. Janko's firstgroup J1 (revisited) In our previous paper [Th] we exploited the fact that away from the prime two

J1

$

behaves like a group with periodic cohomology to calculate H (J1,E [½] ) . With the exception of £ = 11 the £-periods all divide 12, which points to the importance of the [l=ll-representation ~o used originally by Janko to define the group. Indeed the dimension of ~o equals 7 and is minimal for a positive non-trivial representation over any field. The values of ~o on the different conjugacy classes are given by:

class[1

2 3

5 (1)

5 (2)

7 10 (1) 10 (2) 15 (1) 15 (2)

19 (i)

Here A is one of 3 irreducible characters of degree 6 for the normaliser N(JI,19 ) , and al, a 2, a 3 represent three conjugacy classes of elements of order 19.

All elements of order two are conjugate, a 2-Sylow subgroup is elementary abelian of order 8, and any positiverepresentationof J1 must restrictto a direct sum of copies of the trivialand regular representations.For example ~o[J1,2 equals Preg- (1). The calculationsare completed in even dimensions by

T H E O R E M 4 (J.F. Adams) ~[even(Jl,~)(2) may be presented by 5 generators x,y,z,u,v of dimensions 6,8,10,12,14 resuectively,~nd two relations r20 -- 0, r24 = 0, where r20 = x2y + xv + yu + z 2 , r24 = x 4 + x2u + xyz + y3 + zv + u 2 . Proof.

This is a more complicated version of that of Theorem 2, and we again use the symmetric

algebra S(V ) associatedwith the 3---dimensionalvector space V over F 2 . Write S(V ) as a

379

polynomial algebra g:2 [aft,7] , and let K be a subgroup of order 21 in GL(V) ~ GL(3,D:2) acting in the obvious way on S(V ). This is an accurate model for the cohomology of J1 ' since the normaliser of J1,2 is a cyclic-by---cyclicextension of the form 7 : 3. In order to find generators for S(V*) K we embed T2[a,/3,7 ] in IFs[afl,7] , and let ¢ be the Frobenins automorphism as in section 1. Over 0=8 we can find a new basis {A,B,C} of . S(V ) consisting of linearly independent eigenvectors corresponding to the eigenvalues ~?, ~72 and ~74 for an element k E K of order 7. Here ~7 is a primitive 7th root of unity.

Step 1 S(V*) K has an [F2-base consisting of the symmetric sums aij k = AiBJc k + BicJA k + ciAJB k where i + 2 j + 4 k = 0 m o d 7 .

This is proved by showing that monomials of the form AiBJc k are k-invariant, and then taking the sum in order to allow for the group extension. Step 2 Write

x = a l l 1 = ABC y = a130 = AB 3 + B C 3 + C A 3 z = a320 = A3B 2 ÷ B 3 C 2 + C 3 A 2 u = a510 = A 5 B + B 5 C ÷ C 5 A v = a?00 = A 7 + B 7 + C 7 .

Step 3 Use induction on the degree of the symmetric sums aij k to show that the five polynomials above actually do generate the invariant elements. Direct calculation shows that they also belong to g=2[afl,7] , rather than to the polynomial ring over g:8 " Furthermore the two relations r20 and r24 are satisfied. (This can be proved more slickly using Steenrod operations.) Step 4 The relations are exhaustive. We have to show that the ring epimorphism

R = F 2 [x,y,z,u,v]/(r20,r24)

I

' S(v*)K

is a monomorphism. We begin by localising so as to invert x = ABC.

LEMMA 5 The map l:t

~R(x -1) is mono.

Proof. One first shows by successive formation of quotients that the sequence x,y,v,r20,r24 in

380

F 2 [x,y,z,u,v] is regular. From this it follows that multiplication by x is (1-1) on the quotient ring R . Now extend the scalars in the localised ring from ~=2 to ~8 ' noting that we have one generator and one relation less. Replace r24 by

r24

= x4

+ x2u +

y3

+

yzu + z 3 + u 2 , and write x

U = A2/B , V = B 2 / C '

W = C2/X . Then

element h of order 3 i n

K . Write Y / x = U + V + W = g l '

U,V,W

are fixed by k

and permuted by an

Z/x=UV+

VW+WU=g2

and x = UVW = g3 ' Then U/x = U2V + V2W + W2U = g4 ' say, and

R(x-1) = IF2 [gl,g2,g3,g 4] (r12)

(x-1) , where

r12 = g2 -4- (glg 2 + g3)g4 + (g~g3 + g~ -4- g~) and r~4 and r20 can be expressed in terms of it. Given the algebraic independence of U, V and W it is now clear that f is a monomorphism after inversion of x and extension of scalars. Given Lemma 5 the same is true for the original map. COROLLARY 5

Th__~e2-primary part of the Chern subring Ch(J1)(2 ) is properly contained in

Heven(Jl,~)(2 ) • Proof. This is a matter of evaluating the total Chern class of the regular representation of an elementary abelian group of rank 3. It turns out that the only non-vanishing classes are

c4 = a 4 + . . . + a 2 ~ + ... + ~Zv(a + Z + ~) , c6 = 2 ~

c7 = 4 ( ~

+ ... + , Z ~ ( j + $ + 3 + ,Zv) and + 2~) + . . . .

