Complex Cobordism and Stable Homotopy Groups of Spheres (Pure and Applied Mathematics (Academic Pr))

COMPLEX COBORDISM AND STABLE HOMOTOPY GROUPS OF SPHERES

This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks

Editors: Samuel Eilenberg and Hyman Bass The complete listing of books in this series is available from the Publisher upon request.

COMPLEX COBORDISM AND STABLE HOMOTOPY GROUPS OF SPHERES

Douglas C . Ravenel Department of Mathematics University of Washington Seattle, Washington

1986

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

Orlando Sari Diego New York Austin London Montreal Sydney Tokyo Toronto

COPYRIGHT @ 1986 BY ACADEMIC PRESS. INC ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS. EI.ECTRONIC OR MECHANICAL. INCLUDING PHOTOCOPY. RECORDING. OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM. WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. Orlando, Florida 32887

United Kingdom Edition published by

ACADEMIC PRESS INC.

(LONDON) 24-28 Oval Road, London NWI 7DX

LTD.

Library of Congress Cataloging in Publication Data Ravenel, Douglas. Complex cobordism and stable homotopy groups of spheres. (Pure and applied mathematics ; ) Bibliography: p. Includes index. 1. Homotopy groups. 2. Sphere. 3. Spectral sequences (Mathematics) 4. Cobordism theory. 1. Title. 11. Series: Pure and applied mathematics (Academic Press) ; QA3.P8 [QA612.78] 510 s [514’.24] 85-15097 ISBN 0-12-583430-6 (hardcover) (alk. paper) ISBN 0-12-583431-4 (paperback) (a%. paper) PRINTED IN THE UNITED STATES OF AMERICA

86878889

9 8 1 6 5 4 3 2 I

To my wife, Michelle

This Page Intentionally Left Blank

CONTENTS

Preface Notations and Commonly Used Abbreviations

xv xix

Chapter 1. An Introduction to the Homotopy Groups of Spheres

1. Classical Theorems Old and New

3

Homotopy groups. The Hurewicz and Freudenthal theorems. Stable stems. The Hopf map. Serre's finitenesstheorem. Nishida's nilpotence theorem. Cohen, Moore, and Neisendorfer's exponent theorem. Bott periodicity. The J-homomorphism.

7

2. Methods of Computing K(S")

Eilenberg-Mac Lane spaces and Serre's method. The Adams spectral sequence. Hopf invariant one theorems.The Adams - Novikov spectral sequence.Tablesin low dimensions for p = 3.

3. The Adams-Novikov &-Term, Formal Group Laws, and the Greek Letter Construction

16

Formal group laws and Quillen's theorem. The Adams-Novikov &-term as group cohomology. Alphas, betas, and gammas. The Morava- Landweber theorem and higher Greek letters. Generalized Greek letter elements. vii

viii

Contents

4. More Formal Group Law Theory, BP-Theory,

Morava's Point of View, and the Chromatic Spectral Sequence

23

The Brown -Peterson spectrum. Classification of formal group laws. Morava's group action, its orbits and stabilizer groups. The chromatic resolution and chromatic spectral sequence. Bockstein spectral sequences. Use of cyclic subgroups to detect Arf invariant elements. Morava's vanishing theorem. Greek letter elements in the chromatic spectral sequence.

5 . Unstable Homotopy Groups and the EHP Spectral Sequence

29

The EHP sequences. The EHP spectral sequence. The stable zone. The inductive method. The stable EHP spectral sequence. The Adams vector field theorem. James periodicity. The J-spectrum. The spectral sequence for JJRP") and J*(BZp).Relation to the Segal conjecture. The Mahowald invariant.

Chapter 2. Setting Up the Adams Spectral Sequence 1. The Classical Adams Spectral Sequence

48

Mod (p) Eilenberg- Mac Lane spectra. Mod (p)Adams resolutions. Exact couples. The mod ( p )Adams spectral sequence. Differentials.Homotopy inverse limits. Convergence. The extension problem. Examples: integral and mod ( p i )Eilenberg- Mac Lane spectra.

2. The Adams Spectral Sequence Based on a Generalized Homology Theory

57

E,-Adams resolutions. E-completions. The E,-Adams spectral sequence. Assumptions on the spectrum E. EJE) is a Hopf algebroid. The canonical Adams resolution. Convergence. The Adams filtration.

3. The Smash Product Pairing and the Generalized Connecting Homomorphism

62

The smash product induces a pairing in the Adams spectral sequence. A map that is trivial in homology induces a map ofspectral sequences which raises filtration. The relation between the connecting homomorphism in Ext and the boundary map in homotopy.

Chapter 3. The Classical Adams Spectral Sequence 1. The Steenrod Algebra and Some Easy Calculations

69

Milnor's structure theorem for A *. The cobar complex. Multiplication by p in the Em-term.The ASS for a,(MU). Computations for MO, bu, and bo.

Contents

ix

2. The May Spectral Sequence

79

May’s filtration of A , . Nonassociativity of May’s E,-term at odd primes and a modification of his SS that avoids this problem. Computations at p = 2 in dimensions 5 13. Computations with the subalgebra 4 2 ) at p = 2.

3. The Lambda Algebra

90

A as an Adams El-term. The algebraic EHP spectral sequence. Serial numbers. The Curtis algorithm. Computations below dimension 14. James periodicity. The Adams vanishing line. d , is multiplication by I - , . Illustration for S3.

4. Some General Properties of Ext

100

Ext’ for s 5 3. Behavior of elements in Ext2: results of Browder, Mahowald, Ravenel, and Cohen and Goerss. Adams vanishing line (slope $ for p = 2). Periodicity near the vanishing line (above the f line forp = 2). Elements not annihilated by any iterated periodicity operator and their relation to im J. An elementary proof that most of these elements are nontrivial.

5 . Survey and Further Reading

110

Exotic cobordism theories. Decreasing filtrations of A * and the resulting spectral sequences. Application to MSP. Mahowald’s generalizations of A. v,-periodicity in the ASS. Selected references to related work.

Chapter 4. BP-Theory and the Adams - Novikov Spectral Sequence

1. Quillen’s Theorem and the Structure of BP,(BP)

119

Complex cobordism. Complex orientation of a ring spectrum. The formal group law associated to an oriented homology theory. Quillen’s theorem equating the Lazard and complex cobordism rings. Landweber and Novikov’s theorem on the structure of MU,(MU). The Brown - Peterson spectrum BP. Quillen’s idempotent operation and p-typical formal group laws. The structure of BP,(BP).

2. A Survey of BP-Theory

129

Bordism groups ofspaces. The Sullivan - Baas construction. The Johnson - Wilson spectra BP(n). The Morava K-theories K(n).The Landweber filtration and exact functor theorems. The Conner-Floyd isomorphism. K-theory as a functor of complex cobordism. Johnson and Yosimura’s work on invariant regular ideals. Infinite loop spaces associated with MU and BP; Ravenel and Wilson’s Hopf ring. The unstable ANSS of Bendersky, Curtis, and Miller.

3. Some Calculations in BP,(BP)

137

The Morava- Landweber invariant prime ideal theorem. Some invariant regular ideals. A generalization of Witt’s lemma. A formula for the universal p-typical

Contents

X

formal group law. Formulas for the coproduct and conjugation in BP,(BP). A fdtration of BP,(BP)/Z,, and the structure of the associated bigraded object.

4. Beginning Calculations with

150

the Adams-Novikov Spectral Sequence

The ANSS and sparseness.The algebraic Novikov SS of Novikov and Miller. Low dimensional Ext over the algebra of Steenrod reduced powers. Bockstein spectral sequencesleading to the ANSS ,!?,-term. Calculationsat odd primes. Toda's theorem on the first nontrival odd primary Novikov differential. Tables for p = 5 . Calculations and tables for p = 2. Comparison with the Adams spectral sequence.

Chapter 5. The Chromatic Spectral Sequence 1. The Algebraic Construction

172

Greek letter elements and generalizations. The chromatic resolution, spectral sequence, and cobar complex. The Morava stabilizer algebra yn). The change of rings theorem. The Morava vanishing theorem. Signs of Greek letter elements. Computations with p,. Decomposability of y l . Chromatic differentialsat p = 2. Divisibility of (Y

2. Ext'(BP,/Z,) and Hopf Invariant One

184

ExtO(l3P.J. Ext(m). Ext l(BP,). Hopf invariant one elements. The Miller- Wilson calculation of Extl(BP,/Z,,).

3. Ext(M') and the J-Homomorphism Ext(M1).Relation to im J. Patterns of differentials at p the mod (2) Moore spectrum.

4. Ext' and the Thom Reduction

192 = 2.

Computations with

20 1

Results of Miller, Ravenel, and Wilson ( p > 2) and Shimomura ( p = 2) on Ext2(BP,). Behavior of the Thom reduction map. Arfinvariant differentials at p > 2. Results of Mahowald and those of Cohen and Goerss.

5. Periodic Families in Ext'

208

Smith construction of p,. Obstructions at p = 3. Results of Davis and Mahowald, Oka, Smith, and Zahler on permanent cycles in Ext2. Decomposables in Ext2.

6. Elements in Ext3 and Beyond

215

Products of alphas and betas in Ext3.Products of betas in Ext4.A possible obstruction to the existence of V(4).

Contents

xi

Chapter 6. Morava Stabilizer Algebras 1. The Change-of-Rings Isomorphism

220

Theorems of Miller and Ravenel; theorems of Morava. General nonsense about Hopf algebroids. Formal group laws over Artin local rings. Morava’s proof. Miller and Ravenel’s proof.

2. The Structure of X(n)

226

Relation to the group ring for S,,. Recovering the grading via an eigenspace decomposition. A matrix representation of S,,. A splitting of S,, when p X n. Poincark duality and periodic cohomology of S, .

3. The Cohomology of X(n)

23 1

A May filtration of Z(n)and the May SS. The open subgroup theorem. Cohomology of some associated Lie algebras. H 1 and Hz. H,((S(n))for n = 1,2, 3.

4. The Odd Primary Kervaire Invariant Elements

244

The nonexistence of certain elements and spectra. Detecting elements with the cohomology of Z / ( p ) . Differentials in the ASS.

5 . The Spectra T(n)

253

A splitting theorem for certain Thom spectra. Application of the open subgroup theorem. Exto and Ext’.

Chapter 7. Computing Stable Homotopy Groups with the Adams - Novikov Spectral Sequence 1. The Method of Infinite Descent

258

Some Hopf algebroids and extensions. An alternative to the Cartan- Eilenberg spectral sequence. Weak injective comodules. Some useful cochain complexes.

2. The Complex C1,l

262

Poincark series and weak injectives. The structure of CI,I.

3. The Homotopy Groups of a Complex with p Cells

268

Quillen operations and Exto.A useful SES (7.3.5). Another SS (7.3.8).Differentials (7.3.1 1 ). A paradigmatic SS involving P(1).

xii 4. The ‘‘Algorithm’’ and Computations at p = 3

Contents

28 1

The SS for extracting z*(So) from z,(X). The ABC Theorem for n*(X). Table showing ABC at p = 5 . A computational procedure. The input forp = 3. The output through dimension 106.

5. Computations for p = 5

289

d3&) = pi8 at p = 5 and V ( 3 )does not exist. A conjecture for larger primes. The input for p = 5. Computing the output.

Appendix 1. Hopf Algebras and Hopf Algebroids 1. Basic Definitions

309

Hopf algebroids as cogroupoid objects in the category of commutative algebras. Comodules. Cotensor products. Maps of Hopf algebroids. The associated Hopf algebra. Normal maps. Unicursal Hopf algebroids. The kernel of a normal map. Hopf algebroid extensions. The comodule algebra structure theorem. Invariant ideals. Split Hopf algebroids.

2. Homological Algebra

319

Injective comodules. The derived functors Cotor and Ext. Relative injectives and resolutions of same. The cobar resolution and complex. Cup products. Ext isomorphisms for invariant ideals and split Hopf algebroids.

3. Some Spectral Sequences

325

The SS derived from an LES. Filtered Hopf algebroids. Filtration by powers of the unit coideal. The SS associated with a Hopf algebroid map. Change of rings isomorphism. The Cartan-Eilenberg SS. A formulation due to Adams. The E,-term for a cocentral extension of Hopf algebras.

4. Massey Products

335

Definitions of n-fold matrix Massey products and indeterminacy. Defining systems. Juggling theorems (associativity and commutativity formulas). Convergence of Massey products in spectral sequences. A Leibnitz formula for differentials. Differentials and extensions in spectral sequences.

5. Algebraic Steenrod Operations

346

Construction, Cartan formula, and Adem relations. Commutativity with suspension. Kudo transgression theorem.

Contents

xiii

Appendix 2. Formal Group Laws 1. Universal Formal Group Laws and Strict Isomorphisms

355

Definition and examples of formal group laws. Homomorphisms, isomorphisms, and logarithms. The universal formal group law and the h a r d ring. Lazard’s comparison lemma. The Hopf algebroid VT. Proof of the comparison lemma.

2. Classification and Endomorphism Rings

369

Hazewinkel’s and Araki’s generators. The right unit formula. The height of a formal group law. Classification in characteristic p . Finite fields, Witt rings, and division algebras. The endomorphism ring of height n formal group law.

Appendix 3. Tables of Homotopy Groups of Spheres

380

References

395

Index

407

This Page Intentionally Left Blank

PREFACE

My initial inclination was to call this book The Music of the Spheres, but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. The purpose of this book is threefold: (i) to make BP-theory and the Adams - Novikov spectral sequence more accessible to nonexperts, (ii) to provide a convenient reference for workers in the field, and (iii) to demonstrate the computational potential of the indicated machinery for determining stable homotopy groups of spheres. The reader is presumed to have a working knowledge of algebraic topology and to be familiar with the basic concepts of homotopy theory. With this assumption the book is almost entirely self-contained, the major exceptions (e.g., Sections 5.4, 5.4, A1.4, and A1.5) being cases in which the proofs are long, technical, and adequately presented elsewhere. The subject matter is a difficult one and this book will not change that fact. We hope that it will make it possible to learn the subject other than by the only practical method heretofore available, i.e., by numerous exhausting conversations with one of a handful of experts. Much of the material here has been previously published in journal articles too numerous to keep track of. However, a lot of the foundations of the subject, e.g., Chapter 2 and Appendix 1, have not been previously worked out in sufficient generality and the author found it surprisingly difficult to do so. The reader (especially if he is a graduate student) should be warned that many portions of this volume contain more than he is likely to want or need to know. In view of (ii), results are given (e.g., in Sections 4.3,6.3, and Al.4) in greater strength than needed at present. We hope the newcomer to the field will not be discouraged by this abundance of material. xv

xvi

Preface

The homotopy groups of spheres is a highly computational topic. The serious reader is strongly encouraged to reproduce and extend as many ofthe computations presented here as possible. There is no substitute for the insight gained by carrying out such calculations oneself. Despite the large amount of information and techniques currently available, stable homotopy is still very mysterious. Each new computational breakthrough heightens our appreciation of the difficulty of the problem. The subject has a highly experimental character. One computes as many homotopy groups as possible with existingmachinery, and the resulting data form the basis for new conjectures and new theorems, which may lead to better methods of computation. In contrast with physics, in this case the experimentalists who gather data and the theoreticians who interpret them are the same individuals. The core of this volume is Chapters 2-6 while Chapter 1 is a casual nontechnical introduction to this material. Chapter 7 is a more technical description of actual computations of the Adams - Novikov spectral sequence for the stable homotopy groups of spheres through a large range of dimensions. Although it is likely to be read closely by only a few specialists, it is in some sense the justification for the rest of the book, the computational payoff. The results obtained there, along with some similar calculations of Tangora, are tabulated in Appendix 3. Appendixes 1 and 2 are utilitarian in nature and describe technical tools used throughout the book. Appendix 1 develops the theory of Hopf algebroids (of which Hopf algebras are a special case) and useful homologcal tools such as relative injective resolutions, spectral sequences, Massey products, and algebraic Steenrod operations. It is not entertaining reading; we urge the reader to refer to it only when necessary. Appendix 2 is a more enjoyable self-contained account ofall that is needed from the theory of formal group laws. This material supports a bridge between stable homotopy theory and algebraic number theory. Certain results (e.g., the cohomology of some groups arising in number theory) are carried across this bridge in Chapter 6. The house they inhabit in homotopy theory, the chromatic spectral sequence, is built in Chapter 5. The logical interdependence of the seven chapters and three appendixes is displayed in the accompanying diagram. It is a pleasure to acknowledge help received from many sources in preparing this book. The author received invaluable editorial advice from Frank Adams, Peter May, David Pengelley, and Haynes Miller. Steven Mitchell, Austin Pearlman, and Bruce McQuistan made helpful comments on various stages of the manuscript, which owes its very existence to the patient work of innumerable typists at the University of Washington.

Preface

xvii

Finally, we acknowledge financial help from six sources: the National Science Foundation, the Alfred P. Sloan Foundation, the University of Washington, the Science Research Council of the United Kingdom, the Sonderforschungsbereichof Bonn, West Germany, and the Troisikme Sikle of Bern, Switzerland.

This Page Intentionally Left Blank

Notations and Commonly Used Abbreviations

Z ZP Z,P) ZAP)

Q Q p

LES SES

ss

ASS ANSS

css

CESS BSS 0

Integers padic integers Integers localized at p Integers mod (p) Rationals padic numbers Long exact sequence Short exact sequence Spectral sequence Adams spectral sequence (Section 2.1) Adams-Novikov spectral sequence (Section 4.4) Chromatic spectral sequence (Section A 1.3) Cartan - Eilenberg spectral sequence (Section A 1.3) Bockstein spectral sequence (Section 5.1) Cotensor product (Section Al. 1)

Given suitable objects A, B, and C and a mapf: A denoted byf E3 C.

xix

-

B, the evident map A €3 C

-

B €3 Cis

This Page Intentionally Left Blank

CHAPTER 1

AN INTRODUCTION TO THE HOMOTOPY GROUPS OF SPHERES

This chapter is intended to be an expository introduction to the rest of the book. We will informally describe the spectral sequences of Adams and Novikov, which are the subject of the remaining chapters. Our aim here is to give a conceptual picture, suppressing as many technical details as possible. In Section 1 we list some theorems which are classical in the sense that they do not require any of the machinery described in this book. These include the Hurewicz theorem (1.1.2),the Freudenthal suspension theorem (1.1.4),the Serre finiteness theorem (l.l.S), the Nishida nilpotence theorem (1.1.9), and the Cohen-Moore-Neisendorfer exponent theorem (1.1.10). They all pertain directly to the homotopy groups of spheres and are not treated elsewhere here. The homotopy groups of the stable orthogonal group SO are given by the Bott periodicity theorem ( 1 . 1 . 1 1 ) . In 1.1.12 we define S " image ). is given in the J-homomorphism from .iri(SO(n)) to ~ T ~ + ~ ( Its 1.1.13, and in 1.1.14 we give its cokernel in low dimensions. Most of the former is proved in Section 5.3. In Section 2 we describe Serre's method of computing homotopy groups using cohomological techniques. In particular, we show how to find the first element of order p in 77*(S3) (1.2.4). Then we explain how these methods were streamlined by Adams to give his celebrated spectral sequence (1.2.10). The next four theorems describe the Hopf invariant one problem. A table showing the Adams spectral sequence at the prime 3 through dimension 45 is given in 1.2.15. In Chapter 2 we give a more detailed account of how the spectral sequence is set up, including 1

2

1. Introduction to the Homotopy Groups of Spheres

a convergence theorem. In Chapter 3 we make many calculations with it at the prime 2. In 1.2.16 we summarize Adams’s method for purposes of comparing it with that of Novikov. The basic idea is to use complex cobordism (1.2.17) in place of ordinary mod ( p ) cohomology. Figure 1.2.19 is a table of the Adams-Novikov spectral sequence (ANSS) for comparison with Fig. 1.2.15. In the next two sections we describe the algebra surrounding the E2-term of the ANSS. To this end formal group laws (FGLs) are defined in 1.3.1 and a complete account of the relevant theory is given in Appendix 2. Their connection with complex cobordism is the subject of Quillen’s theorem (1.3.4) and is described more fully in Section 4.1. The ANSS &-term is described in terms of FGL theory (1.3.5) and as an Ext group over a certain Hopf algebra (1.3.6). The rest of Section 3 is concerned with the Greek letter construction, a method of producing infinite periodic families of elements in the E2-term and (in favorable cases) in the stable homotopy groups of spheres. The basic definitions are given in 1.3.17 and 1.3.19 and the main algebraic fact required is the Morava-Landweber theorem (1.3.16). Applications to homotopy are given in 1.3.11, 1.3.15, and 1.3.18. The section ends with a discussion of the proofs and possible extensions of these results. This material is discussed more fully in Chapter 5. In Section 4 we describe the deeper algebraic properties of the &-term. We start by introducing BP and defining a Hopf algebroid. The former is a minimal wedge summand of M U localized at a prime. A Hopf algebroid is a generalized Hopf algebra needed to describe the ANSS &-term more conveniently in terms of BP (1.4.2). The algebraic and homological properties of such objects are the subject of Appendix 1. Next we give the Lazard classification theorem for FGLs (1.4.3) over an algebraically closed field of characteristic p, which is proved in Section A2.2. Then we come to Morava’s point of view. Theorem 1.3.5 describes the ANSS E2-termas the cohomology of a certain group G with coefficients in a certain polynomial ring L. Spec(L) (in the sense of abstract algebraic geometry) is an infinite dimensional affine space on which G acts. The points in Spec( L ) can be thought of as FGLs and the G-orbits as isomorphism classes, as described in 1.4.3. This orbit structure is described in 1.4.4. For each orbit there is a stabilizer or isotropy subgroup of G called S,.Its cohomology is related to that of G (1.4.5), and its structure is known. The theory of Morava stabilizer algebras is the algebraic machinery needed to exploit this fact and is the subject of Chapter 6. Our next topic, the chromatic spectral sequence (1.4.8, the subject of Chapter 5 ) , connects the theory above to the ANSS ,??,-term. The Greek letter construction fits into this apparatus very neatly.

3

1. Classical Theorems Old and New

Section 5 is about unstable homotopy groups of spheres and is not needed for the rest of the book. Its introduction is self-explanatory.

1. Classical Theorems Old and New We begin by recalling some definitions. The nth homotopy group of a connected space X , r , ( X ) , is the set of homotopy classes of maps from the n-sphere S" to X . This set has a natural group structure which is abelian for n 2 2 . We now state three classical theorems about homotopy groups of spheres. Proofs can be found, for example, in Spanier [l]. 1.1.1 Theorem.

rl(S') = Z and r,(S')

= 0 for

m > 1.