This calculation serves as a useful check on that in Theorem 4, and the existence of the classes x and z of degrees 6 and 10 shows that there are invariant elements other than Chern classes. Furthermore, and the same argument applies to the Mathieu groups, comparison of spectral sequences shows that the class x (for example) is a universal cycle in the Atiyah-Hirzebruch ^

spectral sequence converging to the completed representation ring R(J1) . Here no localisation of coefficients is involved, and we have yet further examples for which the Grothendieck filtration of R(G) is definitely finer than the topological.

381

References Ad

A. Adem et. al.

The geometry of cohomology of the Mathieu group M12 (these proceedings)

Atlas

J.H. Conway et al.

Atlas of finite groups, Clarendon Press (Oxford) 1985.

Ch

G.R. Chapman

Generators and relations for the cohomology ring of Janko's first group in the first twenty.one dimensions, in "Groups-St. Andrews 1981", Cambridge University Press, 1982.

Z. Janko

A characterisation of the Mathieu simple groups, I & II, J. Algebra 9 (1968) 1-19 and 20-41.

Le

G. Lewis

Integral cohomology rings of groups of order p3, Trans. Amer. Math. Soc. 132 (1968) 501-29

Sw

R.G. Swan

The p-period of a finite group, Ill. J. Math. 4 (1960) 341--6

Td

J.A. Todd

On representations of the Mathieu groups as collineation groups, J. London Math. Soc. 34 (1959) 406"416

Th

C.B. Thomas

Characteristic classes and 2-modular representations for some sporadic simple groups, to appear in Contemporary Mathematics (Proceedings of the Northwestern Homotopy Theory Conference 1988)

Bonn, April 1989

T h e a b e l i a n i z a t i o n o f t h e t h e t a g r o u p in low g e n u s by Steven H. Weintraub

Department of Mathematics Louisiana State University Baton Rouge, LA 70803-4918

Let Sg denote Siegel space of degree g, i.e., the space of g-by-g symmetric complex matrices with positive-definite imaginary part. The classical theta function with characteristic m is the function 0,,, : Sg x cg --+ ¢ (~g consisting of row vectors) defined by the equation

0m(~, z) = ~

e~p(~i[(p + m ' / 2 y '(p + m'/2) + 2(p + m'/2) '(z + m"/2)]).

p6Z g

Here the characteristic m is a row vector m = ( m l , . . . , m g , m l ", . . : , m~) with each entry equal to zero or one. The parity e(m) of a characteristic m is e ( m ) -- ral' m "1 + . . . + mgmg'" 6 Z/2Z and m is called even (odd) as e(m) = 0 (= 1). T h e t a functions satisfy the following transformation law [I]: Theorem.

Let T 6 Sp2g(Z) and write T as a block m a t r i x T=

(A C

Then

or..(z(c~- + B) -1 , ( A t + B)(C~- + D) -1) 1

= 7~(T)det(Cr + D) ~ exp[(=iz)(Cv + D) -I 'z]O.,(v,z) where T.m

= m T -] + ((D tC)o ( B tA)o) (rood 2)

where ( )o denotes the row vector obtained by taking the diagonal elements of the matrix, and 7 ~ ( T ) is an eighth root of unity, known as the theta multiplier o f T t'or the characteristic m .

The action of the symplectic group Sp2g(l) on characteristics (by m ~-~ T • m) preserves parity, and is transitive on characteristics of a given parity. We let F ( m ) be the stabilizer of ra, i.e.

r ( m ) = {T e Sy29(Z) I r . m = m}.

383

We observe that if I'2 is tile principal congruence subgroup of level 2 in Sp2g(Z), i,e.

/'2 = { T e Sp

g(z) I T =

_r(mod2)}

t h e n / ' 2 C / ' ( m ) for every m. (Indeed, E2 = ~ F ( m ) . ) m

The "standard" theta function is the theta function with characteristic m0 = ( 0 , . . . , 0 ) , and the stabilizer/'(rn0) of this characteristic is classically known as the theta subgroup of Sp~g(Z). This subgroup is usually denoted F(1, 2), and consists of the matrices T as above with the g-by-g matrices AtB and CtD having olfly even entries along their diagonals. Since we are interested in what happens for different values of the genus g, we shall often denote this group by/`g(1,2). Let us now explain the connection between these groups and the topology of Riemann surfaces. Let M 2 be a Riemann surface of genus g with intersection pairing I, and choose a symplectic basis {e~, f~} for H I ( M 2 : Z). Then I induces the Z/2Z intersection pairing I(mod 2) on H I ( M 2 : 7_/2][). A quadratic refinement of this Z/2Z intersection pairing is a map q : H I ( M 2 : Z) ~ Z/2Z such that

q(z -4- y) -- q(~) -- q(y) = fix, y] (rood2) for all x , y in H1(M z : Z). In this algebraic situation we may explicitly describe all quadratic refinements of the bilinear form I(mod 2). It is easy to check that they arise as follows: The quadratic refinements q,~ of I(mod 2) are in one-to-one correspondence to the characteristics m (as defined above), where qm : H I ( M 2 : Z) ---* l / 2 l is given by

A lengthy but routine computation also shows: Let f : M 2 -+ M 2 with

T = f . : H~(M 2 : 7) ---+H I ( M 2 : Z) the induced map on homology. Then

qT.m(T(z)) = qm(z) for every x e H I ( M s : Z) where T . m is defined as in 3.1. Thus, in particular, we see that the stabilizer of the form q,~ is the s u b g r o u p / ` ( m )

of Sp~g(Z). (The quadratic refinements of a given non-singular bilinear form are classified up to isomorphism by their Arf invariants (in l / 2 Z ) , and the Arf invariant of q,~ is just the parity e(m).) The quadratic refinements have a geometric interpretation which follows from the work of E. Brown [B], or, alternatively, from that of D. Johnson [J]. Namely, every

384

Riemann surface admits a spin structure, and indeed, exactly 22g spin structures, and these spin structures are in 1 - 1 correspondence with the 22g quadratic refinements q,,~ of I(mod 2). Thus a self-diffeomorplfism f of the Riemann surface M preserves a spin structure wm of M if and only if its induced map f . on H i ( M : Z) satisfies f . E F(m). (The above discussion is taken from [LMW, sections 2 and 3].) The eighth root of unity 7,~(T) which appears above depends on the choice of square root of det(CT + D) in the above transformation law. (This choice can be made independently of r and z.) Because of this choice, the function 7 m : F(m) --+ {eighth roots of unity} is not a homomorphism, but its square (7m) 2 : F(m) ~ {fourth roots of unity}

(*)

is a homomorphism. In [I, theorem 3] Igusa gave a method for computing (7,~)~(T) for T E F2 and m an even characteristic. In case m = m0 the result is particularly simple. Letting 70 = 7m0,

(70)2(T) = (-1)½ for any T C 1-'2 where T is written as a block matrix as above and I denotes the g-byg identity matrix. (Note that for any even m (Tin) 2 : F2 --~ {square roots of unity}.) The work of [JM] contains two main results. The first is an algorithm for computing (70)2(T) for any T E F(1,2). The second, which is our main point of interest, is the following. From (*) we have an epimorphism from F(1, 2) onto Z/4Z, which of course factors through the abelianization of F(1,2). They show T h e o r e m . [JM, theorem 1.1 (i)] For g > 3, the map T H (70)2(T) gives an isomorphism from the abelianization of the genus g theta group Fg(1,2) to Z/4Z. Our main result here, which is considerably easier than theirs, is that this theorem is false for g = 1, 2. To be precise: Theorem

1. i) ii)

The abelianization ofF1(1,2) is isomorphic to Z/4Z @ Z. There is an epimorphism from the abelianization o f F 2 ( 1 , 2 ) onto Z/4Z @ Z/2Z.

In the course of our proof we shall exhibit elements of Fg(1,2), 9 = 1,2, whose images generate the given groups. (As the reader will see, the argument for (i) uses nothing that has not been known been known for decades, and it is certainly possible that this fact has been noticed before. Part (ii) is essentially new.) P r o o f . We have mentioned that I'2 C F(1,2), and that Sp2g(Z) = F1 acts transitively on the even characteristics. For 9 = 1, there are 3 even characteristics, and [/'1 :/'2] = 6, so [F~(1,2) :/'2] = 2. For g = 2, there are 10 even characteristics, and

385

[['1 : F~] = 720, so [/'1(1,2) :/'2] = 72. For convenience we let F = Fg(1,2), the value of g being clear from the context. i) The group F contains the element S = (_~ 10), and S ~ F2, so F is generated by 1"2 and S. We now pass to the projective group PSp2(Z) = Sp2(Z)/+ 1. Then we have a commutative diagram, with tide two vertical arrows being 2 - I maps, 1

~

/'2

~

1

--, PF= ~

F

---*

1/21

--4

1

PF

~

Z/2Z

--,

1

Recall that PSp2(Z) = PSL2(I) acts on $1, the ordinary upper-half plane, by fractional linear transformations. The subgroup PF2 acts freely on S1, so PF2 is isomorphic to 7q(S~/PF2). However (S1/PF2) = p l ( q ~ ) _ {0, 1,c¢} (as is classically known) and so ~rl(81/PF2) is free on two generators, which we choose to be the loops around oo and 0. These loops are represented by the matrices 1~ = (I0 ~) and 0 = (_12 ~) respectively. If .q is the image of S in PP, then matrix multiplication shows $2 = 1 and $ / ~ - 1 = 0 so in terms of generators and relations we find

PV = (R, 0 , 8 I ~2 = 1, oeRoel = 0 ) . =

=

1>

Lifting back t o / ' 1 , the elements R, U, and S lift to elements R, U and S given by the matrices as written. Now however, S is of order 4, so we find r = (n,v, sls'

= 1, s n s

= u)

= (R,S I S 4 =I, RS 2 =S2R) (reflecting the fact that PF is the quotient of F by its center {1, $2}), a n d the abelianization is as claimed. ii) It will be convenient for us to regard Sp2(1) x Sp2(1) as the subgroup of Sp,(Z) consisting of matrices of the form

(al

bl as

cl

b2 dl

Cl

Wi=(~i b) ci

di

ESp2(Z),

i21,2

}

.

d2

We write such a matrix as a pair (W1, W2). As we have observed, the subgroups F(m) are all conjugate for m even. Thus, instead of investigating P we may investigate F ' = F ((1,1,1, 1)), the stabilizer of the even characteristic m = (1,1,1,1). Then by (3.3) and (4.3) of [LW2] (see also section 2.2 of [LWl]), F ' is generated by F2, Sp2(1) × Sp2(1), and the element

386

,)