1.1.2 Hurewicz's Theorem. .rr,(S") = Z and r,(S") = O for m < n. A generator of rn(Sn)is the class of the identity map.

For the next theorem we need to define the suspension homomorphism u:.rr,(S")+

r,+'(Sn+l).

1.1.3 Definition. The kth suspension ZkX of a space X is the quotient of I kx X obtained by collapsing a I kx X onto d Ik,a I' being the boundary of Ik,the k-dimensional cube. Note that Z'X'X = Z"'X and Zy:ZkX + ZkY is the quotient of 1 x f:I kx X + I' x Y. In particular, given f:S"+ S" we have Xf: Sm+'+ S"+', which induces a homomorphism r,(S"')+ r,+l(Sm+l). 1.1.4 Freudenthal Suspension Theorem. The suspension homomorphism U : r n + k ( S " ) + ?Tn+k+l(S"+') defined above is an isomorphism for k < n - 1 and a surjection for k = n - 1. 1.1.5 Corollary.

) only on k The group T , , + ~ ( S "depends

if n > k + 1.

1.1.6 Definition. The stable k-stem or kth stable homotopy group of spheres r t is r, , + k( S ")for n > k + 1. The groups ~ , , + ~ ( are s " )stable i f n > k + 1 and unstable if n 5 k + 1 . When discussing stable groups we will not make any notational distinction between a map and its suspensions.

The subsequent chapters of this book will be concerned with machinery for computing the stable homotopy groups of spheres. We will not be concerned with unstable groups. The groups r : are known at least for

4

1. Introduction to the Homotopy Groups of Spheres

ks45. See Fig. A3.3. Here is a table of k

O

rr?

Z

7r:

4

1

2

3

ZJ2

212

2/24

0

for k s 15: 5

6

0

212

7

8

Zf240

ZJ2

k

9

10

11

12

13

14

15

n?

Zf2OZ/2OZJ2

216

21504

0

Zf3

Zf2OZJ2

ZJ48OOZJ2

This should convince the reader that the groups do not fall into any obvious pattern. Later in the book, however, we will present evidence of some deep patterns not apparent in such a small amount of data. The nature of these patterns will be discussed later in this chapter. When homotopy groups were first defined by Hurewicz in 1935 it was hoped that r , + k ( S " ) = 0 for k > 0, since this was already known to be the case for n = 1 (1.1.1). The first counterexample is worth examining in some detail.

7r3(S2)= Z generated by the class of the Hopf map 7:S 3 + S2 defined as follows. Regard S 2 (as Riemann did) as the complex numbers C with a point at infinity. S3 is by definition the set of unit vectors in R4 = C2. Hence a point in S3 is specified by two complex coordinates ( z l , z 2 ) .Define 7) by if z 2 # 0 7)(Zl, 2 2 ) = : Iz2 if z2 = 0. 1.1.7 Example.

{

It is easy to verify that 7 is continuous. The inverse image under q of any point in S 2 is a circle, specifically the set of unit vectors in a complex line through the origin in C2, the set of all such lines being parameterized by S2. Closer examination will show that any two of these circles in S3 are linked. One can use quaternions and Cayley numbers in similar ways to obtain maps Y : S7+ S4 and u:S15-+ S8, respectively. Both of these represent generators of infinite cyclic summands. These three maps (7,v, and m) were all discovered by Hopf [l] and are therefore known as the Hopf maps. We will now state some other general theorems of more recent vintage. 1.1.8 Finiteness Theorem (Serre [3]). T n + k ( S n ) is finite f o r k > 0 except when n = 2m, k = 2m - 1, and 7r4,,,-,(S2") = ZO F,, where F, isjinite.

=ekz0

The next theorem concerns the ring structure of rr: 7r:, which is induced by composition as follows. Let a E 7rs and p E 7r; be represented + S"", respectively, where n is large. Then by f : S"" + S" and g : Sn+'+l a@ E T:+, is defined to be the class represented by f g: S"+'+l+ S". It can be shown that pa = (-l)'+'ap,so ~i is an anticommutative graded ring.

5

1. Classical Theorems Old and New

1.1.9 Nilpotence Theorem (Nishida [l]). Each element a E r: for k > 0 is nilpotent, i.e., a’ = O for somefinite t. H For the next result recall that 1.1.8 says n-zi+l+j(S2i+1) is a finite abelian group for all j > 0.

1.1.10 Exponent Theorem (Cohen, Moore, and Neisendorfer [ 11). For p 2 5 the p-component of T ~ ~ + , + ~ ( S has ~ ' +exponent ') p i , i.e., each element in it has order ' p i . H This result is also true for p = 3 (Neisendorfer [ 11) as well, but is known to be false for p = 2. It is also known (Gray [l]) to be the best possible, is known to contain elements of order p i for certain j . i.e., 7r2i+l+jS2i+1 We now describe an interesting subgroup of T:, the image of the HopfWhitehead J-homomorphism, to be defined below. Let SO(n ) be the space of n x n special orthogonal matrices over R with the standard topology. SO( n ) is a subspace of SO(n + 1) and we denote Un,O S O ( n ) by SO, known as the stable orthogonal group. It can be shown that .rri(SO)= r i ( S O (n ) ) if n > i + 1. The following result of Bott is one of the most remarkable in all of topology.

1.1.11 Bott Periodicity Theorem (Bott [l]; see also Milnor [l]).

6

r i ( S O ) = Z/2

if i = - l m o d 4 if i = O o r l m o d 8 otherwise. H

We will now define a homomorphism J: ..,(SO( n ) )+ r,,+’( S " ) . Let a E r i ( S O ( n ) )be the class o f f : S' + S O ( n ) . Let D" be the n-dimensional disc, i.e., the unit ball in R".A matrix in SO(n ) defines a linear homeomorphism of D" to itself. We define,f: S' x D" + D" byf(x, y) =f ( x ) ( y ) ,where X E S', Y E D", and ~ ( x ) S EO ( n ) . Next observe that S" is the quotient of D" obtained by collapsing its boundary Sn+' to a single point, so there is a map p : D" + S" which sends the boundary to the base point. Also observe that Sn+', being homeomorphic to the boundary of D'+' x D", is the union of S' x D" and Dif'x S"-' along their common boundary S' x S"-'. We define f :Sn+' + S" to be the extension of pf: S' x D" + S" to S"+' which sends the rest of S"+' to the base point in S".

1.1.12 Definition. The Hopf- Whitehead J-homomorphism J : .rri(SO(n))+ T , + ~ ( S " )sends the class o f f : S ' + S O ( n ) to the class of f : S n + i + S "as described above. H We leave it to the skeptical reader to verify that the above construction actually gives us a homomorphism.

6

1. Introduction to the Homotopy Groups of Spheres

Note that both m i ( S O ( n ) )and m,,+,(S") are stable, i.e., independent of n, if n > i + l . Hence we have J : 7rk(so)+ 7 ~ : . We will now describe its image. 1.1.13 Theorem (Adams [ l ] and Quillen [ l ] ) . J : 7rk(so)'7~: is a monomorphism for k = 0 or 1 mod 8 and J ( r 4 k - l ( S O ) ) is a cyclic group whose 2-component is Z,2,/(8k) and whose p-componenr for p r 3 is Z,,,/(pk) if ( p - 1)(2kand 0 if ( p - 1 ) $ 2 k , where Z, p ) denotes the integers localized at p. In dimensions 1, 3, and 7 , im J is generated by the Hopf maps (1.1.7) 7, v, and a,respectively. If we denote by xk the generator in dimension 4k - 1 , then 7 X 2 k and T 2 X 2 k are the generators of im J in dimensions 8k and 8 k + 1 , respectively. W The order of J ( T ~ ~ - ~ ( Swas O ) determined ) by Adams up to a factor of two, and he showed that the remaining ambiguity could be resolved by proving the celebrated Adams conjecture, which Quillen and others did. Denote this number by a k . Its first few values are tabulated here. k

1

2

3

4

S

6

7

8

9

10

ak

24

240

504

480

264

65,520

24

16,320

28,728

13,200

The number ak has interesting number theoretic properties. It is the denominator of &/4k, where Bk is the kth Bernoulli number, and it is the greatest common divisor of numbers n r ( n ) ( n 2 k - 1for ) n e Z and t ( n ) sufficiently large. See Adams [ l ] and Milnor and Stasheff [ 5 ] for details. Having determined im J, one would like to know something systematic about coker J, i.e., something more than its structure through a finite range of dimensions. For the reader's amusement we record some of that structure now. 1.1.14 Theorem. In dimensions 5 1 5 , the 2-component of coker J has the following generators, each with order 2 : $En;,

q~

E

&,

v G = v 3 E T gs ,

F€?r:,

V2€&

u2€vS4,

K

and

E nS4,

PET;, TK E

=S5.

(There are relations 7 3= 4 v and q2p = 4x,). For p 2 3 the p-component of coker J has the following generators in dimensions 53pq - 6 (where q = 2p - 2), each with order p : s

PIE rpq-2,

where a1= x(p - l ) , 2 ~ 7

P: E 7 F s p q - 4 , .IPl

~

S

@1PlE T(p+l)q-3

is~ thefirst ~ generator of im J, a d :E

E 77s*p+2)q-3,

S

42p+l,q--5

and

9

P2 E T(Zp+l)q-z,

P:E dpq-6.

2. Methods of Computing n*(S")

7

The proof and the definitions of new elements listed above will be given later in the book, e.g., in Section 4.4.

2. Methods of Computing n*(S*) In this section we will informally discuss three methods of computing homotopy groups of spheres, the spectral sequences of Serre, Adams, and Novikov. A fourth method, the EHP sequence, will be discussed in Section 5. We will not give any proofs and in some cases we will sacrifice precision for conceptual clarity, e.g., in our identification of the &-term of the Adams-Novikov spectral sequence. The Serre spectral sequence (SSS) (circa 1951) (Serre [2]) is included here mainly for historical interest. It was the first systematic method of computing homotopy groups and was a major computational breakthrough. It has been used in the past decade by various authors (Toda [4], Oka [ l]),but computations made with it were greatly clarified by the introduction of the Adams spectral sequence (ASS) in 1958 in Adams [3]. In the ASS the basic mechanism of the SSS is substantially streamlined and the information is organized by homological algebra. For the 2-component of T * ( S " ) the ASS is indispensable to this day, but the odd primary calculations were streamlined by the introduction of the Adams-Novikov spectral sequence (ANSS) in 1967 by Novikov [l]. It is the main subject in this book. Its E,-term contains more information than that of the ASS; i.e., it is a more accurate approximation of stable homotopy and there are fewer differentials in the spectral sequence. Moreover, it has a very rich algebraic structure, as we shall see, largely due to the theorem of Quillen [2], which establishes a deep (and still not satisfactorily explained) connection between complex cobordism (the cohomology theory used to define the ANSS; see below) and the theory of formal group laws. Every major advance in the subject since 1969, especially the work of Jack Morava, has exploited this connection. We will now describe these three methods in more detail. The starting point for Serre's method is the following classical result. 1.2.1 Theorem. Let X be a simply connected space with H i ( X )= 0 for i < n for some positive integer n 2 2. Then ( a ) (Hurewicz [l]). r n ( X )= H , ( X ) . ( b ) (Eilenberg and Mac Lane [2]). There is a space K ( T , n ) characterized up to homotopy equivalence by

8

1. Introduction to the Homotopy Groups of Spheres

If X

is above and T = T , ( X ) then there is a map f:X Hn(f) and ~ ~ (are f isomorphisms. ) H

+ K ( T,n )

such that

Let F be the fiber of the map f above. Then

1.2.2 Corollary.

ri(F ) =

(s'"'

for for

i>n+l is n.