-1 F =

(From this description it is also easy to see that [F' : F2] = 72, as stated above.) Let us consider the subgroup A of F ' generated by Sp2(Z) × Sp2 (Z) and F . Then [A :

Sp2(Z) × Sp2(Z)] = 2, but F is an element of order 4 (though F 2 E Sp2(Z) × Sp2(Z)) and conjugation by F gives an automorphism of Spa(Z) × Sp2(Z) of order two given by the formula F(WI, W2)F-~ = (-W~, -W1), so A is a non-split extension of Sp2(Z) x Sp2(Z) by Z/2Z. To avoid confusion we shall denote the principal congruence subgroup of level two of Sp2(Z) by G2. Now the abelianization of F' certainly maps onto the abelianization of F'IF2 = AI(A n F2), and

AnF2={(W~,W2) IW~eG~, i = 1,2}. Note that F 2 E A N/'2 so we have a split extension (with F projecting non-trivially to Z/2Z) 1 --, (,.,¢p~(z)/v~) × (sp~CZ)/c~) ~ A I A n F 2 -..-, Z/2Z ~ 1. It is well-known that Sp2(Z)/G~ is isomorphic to the symmetric group on three symbols, 1 whose abelianization is Z/2Z, and, indeed, the element S = (_~ 0) of Sp2(Z) has nontrivial image under the abelianization map. If we define elements V1, V2 of A by Va = ( - S , / ) and V2 = ( I , - S ) , where I is the 2 x 2 identity matrix, it then follows easily that the abelianization of F'/F2 is the abelianization o f / 3 / ~ n/12, where ~ is the group generated by 1/1, V2, and F . But the quotient group is [ V, =

=

=

V,V

= V V,, _PV, F - '

=

so has abelianization ( l / 2 l ) ¢ (Z/2Z). Thus we have so far that the the abellanization of P ' maps onto (Z/2Z) ~9 (Z/2Z), with the element V~ (or equivalently V2) mapping to (1,0) and F to (0,1). Note, however, that V~ = ( - I , I ) (resp. V~ = ( I , - I ) ) and by [I, theorem 3] (7(1,1,1,1))2(Vh = (7(1.1,1,1))2(V~) = - 1 , so V1 (resp. V2) is an element of order 4, and the theorem follows. Let us write Z additively but identify Z/2Z with { 1 , - 1 } and Z/4Z with {1,i,-1,-i}. Let a = ( a j , a ~ ) : FI(1,2) ~ ZI4Z $ Z and ~' = ( f l ~ , ~ ) : F ' -~ Z/4Z $ Z/2Z be the maps constructed in the proof of the theorem. We have actually stated the theorem for P, not /~t. To obtain an explicit map for F we must conjugate our elements above by a suitable element of Sp4(Z). Such an element is

387

H =

Letting V / = H V i H -1 and

~1 =

1 0 1 0 1 -1 -1 -1 1

0)

= H F H - t we obtain the (fearsome looking) elements

-2 -1 1 2

-I -2 -2 0 2 2 2 0 1

=

2

3

We define fl = (~1,fl~) : F2(1,2) -~ Z / 4 l ~ l / 2 Z by ~(T) = j 3 ' ( H - 1 T H ) . C o r o l l a r y 2. i) T h e m a p a l :/'1(1,2) --* Z/4Z satisfiesal(R) = 1, a l ( S ) = i. Indeed, t h e m a p al agrees with ('},0)2. The map a s : / ' 2 ( 1 , 2 ) ---* I satist~es ct2(R) = 1, a2(S) = O. ii) The map fl~ : F2(1,2) --* l / 4 Z satistles fll(V~) = fit(V2) = i, fit(F) = 1. Indeed, fit agrees with (70) 2. The map f12 : F2(1,2) --~ Z/2Z satisfies fl2(~zt) = ~ ( V 2 ) = 1, f12(_~) = - 1 . Indeed, ~2 agrees with the map T --~ ( - 1 ) a ( H - t r n ) ,

where

T/~14): de~ ( T/'/'21 7n23)(Vt'god2) m32 m34 \ 'D2'41 'D~43

d(M) = de, ( T1212

for M = (Tt~ij) ~ J~t,

P r o o f . All of part i) with the exception of the agreement with (70) 2 follows directly from the proof of tile theorem. The agreement with (70) 2 follows as one may calculate that (70)2(R) = I, (70)2(S) = i, and these two dements are generators. As for part ii), note that in the proof of the theorem fl~ was defined through (7(1j,1j)) 2, so the agreement of fit with (70) 2 is a tautology. Then the given values for (3,0)2 follow by computation. The value of/~2(~) and f12(-~) is given by the proof, and one can easily check that the given map is a homomorphism taking the prescribed values. (The calculation of (70) 2 in i) is quite classical, but the easiest way to calculate (70) 2 in ii) is to use the formula of [JM].) A c k n o w l e d g e m e n t . The author is partially supported by NSF grant DMS-8803552.

388

References

[B]

Brown, E. The Kervaire invariant of a manifold, Proc. Syrup. Pure Math. (AMS) 22(1970), 65-71.

[i]

Igusa, J.-I. On the graded ring of theta-constants, Am. J. Math. 86(1964), 219-246. Johnson, D. Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2) 22(1980), 365-373.

[J] [JM]

Johnson, D. and Millson, J.J. Modular Lagrangians and the theta multiplier, to appear.

[LMW]

Lee, R., Miller, E. Y. and Weintraub, S.H. Rochlin invariants, theta functions, and the holonomy of some determinant line bundles, J. reine angew. Math. 392(1988), 187-218.

[LWI] Lee, R. and Weintraub, S.H. Cohomology of spaces, Topology 24(1985), 391-410.

Sp4(Z) and

related groups and

[LW2] Lee, R. and Weintraub, S.H. On the transformation law for theta-constants, J. Pure Appl. Algebra 44(1987), 273-285.

LIST OF TALKS P L E N A R Y TALKS Kunio Murasugi

Invariants of Graphs with Applications to Knot Theory. F. Thomas Farrell

Topological Rigidity. Alexander B. Goncharov

Projective Geometry and Algebraic K-theory. Hans-Werner Henn

Some Finiteness Results in the Category of Unstable Modules over the Steenrod Algebra. Lowell Jones

The Space of Stable Pseudo-isotopies on a Non-positively Curved Manifolds. Zbigniew Marciniak

Geometric Approach to Units in Group Rings of 1nfinite Groups. Bob Oliver

Self-maps of Classifying Spaces of Compact Lie Groups. Elmer Rees

The Fundamental Groups of Algebraic Varieties. Melvin Rothenberg

Equivariant rational homotopy and classification of G-manifolds. Nobuaki Yagita

BP-cohomology of BG for a Compact Lie Group. S E C T I O N A L TALKS Alexandro Adem

Cohomology of Sporadic Simple Groups. Boris Apanasov

Conformal Structures on Ilyperbolic Manifolds and Varieties of Representations. Dominique Arlettaz

On the Cohomology of Congruence Subgroups. Stanislaw Betley

Homology Groups of GL(R) with Twisted Coefficients. Boris Botwinnik

The Geometrical Point of View on the Adams-Novikov Spectral Sequence. William Browder

Smooth Exotic Actions on Products of Spheres.

390

Frank Connolly

On the Rigidity of Certain G1~ups. Steven R. Costenoble

Application of Equivariant Orientation Theory. Jim Davis

Alexander Polynomials of Periodic Knots. Ryszard Doman

Rational Moore G-spaces and co.Hopf G-spaces. Karl Heinz Dovermann

Topological Invariants of Real Algebraic Group Actions. Giora Dula

Relative Attaching Map in Thorn Spaces. Thomas Fiedler

Knots and the Topology of Complex Curves on Complex Surfaces. Alexander Harshiladze

The Browder-Livsay Groups for Abelian 2-Groups. Jean-Claude Hausmann

Topological Spaces Associated to Robot Arms. Johannes Huebschmann

Perturbation theory and cohomology of groups. Francis E. A. Johnson

Flat Complex Algebraic Manifolds and Flat Kiihler manifolds. Klaus Heinz Kamps

Aspects of Abstract Homotopy Theory. Sung Sook Kim

Characteristic Numbers and Group Actions. John Klippenstein

Applications of a Relationship between K-theory Operations and Cohomology Operations. Andrzej Kozlowski

Characteristic Classes of Transfers of Vector Bundles. Errki J. Laitinen

A Splitting Principle for Fixed Point Functor. Wolfgang Lfick

Analytic and Topological Torsion for Manifolds with Boundary and Symmetries. Mikiya Masuda

Semifree SU(P}-actions on Homology Spheres and the Rochlin lnvariant.

391

Sergiej Matveev Theory of Complexity of $-manifolds. James McClure Topological Hochschild Homology of the bu-Spectrum. Aleksandr S. Mishchenko Fredholm Structures on Infinite-dimensional Manifolds and Their Homologicat Description. Masaharu Morimoto One Fixed Point Actions on Spheres. Hans Jorgen Munkholm On the Boundedly Controlled K-theory over an Open Cone. Roin Nadiradze Realization of Elements in the Sp- and Sc-cobordism Theories. Nguyen Viet Dong On the Cohomology of the Unipotent Subgroup of the General Linear Group GL(3, F(q) ). Nguyen Huynh Phan On the Topology of the Space of Reachable Symmetric Linear Systems. Dietrich Notbohm Maps Between Classifying Spaces. Andrei Pazhitnov On the Exactness of Novikov Inequalities for the Manifold with Free Abelian Fundamental Group. Eric Pedersen Controlled Surgery and Applications to Group Actions. Stewart Priddy The Stable Type of BG. Pham Anh Mingh Transfer Map and the IIochschild-Serre Spectral Sequence. Dieter Puppe Critical Point Theory with Symmetries. Jonathan Rosenberg The KO Assembly Map and Positive Scalar Curvature. Julius Rudiak Orientability of Bundles and Fibrations. Michat Sadowski Equivariant Splittings Induced by Some Total Actions. Reinhard Schultz Positive scalar curvature and spherical space forms.

392

Jolanta Slomiliska Homotopy Colimits over EI-categories. Larry Smith Fake Lie Groups and Maximal Tori. Pawet Traczyk New Criteria for Periodic Knots. Evgenii Troitsky Some Aspects of the C*-index Theorem. Vladimir Vershinin On Spectra Realizing Some Modules over the Steenrod Algebra. Peter Webb The Structure of Mackey Functors. Andrzej Weber A Filtration in the Intersection Homology Groups. Steve Weintraub Cohomology of certain Siegel Modular Varieties.