H

In other words, F has the same homotopy groups as X in dimensions above n, so computing T , ( F ) is as good as computing .rr*(X).Moreover, H,(K(.rr, n ) ) is known, so H , ( F ) can be computed with the SSS applied to the fibration F + X + K ( T,n). Once this has been done the entire process can be repeated: let n'> n be the dimension of the first nontrivial homology group of F and let H,,( F) = T’. Then T,,F = .rr,,(X)= d is the next nontrivial homotopy group X. Theorem 1.2.1 applied to F gives a map f ' : F + K ( T’,n ' ) with fiber F', and 1.2.2 says ri(F ' ) =

{ 2(x) forfor

i > n’ isn'

Then one computes H,( F') using the SSS and repeats the process. As long as one can compute the homology of the fiber at each stage, one can compute the next homotopy group of X . In Serre [3] a theory was developed which allows one to ignore torsion of order prime to a fixed prime p throughout the calculation if one is only interested in the p component of T , ( X ) . For example, if X = S3, one uses 1.2.1 to get a map to K ( Z , 3). Then H , ( F ) is described by: 1.2.3 Lemma.

I f F is theJibre ofthe map f : S 3 + K (Z, 3) given by 1.2.1, then

Hi( F )= 1.2.4 Corollary. prime p .

rcm)

if i = 2 m and otherwise.

m>l

The first p-torsion in 7r,(S3) is Z/(p)in T ~ ~ (forS any ~ )

Proofof 1.2.3. (It is so easy we cannot resist giving it.) We have a fibration QK(Z,3)=K(Z,2)+F-+S3 and H * ( K ( Z , 2 ) ) = H * ( C P " ) = Z [ x ] ,where XEH'(CP") and CP" is an infinite-dimensional complex projective space. We will look at the SSS for H*(F) and use the universal coefficient theorem to translate this to the desired description of H , ( F ) . Let u be the generator of H 3 ( S 3 ) .Then in the SSS we must have d 2 ( x )= fu; otherwise F would not be 3-connected, contradicting 1.2.2. Since d, is a derivation we have d2(x")= & n u - ' . It is

9

2. Methods of Computing rr,(S")

easily seen that there can be no more differentials and we get H'(F )=

I:/(m'

which leads to the desired result.

if i = 2 m + l , m > l otherwise,

W

If we start with X = S " the SSS calculations will be much easier for T , , + ~ ( S "for ) k < n - 1. Then all of the computations are in the stable range, i.e., in dimensions less than twice the connectivity of the spaces involved. K, the SSS gives an LES This means that for a fibration F A X 1.2.5 . * . -

f* H,(F)~H,(X)-H,(K)-

d

H,-,(F)-...,

where d corresponds to SSS differentials. Even if we know H,(X), H , ( K ) , and f*,we still have to deal with the SES. 1.2.6

O+cokerf*+ H,(F)+kerf,+O.

It may lead to some ambiguity in H , ( F ) , which must be resolved by some other means. For example, when computing T * ( S " ) for large n one encounters this problem in the 3-component of n,,+,oand the 2-component of T,,+,~.This difficulty is also present in the ASS, where one has the possibility of a nontrivial differential in these dimensions. These differentials were first calculated by Adams [123, Liulevicius [2], and Shimada and Yamanoshita [3] by methods involving secondary cohomology operations and later by Adams and Atiyah [13] by methods involving K-theory (see 1.2.12 and 1.2.14). The Adams spectral sequence of Adams [3] begins with a variation of Serre's method. One works only in the stable range and only on the p-component. Tnstead of mapping X to K ( T , n ) as in 1.2.1, one maps to K =n,,, K ( H J ( X ;Z / ( p ) ) , j )by a certain map g which induces a surjection in mod ( p ) cohomology. Let X , be the fiber of g. Define spaces Xi and K, inductively by K , = nlsO K(H'(X,; Z/( p ) ) ,j ) and X,,, is the fiber of g,: X , + K , (this map is defined in Section 2.1, where the ASS is discussed in more detail). Since H * ( g , ) is onto, the analog of 1.2.5 is an SES in the stable range 1.2.7

O + H*(X,)+ H * ( K , ) + H*(EX,+,)+O,

where all cohomology groups are understood to have coefficients Z / ( p ) . Moreover, H * ( K,) is a free module over the mod ( p ) Steenrod algebra A, so if we splice together the SESs of 1.2.7 we get a free A-resolution of H*( X ) 1.2.8

o+ H * ( X )t H * ( K )+ H * ( C ' K , )t H * ( C 2 K 2 ) + . .

Each of the fibrations X , , ,

+X ,+

'

K , gives an LES of homotopy groups.

10

1. Introduction to the Homotopy Groups of Spheres

Together these LESS form an exact couple and the associated spectral sequence is the ASS for the p-component of m*(X). If X has finite type, the diagram 1.2.9 K - ~ z - ' K ~ + z - ~ K. ~ . +. (which gives 1.2.8 in cohomology) gives a cochain complex of homotopy groups whose cohomology is ExtA(H*(X);Z / ( p ) ) .Hence one gets 1.2.10 Theorem (Adams [3J). There is a spectral sequence converging to the p-component of r , + k ( S " ) for k < n - 1 with E:'=Ext?(Z/(p), Z / ( p ) ) = H " ' ( A ) and 4,: E:' E s r+ r , t + r - 1 . Here the groups Ek' for t - s = k form the associated graded group to a $filtration of the p-component of 7 ~ , , + k ( S " ) . +

Computing this E,-term is hard work, but it is much easier than making similar computations with SSS. The most widely used method today is the spectral sequence of May [ l , 21 (see Section 3.2). This is a trigraded spectral sequence converging to H * * ( A ) , whose E2-term is the cohomology of a filtered form of the Steenrod algebra. This method was used by Tangora [ 1J to compute E:' for p = 2 and t - s 5 70. Most of his table is reproduced here in Figure A3.1. Computations for odd primes can be found in Nakamura [2]. As noted above, the Adams E2-term is the cohomology of the Steenrod algebra. Hence El** = H ' ( A ) is the indecomposables in A. For p = 2 one knows that A is generated by Sq" for i 3 0; the corresponding elements in Els* are denoted by hi E E:92’. For p > 2 the generators are the Bockstein /3 and p p ' for i z 0 and the corresponding elements are a o E E>' and hi E E:@, where q = 2p - 2. For p = 2 these elements figure in the famous Hopf invariant one problem. 1.2.11 Theorem (Adams [ 121).

The following statements are equivalent.

( a ) S2'-' is paraflelizable, Le., it has 2' - 1 globally linearly independent tangent vector fields. ( b ) There is a division algebra (not necessarily associative) over R of dimension 2'. S2' of Hopf invariant one (see 1.5.2). ( c ) There is a map S2.2'-'+ There is a 2-cell complex X = S2' u e2'+'[the cojiber of the map in (d) ( c ) ]in which thegeneratorofH"+'(X)is thesquareof thegeneratorofH2'(X). is a permanent cycle in the ASS. H ( e ) The element hi E

Condition (b) is clearly true for i=O, 1, 2, and 3, the division algebras being the reals R, the complexes C, the quaternions H, and the Cayley numbers, which are nonassociative. The problem for i 2 4 is solved by

2. Methods of Computing m*(Sn)

11

1.2.12 Theorem (Adams [12]). The conditions of 1.2.11 are false for i 2 4 and in the ASS one has d2(h i )= h,h?-, # 0 for i 2 4.

For i = 4 the above gives the first nontrivial differential in the ASS. Its target has dimension 14 and is related to the difficulty in Serre's method referred to above. The analogous results for p > 2 are 1.2.13 Theorem (Liulevicius [2] and Shimada and Yamanoshita [3]). foilowing are equiualent.

The

( a ) There is a map SzP"l-'+ gZp'with Hopf invariant one (see 1.5.3 for the dejnition of the Hopf invariant and the space 9"'). ( b ) There is a p-cell complex X = S2p'v e4p'v e6p'v ' v ezpi+'[the cojber of the map in ( a ) ]whose mod ( p ) cohomology is a truncatedpolynomial algebra on one generator. (c) The element hi E E:,qpiis a permanent cycle in the ASS. The element h, is the first element in the ASS above dimension zero so it is a permanent cycle. The corresponding map in (a) suspends to the element of T ~ ~ (given S ~ by ) 1.2.4. For i -> 1 we have 1.2.14 Theorem (Liulevicius [2] and Shimada and Yamanoshita [3]). The conditions of 1.2.13 are false for i z 1 and d,(hi)= aobi-,, where biPl is a generator of E:,w' (see Section 5.2).

For i = 1 the above gives the first nontrivial differential in the ASS for p > 2. For p = 3 its target is in dimension 10 and was referred to above in our discussion of Serr's method. Figure 1.2.15 shows the ASS for p = 3 through dimension 45. We present it here mainly for comparison with a similar figure (1.2.19) for the ANSS. E;' is a Z / ( p ) vector space in which each basis element is indicated by a small circle. Fortunately in this range there are just two bigradings [(5,28) and (8,43)] in which there is more than one basis element. The vertical coordinate is s, the cohomological degree, and the horizontal coordinate is t - s, the topological dimension. A d, is indicated by a line which goes up by r and to the left by 1. The vertical lines represent multiplication by a. E E;,' and the vertical arrow in dimension zero indicates that all powers of a, are nonzero. This multiplication corresponds to multiplication by p in the corresponding homotopy group. Thus from the figure one can read off n 0 = Z , r I l = T ~ ~ = Z / (~~~)~, = Z / ( 9 ) 0 2 / (and 3 ) , ~ ~ ~ = 2 / ( 2Lines 7). that go up 1 and to the right by 3 indicate multiplication by hoe E:94,while those that go to the right by 7 indicate the Massey product ( h o ,ho,-) (see A1.4.1). The elements a, and hi for i=O, 1, 2 were defined above and the elements bo€ E$'*, koe E:,'*, and b, E E$36 are up to sign the Massey

11 10 9 8 7

!

6

5

4 3

2

--

1

n

I

n

5

I

I

15

20

1

1

1

I

I

25

3n

35

40

L5

I

10

t -

Figure 1.2.15

s-

The Adam spectra sequence for p = 3, r - s 4 45.

1

n

2. Methods of Computing m*(S”)

13

products (h,,, ha, ho), ( h a ,h , , h,), and ( h , , h , , hl), respectively. The unlabeled elements in E F ‘ - ’ for i 2 2 (and hog E Y ) are related to each other by the Massey product ( h a ,ao,-). This accounts for all of the generators except those in Ei*26,Ei*45,and E:.”, which are too complicated to describe here. We suggest that the reader take a colored pencil and mark all of the elements which survive to Em, i.e., those which are not the source or target of a differential. There are in this range 30 differentials which eliminate about two-thirds of the elements shown. Now we consider the spectral sequence of Adams and Novikov (ANSS), which is the main object of interest in this book. Before describing its construction we review the main ideas behind the ASS. They are the following.