CURRENT

ADDRESSES

OF P A R T I C I P A N T S A N D A U T H O R S

Alexandro Adem Department of Mathematics University of Wisconsin MADISON, WI 53706 U.S.A.

Stanislaw Betley Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Andrzej Dawidowicz Zaklad Matematyki Wy~sza Szkola Pedagogiczna PL-10-561 OLSZTYN Poland

Piotr Akhmetev Steklov Institute Soviet Academy of Sciences MOSCOW 117333 Soviet Union

Agnieszka Bojanowska Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Ryszard Doman Instytut Matematyki Uniwersytet ira. A. Mickiewicza PL-60-769 POZNAI~I Poland

Christopher Allday Department of Mathematics University of Hawaii at Manoa HONOLULU, HI 96822 U.S.A.

Boris Botwinnik Computer Center Soviet Academy of Sciences KHABAROVSK 680063 Soviet Union

Woj ciech Dorabiala Instytut Matematyki Uniwersytet Szczecifiski PL-70-451 SZCZECIN Poland

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Cezary Bowszyc Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Ronald M. Dotzel Department of Mathematics University of Missouri-St.Louis ST.LOUIS, MO 63121 U.S.A.

Boris Apanasov Institute of Mathematics Soviet Academy of Sciences NOVOSIBIRSK 630090 Soviet Union

William Browder Department of Mathematics Princeton University PRINCETON, NJ 08544 U.S.A.

Karl-Heinz Dovermann Department of Mathematics University of Hawaii at Manoa HONOLULU, HI 96822 U.S.A.

Dominique Arlettaz Department of Mathematics Universite de Lausanne CH-1015 LAUSANNE Switzerland

Frank Connolly Department of Mathematics University of Notre Dame NOTRE DAME, IN 46556 U.S.A.

Emmanuel Dror-Farjoun Department of Mathematics The Hebrew University 91904 JERUSALEM Israel

Amir Assadi Department of Mathematics University of Wisconsin MADISON, WI 53706 U.S.A.

R. Costenoble Department of Mathematics Hofstra University HEMPSTEAD, NY 11550 U.S.A.

Giora Dula Department of Mathematics Purdue University WEST LAFAYETTE, IN 47907 U.S.A.

Hans Joachim Baues Max-Planck-lnstitut fiir Mathematik D-5300 BONN 3 Germany

James F. Davis Department of Mathematics Indiana University BLOOMINGTON, IN 47405 U.S.A.

Grzegorz Dylawerski Instytut Matematyki Uniwersytet Gdafiski PL-80-308 GDAI~SK Poland

394

Zdzislaw Dzedzej Instytut Matematyki Uniwersytet Gdafiski PL-80-308 GDAl~SK Poland

Andrzej Granas Department of Mathematics Universite de Montreal MONTREAL, Quebec H3G 3J7 Canada

Stefan Jackowski Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

John Ewing Department of Mathematics Indiana University BLOOMINGTON, IN 47405 U.S.A.

Boguslaw Hajduk Instytut Matematyki Uniwersytet Wroclawski PL-50-384 WROCLAW Poland

Jan Jaworowski Department of Mathematics Indiana University BLOOMINGTON, IN 47405 U.S.A.

F. Thomas Farrell Department of Mathematics Columbia University NEW YORK, NY I0027 U.S.A.

Alexander Harshiladze I.Z.M.I.R. Soviet Academy of Sciences TROITSK 142092 Soviet Union

Jerzy Jezierski Katedra Zastosowafi Matematyki S.G.G.W. PL-02-766 WARSZAWA Poland

Thomas Fiedler Institute of Mathematics Akademie der Wissenschaften D-1086 BERLIN Germany

Akiro Hattori Department of Mathematics University of Tokyo TOKYO 113 Japan

Jerzy Jodel Instytut Matematyki Uniwersytet Gdafiski PL-80-308 GDAl~SK Poland

Pawel Gajer Instytut Matematyki Uniwersytet Wroctawski PL-50-384 WROCLAW Poland

Jean-Claude Hausmann Department of Mathematics Universite de Geneve CH-1211 GENEVE 24 Switzerland

Francis E. A. Johnson Department of Mathematics University College LONDON WC1E 6BT Great Britain

Andrzej Gaszak Instytut Matematyki Uniwersytet im. A. Mickiewicza PL-60-769 POZNAl~I Poland

Hans-Werner Henn Department of Mathematics Universit~it Heidelberg D-6900 HEIDELBERG Germany

Lowell Jones Department of Mathematics State University of New York STONY BROOK, NY 11790 U.S.A.

Charles H. Giffen Department of Mathematics University of Virginia CHARLOTTESVILLE, VA22903 U.S.A.