1.2.16 Procedure. (i) Use mod (p)-cohomology as a tool to study the p component of r * ( X ) . (ii) Map X to an appropriate Eilenberg-Mac Lane space K , whose homotopy groups are known. (iii) Use knowledge of H * ( K ) , i.e., of the Steenrod algebra, to get at the3ber of the map in (ii). (iv) Iterate the above and codify all information in an SS as in 1.2.10. H An analogous set of ideas lies behind the ANSS, with mod ( p ) cohomology being replaced by complex cobordism theory. To elaborate, we first remark that “cohomology” in 1.2.16(i) can be replaced by “homology” and 1.2.10 can be reformulated accordingly; the details of this reformulation need not be discussed here. Recall that singular homology is based on the singular chain complex, which is generated by maps of simplices into the space X . Cycles in the chain complex are linear combinations of such maps that fit together in an appropriate way. Hence H , ( X ) can be thought of as the group of equivalence classes of maps of certain kinds of simplicia1 complexes, sometimes called “geometric cycles,” into X . Our point of departure is to replace these geometric cycles by closed complex manifolds. Here we mean “complex” in a very weak sense; the manifold M must be smooth and come equipped with a complex linear structure on its stable normal bundle, i.e., the normal bundle of some embedding of M into a Euclidean space of even codimension. The manifold M need not be analytic or have a complex structure on its tangent bundle, and it may be odd-dimensional. The appropriate equivalence relation among maps of such manifolds into X is the following. 1.2.17 Definition. Maps J :M i+ X ( i = 1,2) of n-dimensional complex ( i n the above sense) manifolds into X are cobordant i f there is a map g : W + X where W is a complex manifold with boundary a W = M , u M2 such that

14

1. Introduction to the Homotopy Groups of Spheres

g / M i=A. ( T o be correct we should require the restriction to M 2 to respect the complex structure on M2 opposite to the given one, but we can ignore such details here.) W

One can then define a graded group M U , ( X ) , the complex bordism of X , analogous to H , ( X ) . It satisfies all of the Eilenberg-Steenrod axioms except the dimension axiom, i.e., MU,(pt.), is not concentrated in dimension zero. It is by definition the set of equivalence classes of closed complex manifolds under the relation of 1.2.17 with X = pt., i.e., without any condition on the maps. This set is a ring under disjoint union and Cartesian product and is called the complex cobordism ring. Remarkably, its stiucture is known, as are the analogous rings for several other types of manifolds; see Stong [ l ] . 1.2.18 Theorem (Thom [ 11, Milnor [4], Novikov [2]). The complex cobordism ring, MU,(pt.), is Z [ x , , x 2 , . . .] where dim x, =2i. W

Now recall 1.2.16. We have described an analog of (i), i.e., a functor MU,(-) replacing HJ-). Now we need to modify (ii) accordingly, e.g., to define analogs of the Eilenberg-Mac Lane spaces. These spaces (or rather the corresponding spectrum MU) are described in Section 4.1. Here we merely remark that Thom's contribution to 1.2.18 was to equate MUi(pt.) with the homotopy groups of certain spaces and that these spaces are the ones we need. To carry out the analog of 1.2.16(iii) we need to know the complex bordism of these spaces, which is also described (stably) in Section 4.1. The resulting spectral sequence is formally introduced in Section 4.4, using constructions given in Section 2.2. We will not state the analog of 1.2.10 here as it would be too much trouble to develop the necessary notation. However we will give a figure analogous to 1.2.15. The notation of Fig. 1.2.19 is similar to that of Fig. 1.2.15 with some minor differences. The &-term here is not a Z/(3)-vector space. Elements of order > 3 occur in E:.' (an infinite cyclic group indicated by a square), and in Ei*'2' and E:348,in which a generator of order 3k+' is indicated by a small circle with k parentheses to the right. The names a,, PI, and PSI, will be explained in the next section. The names a3, refer to elements of order 3 in, rather than generators of, In E:*48the product a,P3 is divisible by 3. One sees from these two figures that the ANSS has far fewer differentials than the ASS. The first nontrivial Adams-Novikov differential originates in dimension 34 and leads to the relation a& in r*(So). It was first established by Toda [2,3].

I

I

a ,

rn I

3 1

c? 1

CJ

1 ,

A

I I

I

I

m

1

,

1

I

r

m

I

,I

r

I

I

*

3

,

I

c

16

I . Introduction to the Homotopy Groups of Spheres

3. The Adams-Novikov &-term, Formal Group Laws, and the Greek Letter Construction In this section we will describe the &-term of the ANSS introduced at the end of the previous section. We begin by defining formal group laws (1.3.1) and describing their connection with complex cobordism (1.3.4). Then we characterize the E2-term in terms of them (1.3.5 and 1.3.6). Next we describe the Greek letter construction, an algebraic method for producing periodic families of elements in the E2-term.We conclude by commenting on the problem of representing these elements in .rr*(S). Suppose T is a one-dimensional commutative analytic Lie group and we have a local coordinate system in which the identity element is the origin. Then the group operation T x T + T can be described locally as a real-valued analytic function of two variables. Let F ( x , y ) E R [ [ x ,y ] ]be the power series expansion of this function about the origin. Since 0 is the identity element we have F ( x , 0 ) = F ( 0 , x) = x. Commutativity and associativity give F ( x , y ) = F ( y , x ) and F ( F ( x , y ) , z) = F ( x , F ( y , z)), respectively.

1.3.1 Definition. A formal group law ( F G L ) ouer a Commutative ring with unit R is a power series F ( x , y ) E R [ [ x ,y ] ] satisfying the three conditions W above. Several remarks are in order. First, the power series in the Lie group will have a positive radius of convergence, but there is no convergence condition in the definition above. Second, there is no need to require the existence of an inverse because it exists automatically. It is a power series i ( x )E R [ [ x ] ] satisfying F ( x , i ( x ) )= 0; it is an easy exercise to solve this equation for i ( x ) given F. Third, a rigorous self-contained treatment of the theory of FGLs is given in Appendix 2. Note that F ( x , 0) = F ( 0 , x ) = x implies that F = x + y mod ( x , y)' and that x + y is therefore the simplest example of an FGL; it is called the additive FGL and is denoted by Fa. Another easy example is the multiplicative FGL, F,,, = x + y + rxy for r E R. These two are known to be the only FGLs which are polynomials. Other examples are given in A2.1.4. To see what FGLs have to do with complex cobordism and the ANSS, consider MU*(CP"), the complex cobordism of infinite-dimensional complex projective space. Here MU*(-) is the cohomology theory dual to the homology theory MU,(-) (complex bordism) described in Section 2. Like ordinary cohomology it has a cup product and we have 1.3.2 Theorem. Thereisan elementx E MU2(CPa')such that M U * ( C P w )= Mu*(pt.)[[x]] and M u * ( c P " x c ~ " ) =MU(pt.)[[xOl, ~ O I ] ] .

3. The Adams-Novikov E,-term, Formal Group Laws

17

Here MU*(pt.) is the complex cobordism of a point; it differs from MU,(pt.) (described in 1.2.18) only in that its generators are negatively graded. The generator x is closely related to the usual generator of H 2 ( C P m ) , which we also denote by x. The alert reader may have expected M U * ( C P m ) to be a polynomial rather than a power series ring since H * ( C P m ) is traditionally described as Z[x]. However, the latter is really Z[[x]] since the cohomology of an infinite complex maps onto the inverse limit of the cohomologies of its finite skeleta. [ M U * ( C P " )like , H * ( C P " ) ,is a truncated polynomial ring.] Since one usually considers only homogeneous elements in H*(CP"), the distinction between Z[x] and Z[[x]] is meaningless. However, one can have homogeneous infinite sums in M U * ( C P m )since the coefficient ring is negatively graded. Now CP" is the classifying space for complex line bundles and there is a map y : CP" x C P m + CP" corresponding to the tensor product; in fact, CP" is known to be a topological abelian group. By 1.3.2 the induced map y * in complex cobordism is determined by its behavior on the generator x E M U 2 ( C P m )and one easily proves, using elementary facts about line bundles,

1.3.3 Proposition. For the tensor product map f : CP" x C P" -+ C P", p * ( x ) = F u ( x O l , 1 0 ~MU*(pt.)[[x@l, ) ~ 1 0 x 1 1 is an FGL over MU*(pt.). A similar statement is true of ordinary cohomology and the FGL one gets is the additive one; this is a restatement of the fact that the first Chern class of a tensor product of complex line bundles is the sum of the first Chern classes of the factors. One can play the same game with complex K-theory and get a multiplicative FGL. CP" is a good test space for both complex cobordism and K-theory. One can analyze the algebra of operations in both theories by studying their behavior in CP" (see Adams [5]) in the same way that Milnor [2] analyzed the mod (2) Steenrod algebra by studying its action on H*(RPm;Z/(2)). (See also Steenrod and Epstein [l].) The FGL of 1.3.3 is not as simple as the ones for ordinary cohomology or K-theory; it is complicated enough to have the following universal property. 1.3.4 Theorem (Quillen [2]). For any FGL F over any commutative ring with unit R there is a unique ring homomorphism 0: MU*(pt.) + R such that F(x, Y ) = OFU(X,y ) . We remark that the existence of such a universal FGL is a triviality. Simply write F(x, y ) = 2 ai,jxiyiand let L = Z[ai,j]/I, where I is the ideal generated by the relations among the imposed by the definition 1.3.1 of

18

1. Introduction to the Homotopy Group of Spheres

an FGL. Then there is an obvious FGL over L having the universal property. Determining the explicit structure of L is much harder and was first done by Lazard [ 11. Quillen's proof of 1.3.4 consisted of showing that Lazard's universal FGL is isomorphic to the one given by 1.3.3. Once Quillen's theorem 1.3.4 is proved, the manifolds used to define complex bordism theory become irrelevant, however pleasant they may be. All of the applications we will consider follow from purely algebraic properties of FGLs. This leads one to suspect that the spectrum M U can be constructed somehow using FGL theory and without using complex manifolds or vector bundles. Perhaps the corresponding infinite loop space is the classifying space for some category defined in terms of FGLs. Infinite loop space theorists, where are you? We are now just one step away from a description of the ANSS E,-term. Let G = { f ( x )E Z [ [ x ] ] l f ( x = ) x mod (x)'}. Here G is a group under composition and acts on the Lazard/complex cobordism ring L = MU,(pt.) as follows. For g E G define an FGL F,over L by F,(x, y ) = g - ' F , ( g ( x ) , g ( y ) ) . By 1.3.4F, is induced by a homomorphism 8,: L + L. Since g is invertible under composition, 8, is an automorphism and we have a G-action on L. Note that g ( x ) defines an isomorphism between F and Fg. In general, isomorphisms between FGLs are induced by power series g ( x ) with leading term a unit multiple (not necessarily one) of x. An isomorphism induced by a g in G is said to be strict. 1.3.5 Theorem. to H * * ( G ; L ) .

The E2-terrn ofthe ANSS converging to .?r*(S)is isomorphic

There is a difficulty with this statement: since G does not preserve the grading on L, there is no obvious bigrading on H*(G; L). We need to reformulate in terms of L as a comodule over a certain H'opf algebra B defined as follows. Let g E G be written as g ( x ) =Z,20 b,x'+' with bo= 1. Each b, for i > 0 can be thought of as a Z-valued function on G and they generate a graded algebra of such functions B = Z [ b , , b 2 , . . .] with dim b, =2i. (Do not confuse this ring with L, to which it happens to be isomorphic.) The group structure on G corresponds to a coproduct A :B + BOB on B given by A ( ~ I ) = X b, '~+~' O b , , where b =Z,20 b, and bo= 1 as before. To see this suppose g ( x ) = g"'(g'2'(x)) with g ' k ' ( x )= Z bjk)x'+'.Then we have 1 b,x'+' = 1 bl"(1 bj2)xJc"')'+' from which the formula for A follows. This coproduct makes B into a graded connected Hopf algebra over which L is a graded comodule. We can restate 1.3.5 as

19

3. The AdarnsNovikov &-term, Formal Group Laws

1.3.6 Theorem. The E,-term of the ANSS converging to .rr,(S) is given by E ; ' = Extg'(Z, L ) . The definition of this Ext is given in A1.2.3; all of the relevant homological algebra is discussed in Appendix 1. Do not be alarmed if the explicit action of G (or coaction of B ) on L is not obvious to you. It is hard to get at directly and computing its cohomology is a very devious business. Next we will describe the Greek letter construction, which is a method for producing lots (but by no means all) of elements in the E,-term, including the a,'s and p,'s seen in 1.2.19. We will use the language suggested by 1.3.5; the interested reader can translate our statements into that of 1.3.6. Our philosophy here is that group cohomology in positive degrees is too hard to comprehend, but H o ( G ; M ) (the G-module M will vary in the discussion), the submodule of M fixed by G, is relatively straightforward. Hence our starting point is

1.3.7 Theorem. Ho(G; L ) = Z concentrated in dimension 0. This corresponds to the 0-stem in stable homotopy. Not a very promising beginning you say? It does give us a toehold on the problem. It tells us that the only principal ideals in L which are G-invariant are those generated by integers and suggests the following. Fix a prime number p and consider the SES of G-modules 1.3.8

o+ L

p - L+ L / ( p ) + O .

We have a connecting homomorphism 6o:H '( G ; L / ( p ) ) + H

(G; L).

1.3.9 Theorem. Ho(G ; L( p ) ) = Z/( p ) [ v , ] , where v , E L has dimension q = 2(p-1). 1.3.10 Definition. For t > 0 let a, = 6(, v : )E E:.". It is clear from the LES in cohomology associated with 1.3.8 that a, f 0 for all t > 0, so we have a collection of nontrivial elements in the ANSS E,-term. We will comment below on the problems of constructing corresponding elements in .rr*(S);for now we will simply state the result. 1.3.11 Theorem. ( a ) (Toda[4, IV]) For p > 2 each a , is represented by an element of order p in T ~ , - ~ (which S ) is in the image of the J-homomophism (1.1.18). For p = 2 a , is so represented provided t 93 mod (4). If t = 2 mod (4) (b) then the element has order 4; otherwise it has order 2. It is in im J (1.1.12) i f t is even. H

20

1. Introduction to the Homotopy Groups of Spheres

Theorem 1.3.9 tells us that 1.3.12 o + = / ( p ) ~ L / ( P ) + L / ( P ,v,)+O is an SES of G-modules and there is a connecting homomorphism 8 , : H ' ( G ;L / ( P ,01))'

H ' + ' ( G ;L / ( p ) ) .

The analogs of 1.3.9 and 1.3.10 are 1.3.13 Theorem. 2(p2-1). 1.3.14 Definition.

Ho( G; L / ( p, v , ) )= Z / ( p ) [u2] where v2E L has dimension

For t > 0 let P, = 6081(v;) E E:."p+l)q-q.

More work is required to show that these elements are nontrivial for p > 2, and PI = 0 for p = 2. The situation in homotopy is 1.3.15 Theorem (Smith [l]). For p 2 5 Pi is represented by a nontrivial

element of order p in

T

(

~

+

~

)

(SO). ~ - ~

~

-

~

Your are probably wondering if we can continue in this way and construct y,, S,, etc. The following results allow us to do so. 1.3.16 Theorem (Morava [3], Landweber [4]).

( a ) There are elements v, E L of dimension 2( p" - 1) such that I,, = ( p , ul, v2, . . . , v,,-,) c L is a G-invariant prime ideal for all n > 0. ( 6 ) 0 + X2(P"-lJL/l,,% L/l,,+ L/I,+,+O is an SES of modules with connecting homorphism 6,: H ' ( G ; L/I,,+,)+ H " ' ( G ; LJI,,).

( c ) H o w ;L / k )=Z/(p)[ufll. ( d ) The only G-invariant prime ideals in L are the I,, for 0 < n s ~3 for all primes p. a Part (d) above shows how rigid the G-action on L is; there are frightfully many prime ideals in L, but only the I,, for various primes are G-invariant. Using (b) and (c) we can make 1.3.17 Definition.

For

t,

n > 0 let a:")= So 6,

6 , , - , ( u ~E) E?*.

Here a(’’’stands for the nth letter of the Greek alphabet, the length of which is more than adequate given our current state of knowledge. The only other known result comparable to 1.3.11 or 1.3.15 is 1.3.18 Theorem.

( pnontrivial +2) ( a ) (Miller, Ravenel, and Wilson [ 1 3 ) y, E E : * r q ( p 2 + p + 1 ) - q IS for all t > 0 and p > 2. ( 6 ) (Toda [ 13) For p 2 7 each y, is represented 6y a nontrivial element of orderp in T ~ ~ , ~ ~ + ~ + ~ ~ - ~ ( ~ + ~ ) - ~ ( S ~ ) . '

3. The AdamsNovikov El-term, Formal Group Laws

21

It is known that not all yI exist in homotopy for p = 5 (see 7.5.1). Part (b) above was proved several years before part (a). In the intervening time there was a controversy over the nontriviality of y 1 which was unresolved for over a year, ending in 1974 (see Thomas and Zahler [l]).This unusual state of affairs attracted the attention of the editors of Science [ l ] and the New York Times [l], who erroneously cited it as evidence of the decline of mathematics. We conclude our discussion of the Greek letter construction by commenting briefly on generalized Greek letter elements. Examples are /3313and P3/2 (and the elements in E:** of order >3) in 1.2.19. The elements come via connecting homomorphisms from H o ( G ;L / J ) , where J is a Ginvariant regular (instead of prime) ideal. Recall that a regular ideal (xo, xl, . . . ,x " - ~ c ) L is one in which each xi is not a zero divisor modulo (x?, . . ~, x i P l ) ,Hence . prime ideals are regular as are ideals of the form ( p ' o , u?, . . . , u>:\). Many but not all G-invariant regular ideals have this form. 1.3.19 Definition. P S I , (for appropriate s and t ) is the image of Ho(G; L / ( p , u : ) ) and as,lis the image o f u s e H o ( G ; L / ( p * ) ) .

USE

Hence p a s I I= a,/, = as,and p l I l= PI by definition. Now we will comment on the problem of representing these elements in the E2-term by elements in stable hornotopy, e.g., on the proofs of 1.3.11, 1.3.15, and 1.3.18(b). The first thing we must do is show that the elements produced are actually nontrivial in the E,-term. This has been done only for a's, P ' s , and 7's. For p = 2, PI and y1 are zero but for t > 1 Pt and yt are nontrivial; these results are part of the recent computation of E?* at p = 2 by Shimomura [ l ] , which also tells us which generalized p's are defined and are nontrivial. The corresponding calculation at odd primes was done in Miller, Ravenel, and Wilson [l], as was that of E:,* for all primes. The general strategy for representing Greek letter elements geometrically is to realize the relevant SESs [e.g., 1.3.8, 1.3.12, and 1.3.16(b)] by cofiber sequences of finite spectra. For any connective spectrum X there is an ANSS converging to r * ( X ) . Its E2-term [denoted by E 2 ( X ) ]can be described as in 1.3.5 with L = MU,(So) replaced by M U , ( X ) , which is a G-module. For 1.3.8 we have a cofiber sequence

soP, so+=

V(O),

where V(0) is the mod ( p ) Moore spectrum. It is known (2.3.4) that the LES of homotopy groups is compatible with the LES of &-terms. Hence the elements v : of 1.3.9 live in E:74r(V(0)) and for 1.3.11 (a) [which says a , is represented by an element of order p in T ~ ~ - ~ ( for S O p) > 2 and t > 01

22

1. Introduction to the Homotopy Groups of Spheres

it would suffice to show that these elements are permanent cycles in the ANSS for P*( V(0)) with p > 2. For t = 1 (even if p = 2) one can show this by brute force; one computes E,( V(0)) through dimension q and sees that there is no possible target for a differential coming from u1E E:". Hence v1 is realized by a map

S"+ V(0). If we can extend it to C4V(0),we can iterate and represent all powers of vl. We can try to do this either directly, using obstruction theory, or by showing that V(0) is a ring spectrum. In the latter case our extension a would be the composite

s" A v(0) + v(0) A v(0)

-j

v(o),

where the first map is the original map smashed with the identity on V(0) and the second is the multiplication on V(0). The second method is generally (in similar situations of this sort) easier because it involves obstruction theory in a lower range of dimensions. In the problem at hand both methods work for p > 2 but both fail for p = 2. In that case V(0) is not a ring spectrum and our element in n2(V(0)) has order 4, so it does not extend to X2V(0). Further calculations show that v: and v: both support nontrivial differentials (see 5.3.13) but v: is a permanent cycle represented by map S 8 + V(O), which does extend to X8V(0). Hence iterates of this map produce the homotopy elements listed in 1.3.11(b) once certain calculations have been made in dimensions 5 8 . For p > 2 the map a: Z"V(0) + V(0) gives us a cofibre sequence C"V(0)-5

V(O)+ V(1),

realizing the SES 1.3.12. Hence to arrive at 1.3.15 (which describes the P's in homotopy) we need to show that U ~ EE2 ( P + 1 ) qV(1)) ( is a permanent cycle represented by a map which extends to 0 :E(p+l'qV(1) + V( 1). We can do this for p 2 5 but not for p = 3. Some partial results for P's at p = 3 and p = 2 are described in Section 5.5. The cofiber of the map P (corresponding to u2) for p 2 5 is called V( 2) by Toda [l]. In order to construct the 7's [1.3.18(b)] one needs a map y:X2(~3-1)

V(2)+ V(2)

corresponding to u3. Toda [ 11 produces such a map for p 2 7 but it is known not to exist for p = 5 (see 7.5.1). Toda [l] first considered the problem of constructing the spectra V(n) above, and hence of the representation of Greek letter elements in P * ( S ) , although that terminology (and 1.3.16) was not available at the time. While the results obtained there have not been surpassed, the methods used leave

4. More Formal Group Law Theory, BP-Theory

23

something to be desired. Each positive result is proved by brute force; the relevant obstruction groups are shown to be trivial. This approach can be pushed no further; the obstruction to realizing u, lies in a nontrivial group for all primes (5.6.13). Homotopy theorists have yet to learn how to compute obstructions in such situations. The negative results of Toda [ l ] are proved by ingenious but ad hoc methods. The nonexistence of V( 1) for p = 2 follows easily from the structure of the Steenrod algebra; if it existed its cohomology would contradict the Adem relation Sq2Sq2= Sq’Sq2Sq’.For the nonexistence of V(2) at p = 3 Toda uses a delicate argument involving the nonassociativity of the mod (3) Moore spectrum, which we will not reproduce here. We will give another proof (5.5.1) which uses the multiplicative structure of the ANSS &-term to show that the nonrealizability of p4E E:,6', and hence of V(2), is a formal consequence of that of p3,3E E:,36. This was shown by Toda [2,3] using an extended power construction, which will also not be reproduced here. Indeed, all of the differentials in the ANSS for p = 3 in the range we consider are formal consequences of that one in dimension 34. A variant of the second method used for V(2) at p = 3 works for V(3) (the cofiber of y ) at p = 5.