Johannes Hfibschmann Department of Mathematics Universit/it Heidelberg D-6900 HEIDELBERG Germany

Yoshinobu Kamishima Department of Mathematics Hokkaido University SAPPORO 060 Japan

Jacek Goclowski Zaklad Matematyki Wyisza Szkola Pedagogiczna PL-10-561 OLSZTYN Poland

Soren Illman Department of Mathematics University of Helsinki SF-00100 HELSINKI 10 Finland

Klaus Heiner Kamps Department of Mathematics Fernuniversit/it D-5800 HAGEN Germany

Marek Golasifiski Instytut Matematyki Uniwersytet im. M. Kopernika PL-87-100 TORUiQ Poland

Paul Iqodt Department of Mathematics K.U.L. B-8500 KORTRIJK Belgium

Cherry Kearton Department of Mathematics University of Durham DURHAM DH1 3LE Great Britain

Alexander B. Goncharov Steklov Institute Soviet Academy of Sciences MOSCOW 117133 Soviet Union

Marek Izydorek Instytut Matematyki Politechnika Gdatiska PL-80-952 GDAlqSK Poland

Sung Sook Kim Department of Mathematics Korea Institute of Technology TAEJON, 305-701 South Korea

395

John Klippenstein Department of Mathematics University of British Columbia VANCOUVER, B.C. V6T 1Y4 Canada

Wladyslaw Lorek Instytut Matematyki Uniwersytet Wroctawski PL-50-384 WROCLAW Poland

Aleksandr S. Mishchenko Department of Mathematics Moscow State University MOSCOW 129344 Soviet Union

Julius Korbas Institute of Mathematics Slovak Academy of Sciences CS-81473 BRATISLAVA Czechoslovakia

Wolfgang Liick Department of Mathematics University of Kentucky LEXINGTON, KY40506 U.S.A.

Masaharu Morimoto Department of Mathematics Okayama University OKAYAMA 700 Japan

Ulrich Koschorke Department of Mathematics Universit~it Siegen D-5900 SIEGEN 21 Germany

Oleg W. Manturov Department of Mathematics Moscow State University MOSCOW 129344 Soviet Union

Hans Jorgen Munkholm Department of Mathematics Odense Universitet DK-5320 ODENSE Denmark

Andrzej Kozlowski Department of Mathematics Wayne State University DETROIT,MI 48202 U.S.A.

Ewa Marchow Instytut Matematyki Uniwersytet ira. A. Mickiewicza PL-60-769 POZNAI~ Poland

Kunio Murasugi Department of Mathematics University of Toronto TORONTO, Ontario M5S 1A1 Canada

Tadeusz Kolniewski Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Zbigniew Marciniak Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Roin Nadiradze Institute of Mathematics Georgian Academy of Sciences TBILISI 380093 Soviet Union

JSzef Krasinkiewicz Instytut Matematyczny Polska Akademia Nauk PL-00-950 WARSZAWA Poland

Tadeusz Marx Katedra Ekonometrii i Inform. S.G.G.W. PL-02-528 WARSZAWA Poland

Ikumitsu Nagasaki Department of Mathematics Osaka University OSAKA 560 Japan

Jan Kubarski Instytut Matematyki Politechnika LSdzka PL-93-590 L6DZ Poland

Mikiya Masuda Department of Mathematics Osaka City University OSAKA 558 Japan

Adam Neugebauer Instytut Matematyki Uniwersytet im. A. Mickiewicza PL-60-769 POZNA/~ Poland

Errki J. Laitinen Department of Mathematics University of Helsinki SF-00100 HELSINKI Finland

Sergiej Matveev Department of Mathematics Chelabinsk University CHELABINSK 454014 Soviet Union

Nguyen Viet Dong Department of Mathematics University of Hanoi HANOI Vietnam

L. Gaunce Lewis Department of Mathematics Syracuse University SYRACUSE, NY 13244 U.S.A.

James McClure Department of Mathematics University of Kentucky LEXINGTON, KY 40506 U.S.A.

Nguyen Huynh Phan Department of Mathematics Pedagogical Univ. of Vinh NGHE TINH Vietnam

Marek Lewkowicz Instytut Matematyki Uniwersytet Wroctawski PL-50-384 WROCLAW Poland

Piotr Mikrut Instytut Matematyki Uniwersytet Wroclawski PL-50-384 WROCLAW Poland

Dietrich Notbohm SFB 170 Georg Aug.ust Universit~it D-3400 GOTTINGEN Germany

396

Krzysztof Nowifiski Instytut Matematyki Stosowanej Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Volker Puppe Department of Mathematics Universit~t Konstanz D-7750 KONSTANZ Germany

Roland Schw~inzl Department of Mathematics Universit~t Osnabrfick D-4500 OSNABRUCK Germar,y

Robert Oliver Department of Mathematics Aarhus Universitet DK-8000 AARHUS C Denmark

Elmer Rees Department of Mathematics Edinburgh University EDINBURGH EH9 3JZ Great Britain

Jolanta Slomifiska Instytut Matematyki Uniwer~ytet ira. M. Kopernika PL-87-100 TORUI~ Poland

Krzysztof Pawalowski Instytut Matematyki Uniwersytet im. A. Mickiewicza PL-60-769 POZNAI~ Poland

Jonathan Rosenberg Department of Mathematics University of Maryland COLLEGE PARK, MD 20742 U.S.A.

Larry S:mith Departraent of Mathematics Georg Aug..ust Universit~t D-3400 GOTTINGEN Germany

Andriej Pazhitnov Institute of Chemical Physics Soviet Academy of Sciences MOSCOW 117977 Soviet Union

Shmuel Rosset Department of Mathematics Tel Aviv University 69978 RAMAT AVIV Israel

Stanislaw Spiel Instytut Matematyczny Polska Akademia Nauk PL-00-950 WA RSZAWA Poland

Eric Pedersen Department of Mathematics State University of New York BINGHAMTON, NY 13901 U.S.A.

Melvin Rothenberg Department of Mathematics University of Chicago CHICAGO, IL 60637 U.S.A.