4. More Formal Group Law Theory, BP-Theory, Morava's Point of View, and the Chromatic Spectral Sequence We begin this section by introducing BP-theory, which is essentially a p-local form of MU-theory. With it many of the explicit calculations behind our results become a lot easier. Most of the current literature on the subject is written in terms of BP rather than MU. On the other hand, B P is not essential for the overall picture of the E,-term we will give later, so it could be regarded as a technicality to be passed over by the casual reader. Next we will describe the classification of FGLs over an algebraically closed field of characteristic p . This is needed for Morava's point of view, which is a useful way of understanding the action of G on L (1.3.5). The insights that come out of this approach are made computationally precise in the chromatic spectral sequence (CSS), which is the pivotal idea in this book. Technically the CSS is a trigraded spectral sequence converging to the ANSS &-term; heuristically it is like a spectrum in the astronomical sense in that it resolves the &-term into various components each having a different type of periodicity. In particular, it incorporates the Greek letter elements of the previous section into a broader scheme which embraces the entire E,-term.

24

1. Introduction to the Homotopy Groups of Spheres

BP-theory began with Brown and Peterson [ 13 (after whom it is named), who showed that after localization at any prime p , the MU spectrum splits into an infinite wedge suspension of identical smaller spectra subsequently called BP. One has 1.4.1

.rr*(BP) = Z(,,Eulv2,. , * .I,

where Z(,, denotes the integers localized at p and the u,'s are the same as the generators appearing in the Morava-Landweber theorem 1.3.16. Since dim L), = 2( p" - l ) , this coefficient ring, which we will denote by BP,, is much smaller than L = T * (M U ) , which has a polynomial generator in every even dimension. Next Quillen [2] observed that there is a good FGL theoretic reason for this splitting. A theorem of Cartier [ l ] (A2.1.18) says that every FGL over a Z, ,,-algebra is canonically isomorphic to one in a particularly convenient form called a p-typical FGL (see A2.1.17 and A2.1.22 for the definition, whose details need not concern us now). This canonical isomorphism is reflected topologically in the above splitting of the localization of MU. This fact is more evidence in support of our belief that MU can somehow be constructed in purely FGL theoretic terms. There is a p-typical analog of Quillen's theorem 1.3.4; i.e., BP*(CP") gives us a p-typical FGL with a similar universal property. Also, there is a BP analog of the ANSS, which is simply the latter tensored with Z, ,,; i.e., its &-term is the p-component of H*(G; L ) and it converges to the p component of T * ( S ) . However, we encounter problems in trying to write an analog of our methaphor 1.3.5 because there is no p-fypical analog of the group G. In other words there is no suitable group of power series over Z, which will send any p-typical FGL into another. Given a p-typical FGL F over Z(,) there is a set of power series g E Z,,,[[x]] such that g - ' F ( g ( x ) , g ( y ) ) is also p-typical, but this set depends on F. Hence Hom(BP,, K ) the set ofp-typical FGLs over a Z, ,,-algebra K , is acted on not by a group analogous to G, but by a groupoid. Recall that a groupoid is a small category in which every morphism is an equivalence, i.e., invertible. A groupoid with a single object is a group. In our case the objects are p-typical FGLs over K and the morphisms are isomorphisms induced by power series g ( x ) with leading term x. Now a Hopf algebra, such as B in 1.3.6, is a cogroup object in the category of commutative rings R, which is to say that Hom( B, R ) = GR is a groupvalued functor. In fact GR is the group (under composition) of power series f ( x ) over R with leading term x. For a p-typical analog of 1.3.6 we need to replace B by cogroupoid object in the category of commutative Z(,)algebras K . Such an object is called a Hopf algebroid (Al.l.1) and consists

,,

4. More Formal Group

Law Theory, BP-Theory

25

of a pair (A, r) of commutative rings with appropriate structure maps so that Hom(A, X ) and Hom(T, K ) are the sets of objects and morphisms, respectively, of a groupoid. The groupoid we have in mind, of course, is that of p-typical FGLs and isomorphisms as above. Hence BP, is the appropriate choice for A ; the choice for r turns out to be B P J B P ) , the BP-homology of the spectrum BP. Hence the p-typical analog of 1.3.6 is 1.4.2 Theorem. The p-component of the E,-term of the ANSS converging to n,(S) is

Ex~BP*(BP)(BP*, BP*). Again this Ext is defined in A1.2.3 and the relevant homological algebra is discussed in Appendix 1. We will now describe the classification of FGLs over an algebraically closed field of characteristic p. First we define power series [ m I F (x ) associated with an FGL F and natural numbers m. We have [ 0 l F ( x )= 0, [ 1 I F ( x )= x,and [ m ] F ( x )= F ( x , [ m - 1 I F ( x ) ) An . easy lemma (A2.1.6) says that if F is defined over a field of characteristic p, then [ p ] F ( X )is in fact a power series over xp" with leading term axpn,a # 0, for some n > 0, provided F is not isomorphic to the additive FGL, in which case [ p ] F ( x )= 0. This integer n is called the height of F, and the height of the additive FGL is defined to be CO. Then we have 1.4.3 Classification Theorem (Lazard [2]). ( a ) Two FGLs dejned over the algebraic closure of F are isomorphic iff they have the same height. ( b ) If F is nonadditive, its height is the smallest n such that O( v , ) # 0, where 0 : L+ K is the homomorphism of 1.3.4 and v,, E L is as in 1.3.16.

Now we come to Morava's point of view. Let K = pp, the algebraic closure of the field with p elements, and let GKc K [ [ x ] ] be the group (under composition) of power series with leading term x. We have seen that GK acts on Hom(L, K ) , the set FGLs defined over K. Since L is a polynomial ring, we can think of Hom( L, K ) as an infinite-dimensional vector space V over K ; a set of polynomial generators of L gives a basis of V. For a vector V E V, let F, be the corresponding FGL. Two vectors in V are in the same orbit iff the corresponding FGLs are strictly isomorphic (strict isomorphism was defined just prior to 1.3.5), and the stabilizer group of V E V (i.e., the subgroup of GK leaving V fixed) is the strict automorphism group of F,. This group S, (where n is the height) can be described explicitly (A2.2.17); it is a profinite group of units in a certain p-adic division algebra, but the details need not concern us here. Theorem 1.4.3 enables us to describe the orbits explicitly.

26

1 . Introduction to the Homotopy Croup of Spheres

1.4.4 Theorem. There is one GK-orbit of V for each height as in 1.4.3. The height n orbit V,, is the subset defined by v, = 0 for i < n and v, # 0. H

Now observe that V is the set of closed points in Spec( L O K ) , and V,, is the set of closed points in Spec(L,OK), where L, = v;'L/I,,. Here V,, is a homogeneous GK-spaceand a standard change of rings argument gives 1.4.5 Change-of-Rings Theorem. H*(GK; L , O K ) = H * ( S , ; K ) .

H

We will see in Chapter 6 that a form of this isomorphism holds over F, as well as over K. In it the right-hand term is the cohomology of a certain Hopf algebra [called the nth Morava stabilizer algebra X( n)] defined over F,, which, when tensored with F,", becomes isomorphic to the dual of Fpn[Sn],the F,n-group algebra of S,. Now we are ready to describe the central construction of this book, the chromatic spectra sequence, which enables us to use the results above to get more explicit information about the ANSS &-term. We start with a LES of G-modules, called the chromatic resolution 1.4.6

0 - LOZ,,,+ M o + M ' + *

* *

defined as follows. M o = L O Q , and N' is the cokernel in the SES

O + LOZ(,,+ Mo-+ N ' + 0 . M" and N" are defined inductively for n > 0 by SESs 1.4.7

0 + N " + M " + N"+' + 0,

where M"=v;'N".Hencewehave N ' = L O Q / Z , , , = @ t / ( p ' ) = L/(p") and N"+' =lim L/(p', v i , . . . , u : ) = L/(p", v y , . . . , v:). The fact that these are SESs o f z m o d u l e s is nontrivial. The LES 1.4.6 is obtained by splicing together the SESs 1.4.7. In Chapter 5, where the CSS is described in detail, M " and N" denote the corresponding objects defined in terms of BP,. In what follows here Ext,(Z, M) will be abbreviated by Ext(M) for a Bmodule (e.g., G-module) M. Standard homological algebra (A1.3.2) gives There is a spectral sequence converging to Ext( L@Z(,)) with E:" = ExfS(Mn),d,: E:"+ E:+r*s-r+', and d,:Ext(M")+Ext(M"+') being induced by the maps M"-+ M"" in 1.4.6. [ E Z is a subquotient of H Extnts(LOZ(,,).] 1.4.8 Proposition.

This is the chromatic spectral sequence. We can use 1.4.5 to get at its El term as follows. Define G-modules M : for 01 i In by M," = M", and M : is the kernel in the SES 1.4.9

o-+M Y + MY-'_,';".

where vo=p. This gives M : = L,

M;-'+o,

= u;'L/In,

so the F,-analog of 1.4.5

27

4. More Formal Group Law Theory, BP-Theory

describes Ext(Mz) in terms of the cohomology of the stabilizer group S,,. Equation 1.4.9 gives an LES of Ext groups of a Bockstein spectral sequence (BSS) computing Ext(My-,) in terms of Ext(M:). Hence in principle we can get from H * ( S , ) to Ext( M " ) , although the BSSs are difficult to handle in practice. Certain general facts about H * ( S,,) are worth mentioning here. If ( p - 1) divides n then this cohomology is periodic (6.2.10); i.e., there is an element c E H * ( S , ;FP) such that H * ( S , ;FP) is a finitely generated free module over Fp[c]. In this case S,, has a cyclic subgroup of order p to whose cohomology c restricts nontrivially. This cohomology can be used to detect elements in the ANSS E,-term of high cohomological degree, e.g., to prove 1.4.10 Theorem. For p > 2 , all monomials in the trivial.

p p ~ l (1.3.19) p~

are non-

If n is not divisible by p - 1 then S,, has cohomological dimension n2; i.e., H ' ( S , ) = 0 if i > n 2 , and H*(S,,) has a certain type of Poincart duality (6.2.10). It is essentially the cohomology of a certain n-stage nilpotent Lie algebra (6.3.5), at least for n < p - 1. The cohomological dimension implies 1.4.11 Morava Vanishing Theorem. If ( p - 1 ) j n, then in the CSS (1.4.8) E?" = 0 for s > n2.

It is also known (6.3.6) that every sufficiently small open subgroup of S, has the same cohomology as a free abelian group of rank n2. This fact can be used to get information about the ANSS &-term for certain Thom spectra (6.5.6). Now we will explain how the Greek letter elements of 1.3.17 and 1.3.18 appear in the CSS. If J is a G-invariant regular ideal with n generators then L/ J is a sub[e.g., the invariant prime ideal 1, = ( p , 0 , , . . . , q-,)], module of N" and M", so Exto(L / J ) c Exto(N")c Exto(M")= E?'. Recall that the Greek letter elements are images of elements in Ext'(J) under the appropriate composition of connecting homomorphisms. This composition corresponds to the edge homomorphism E?'+ En,’ in the CSS. [Note that every element in the chromatic E:' is a permanent cycle; i.e., it supports no nontrivial differential although it may be the target of one. Elements in E?' coming from Ext(L/J) lift to Ext(N") are therefore in ker d , and live in I!?;’ The .] module N " is the union of the L/ J over all possible invariant regular ideals J with n generators, so Ext'(N") contains all possible nth Greek letter elements. To be more specific about the particular elements discussed in Section 3 we must introduce chromatic notation for elements in N" and M". Such "-1 elements will be written as fractions x / y with x E L and y = p'ov't . . - u'"-' with all exponents positive, which stands for the image of y in L/ J G N"

28

1. Introduction to the Homotopy Groups of Spheres

where J = ( p b , u ) , . . . , u > z ; ) . Hence x / y is annihilated by J and depends only on the mod J reduction of x. The usual rules of addition, subtraction, and cancellation of fractions apply here.