Mihail Stanko Steklov Institute Soviet Academy of Sciences MOSCOW 117133 Soviet Union

Charya Peterson SFB 170 Georg-Au~[ust Universitaet D-3400 GOTTINGEN Germany

Julius Rudiak M.I.S.I. MOSCOW 129337 Soviet Union

Boris Sternin M.I.M.S. MOSCOW 109028 Soviet Union

Franklin Peterson Department of Mathematics Massachusets Institute of Techn. CAMBRIDGE, MA 02139 U.S.A.

Stawomir Rybicki Instytut Matematyki Politechnika Gdafiska PL-80-952 GDAI~SK Poland

Andrzej Szczepafiski Instytut Matematyki Uniwersytet Gdafiski PL-80-308 GDAl~SK Poland

Pham Anh Mingh Department of Mathematics University of Hanoi HANOI Vietnam

Michal Sadowski Instytut Matematyki Uniwersytet Gdafiski PL-80-308 GDAl~SK Poland

Laurence Taylor Department of Mathematics University of Notre Dame NOTRE DAME, IN 46556 U.S.A.

Stewart Priddy Department of Mathematics Northwestern University EVANSTON, IL 60208 U.S.A.

Jan Samsonowicz Instytut Matematyki Politechnika Warszawska PL-00-661 WARSZAWA Poland

Pawel Traczyk Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Dieter Puppe Department of Mathematics Universit~it Heidelberg D-6900 HEIDELBERG Germany

Reinhard Schultz Department of Mathematics Purdue University WEST LAFAYETTE, IN 47907 U.S.A.

Krzysztof Trautman Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

397

Evgenii Troitsky Department of Mathematics Moscow State University MOSCOW 129344 Soviet Union

Peter Webb Department of Mathematics University of Oregon EUGENE, OR 97403 U.S.A.

Michael Weiss Department of Mathematics Aarhus Universitet DK-8000 AARHUS Denmark

Katsuhiro Uno Department of Mathematics Osaka University OSAKA, 560 Japan

Andrzej Weber Instytut Matematyki Uniwersytet Warszawski PL-00-913 WARSZAWA 59 Poland

Urs Wiirgler Institute of Mathematics Universitgt Bern CH-3012 BERN Switzerland

Vladimir Vershinin Institute of Mathematics Soviet Academy of Sciences NOVOSIBIRSK 630090 Soviet Union

Steven Weintraub Department of Mathematics Louisiana State University BATON ROUGE, LA 70803 U.S.A.

Nobuaki Yagita Department of Mathematics Musashi Institute of Technology TOKYO 158 Japan

Rainer Vogt Department of Mathematics Universit/it Osnabriick D-4500 OSNABRfJCK Germany

Algebraic Topology, Poznan 1989: Proceedings of a Conference Held in Poznan, Poland, June 22-27, 1989 (Lecture Notes in Mathematics)

Algebraic Topology, Poznan 1989

Lecture notes in algebraic topology

Lecture notes in algebraic topology

Algebraic K-theory: Proceedings of a conference held at Oberwolfach, June 1980, Part I (Lecture notes in mathematics)

Lecture Notes in Algebraic Topology (Graduate Studies in Mathematics, 35)

Algebraic Topology: Homotopy and Group Cohomology (Lecture Notes in Mathematics)

Recursion Theory Week. Proceedings Conference Oberwolfach, 1989

Unipotent Algebraic Groups (Lecture Notes in Mathematics)

Algebraic Groups: Lecture Notes in Mathematics

Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978 (Lecture Notes in Mathematics)

1989)

Preconditioned Conjugate Gradient Methods: Proceedings of a Conference Held in Nijmegen, The Netherlands, June 19-21, 1989

19, 1989, Proceedings

Autobahn nach Poznan

Topology: General and Algebraic Topology and Applications. Proceedings of the International Topological Conference held in Leningrad, August 23-27, 1983 (Lecture Notes in Mathematics)

Church and State in Communist Poland: A History, 1944-1989

Geometry of Banach Spaces: Proceedings of the Conference Held in Strobl, Austria 1989 (London Mathematical Society Lecture Note Series)

Transformation Groups, Poznan 1985

Complex Analysis and Algebraic Geometry: Proceedings of a Conference, Held in Gottingen, June 25 - July 2, 1985 (Lecture Notes in Mathematics Mathematica Gottingensis)

Autobahn Nach Poznan

Dynamic Bifurcations: Proceedings of a Conference held in Luminy, France, March 5-10, 1990 (Lecture Notes in Mathematics)

Conjugacy classes in algebraic groups (Lecture notes in mathematics 366)

Real Algebraic Geometry: Proceedings of the Conference held in Rennes, France, June 24-28, 1991 (Lecture Notes in Mathematics) (English and French Edition)

Lecture Notes in Mathematics. Conference in Mathematical Logic - London '70

Democratic Government in Poland: Constitutional Politics Since 1989

Democratic Government in Poland: Constitutional Politics Since 1989

Democratic Government in Poland: Constitutional Politics Since 1989

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Category Theory: Proceedings of the International Conference held in Como, Italy, July 22-28, 1990 (Lecture Notes in Mathematics)

Algebraic Topology, Poznan 1989: Proceedings of a Conference Held in Poznan, Poland, June 22-27, 1989 (Lecture Notes in Mathematics)