1.4.12 Proposition. Up to sign the elements a!") (1.3.17), a.v/,and psi, (1.3.19) are represented in the CSS by u ~ / p u., . * u,-' E E,".', u ; / p ' E E:.', and uq/pu: E E:,', respectiuely. The signs here are a little tricky and come from the double complex used to prove 1.4.8 (see 5.1.18). The result suggests elements of a more complicated nature; e.g., ps/i2,,,stands for uq/pilu?, with the convention that if i, = 1 it is omitted from the notation. The first such element with il > 1 is p p 2 1 p , 2 . We also remark that some of these elements require correcting terms ~ ) not / ~ ~ : / 2 ~is)in Exto(N1)and in their numerators; e.g., ( u ~ + ~ u , u(but represents a4,4,which corresponds to the generator U E .rr,(So). We will describe E :* for n 5 1 at p > 2. For all primes E F = Q (concentrated in dimension 0) and E:" = 0 for s > 0. For p > 2, E:.' = 0 for s > 1 and E:*'=Q/Z(,, concentrated in dimension 0, and E:.' is trivial in dimension not divisible by q = 2( p - 1) = dim ul and is generated by all elements of the form u : / p t for t E Z. Hence if p' is the largest power of p dividing t, then E:.O.= Z/( pi+') in dimension qf, and in dimension 0, E:*O=

Q/z(

p)

The differential d , : EY+E:.O is the usual map Q + Q / Z ( , , . Its kernel Z ( p )is Exto(LOZ(,,). On E;.' = Q/Z(,, the kernel of d , is trivial, so E:.' = E$2= 0 and Ext2(LOZ, ,,) = E:.'. On E the kernel of d , consists of all elements in nonnegative dimensions. Since the Q/Z(,, in dimension 0 is hit by d , , E;*O consists of the positive dimensional elements in l 3 t . O and this group is Ext'(LOZ(,,). In .rr*(So)it is represented by the p-component of im J. Now the chromatic El-term is periodic in the following sense. M " = lim u;'L/J, where the limit is over all invariant regular ideals J with n generators. For each J, Ext0u,'L/J contains some power of u,, say u,.k Then Ext(u;'L/J) is a module over Z ( , , [ u i , u i k ] , i.e., multiplication by u l is an isomorphism, so we say that this Ext is u,-periodic. Hence E : * = Ext(M") is a direct limit of such groups. We may say that an element in the ANSS E2-term is u,-periodic if it represents an element in E2* of the CSS. Hence the CSS &-term is the trigraded group associated with the filtration of Ext( LOZ, ,) by u,-periodicity. This filtration is decreasing and has an infinite number of stages in each cohomological degree. One sees this from the diagram

-

Ext"( N o )t Ext"-'( N ' ) +. . .Exto(N " ) where N o= LO Z, ), ; the filtration of Ext( N o ) is by images of the groups

29

5. Unstable Homotopy Groups and the EHP Spectral Sequence

Ext(Nn). This local finiteness allows us to define an increasing filtration on Ext( N o ) by F, Ext"( N " ) = im Exti(N s - i ) for 0 5 i I:s, and Fo Ext( N ) is the subgroup of Greek letter elements in the most general possible sense.

5. Unstable Homotopy Groups and the EHP Spectral Sequence In this section we will describe the EHP sequence, which is an inductive method for computing n , , + k ( S n ) beginning with our knowledge of T*( S ' ) (1.1.7). We will explain how the Adams vector field theorem, the Kervaire invariant problem, and the Segal conjecture are related to the unstable homotopy groups of spheres. We will not present proofs here or elsewhere in the book, nor will we pursue the topic further except in Section 3.3. We are including this survey here because no comparable exposition exists in the literature and we believe these results should be understood by more than a handful of experts. In particular, this section could serve as an introduction to Mahowald [4]. The EHP sequences are the LESS of homotopy groups associated with certain fibration constructed by James [ l ] and Toda [ 5 ] . There is a different set of fibrations for each prime p. All spaces and groups are assumed localized at the prime in question. We start with p = 2. There we have a fibration

Sn

1.5.1

~

n s n + ln~ ~ 2 n + l

(where the spaces are localized at 2 if n is even), which gives the LES 1.5.2

-

E

' ' '+

H

nn+k(s")-

Tn+k+l(S7n+')-

'rrn+k+l(S"+')

P

nn+k-l(S")+.

.

'

where the groups are localized at 2 if n is even. Here E stands for Einhangung (suspension), H for Hopf invariant, and P for Whitehead product. If n is odd the fibration is valid for all primes and it splits at odd primes, so for p > 2 we have nzm+k(S2m)

= n2m+k-l(Szrn-I)@

T~M+~(S~~-').

This means that even-dimensional spheres at odd primes are uninteresting. Instead one considers the fibration 1S.3

22,

~

fiSZm+l+

fis2pm+l,

30

1. Introduction to the Homotopy Groups of Spheres

where the second map is surjective in H,( ; Z,,,), and g2"'is the (2mp - 1)-skeleton o f fiS2"'+',which is a CW-complex with p - 1 cells of the form Sz" v e4mu . u e2(p - l ' m . The corresponding LES is 1

-

1.5.4

'

'+T2m+k(SZm)

E

r2m+k+[(S2mf')

H _.)

There is also a fibration

which gives

1.5.4 and 1.5.6 are the EHP sequences for odd primes. Note that for p = 2, = S2" and both sequences coincide with Eq. 1.5.2. For each prime these LESS fit together into an exact couple (2.1.6) and

$m

we can study the associated spectral sequence, namely

1.5.7 Proposition. ( a ) For p = 2 there is a spectral sequence converging to T: (stable homotopy) with E!" = T k + n ( S 2 n - 1 ) and d,: E?" + E f - 1 7 n.- rE2k is the sub quotient im .ir,+k(S")/im??,,+k-I( S - ' ) of T;. There is a sirnifar spectral sequence converging to r*(S J ) with EF" as above for n 5 j and EF" = 0 for

n>j.

( b ) For p > 2 there are similar spectral sequences with E Y m + ' = ( S Z p m + l )and E?" = Tk+zm( SZp"-l). The spe5tral sequence with E Fn = 0 for n >j converges to r * ( S J )ifi is odd and to n-*( S') iJj is even. rk+2m+1

This is the EHP spectral sequence (EHPSS). We will explain below how it can be used to compute ? r n + k ( s " ) [or T , , + k ( S nif) n is even and p is odd] by double induction on n and k. First we make some easy general observations. 1.5.8 Proposition.

(a) (b) (c) (d)

F o r a l l p r i m e s E : l = r l + k ( S 1 ) whichisz,,, , f o r k = O a n d 0 fork>O. Forp=2, EF"=O f o r k < n - 1 . Forp=2, EFn=~.rrSk-,+,fork2, E F 2 m + ' = 0f o r k < g m a n d E F 2 " = 0 f o r k c q m - 1 , where

q = 2 ( p - 1).

( e ) F o r p > 2 , E Y m + ' = 7'rk-qm for k < q( p m + rn + 1) - 2, and E F2m = for k < q ( p m + m ) - 3 .

S ' Tk+l-qm

5. Unstable Homotopy Groups and the EHP Spectral Sequence

31

Part (b) follows from the connectivity of the ( 2 n - 1)-sphere, while (c) and (d) follow from the fact that 7r2m-l+k(S2m-' ) = r; for k < qm - 2 , which is in turn a consequence of 1.5.7. We will refer to the region where n - 1 5 k and EFn is a stable stem as the stable zone. Now we will describe the inductive aspect of the EHPSS. Assume for the moment that we know how to compute differentials and solve the group extension problems. Also assume inductively that we have computed E? for all ( i , j ) with i < k and all ( k , j ) for j > n . For p = 2 we have E:"= rn+k(S2"-'). This group is in the (k-nS1)-stem. If n = 1, this group is r l + k ( S ' ) , which is known, so assume n > 1. If n = 2 this group is r 2 + k ( S 3 ) , which is 0 for k = 0,Z for k = 1 , and for k > 1 is the middle term in the SES 0 + Et-132 + T k + 2 ( S3)+ ker d, c Et-1*3 -+ 0. Note that E,k-1,2is the cokernel of the d , coming from EF3 and is therefore known by induction. Finally, if n > 2 , EF"= r , , + k ( S Z n - ' ) can be read off from the already computed portion of the EHPSS as follows. As in 1.5.7 one obtains an SS for T * ( S ~ ~ -by' )truncating the EHPSS, i.e., by setting all E;" = 0 for m > 2n - 1 . The group r n + k ( S 2 " - ' ) lies in a stem which is already known, so we have EF". Similar remarks apply to odd primes. We will illustrate the method in detail forp = 2 by describing what happens for 0 5 k 5 7 in Fig. 1.5.9. By 1.5.8(c) we have EFk+' = ri = Z. Let x k denote the standard generator of this group. We will see below (1.5.13) that d,( xk) = 2xk-1 for even positive k and dl(Xk)= 0 otherwise. Hence E l s 2= E k 2 = r r s = Z / ( 2 ) , so E F k = Z / ( 2 ) for all k r 2 . We denote the generator of each of these groups by 1 to indicate that, if the generator is a permanent cycle, it corresponds to an element whose Hopf invariant suspends to the element corresponding to xl.Now the first such generator, that of E:*2,is not hit by a differential, so we have EFk-'= ~ T ~ ~ - = ~ 2/(2) ( S ~for~ all - ~ ) k 2 3 . We denote these generators by 11, to indicate that their Hopf invariants each desuspend to elements with Hopf invariant x,. In general we can specify an element of a E ? r n + k ( s n ) by a sequence of integers adding up to k as follows. Desuspend (Y as far as possible, say to Sm+'.The first integer is then m (necessarily s k ) and the desuspended a has a Hopf invariant p E r , + l + k ( SZm+').To get the second integer we desuspend p, and so forth. After a finite number of steps we get an element with Hopf invariant in the zero stem and stop the process. Of course there is some indeterminacy in desuspending but we can ignore it for now. We call this sequence of integers the serial number of a. In Fig. 1.5.9 we indicate each element of EFn = ? T , + k ( S Z n - ' ) by its serial number. In almost all cases if p a # 0, its serial number differs from that of a itself. To get back to Fig. 1.5.9, we now have to determine the groups E F k P 2 = 7 r 2 k - 2 ( S 2 k - 5 ) for k 2 4 , which means examining the 3-stem in detail. The

.r

10

P

m

N

c

c

0 X

c

N

c

-

P X

'- /

TIn

ln

+ x

In

v)

u)

-

t -

m

33

5. Unstable Homotopy Groups and the EHP Spectral Sequence

groups E:.2 and E i V 3are not touched by differentials, so there is an SES

o + E : , ~ +7 r 6 ( ~ 3 ) + ~ : , 3 + 0 . The two end terms are Z/(2) and the group extension can be shown to be nontrivial, so E:,2 = 7r6(S3)= Z/(4). Using the serial number notation, we denote the generator by 21 and the element of order 2 by 111. Similarly one sees r s ( S 2 ) = Z / ( 2 ) , 7r7(S4) = Z O Z / ( 4 ) , and there is an SES

o+ 7r6(S3)+r * ( S S ) +E:,4+0. Here the subgroup and kernel are Z/(4) and Z/(2), respectively, and the group extension is again nontrivial, so 7r8(S5)= EFk-2= Z/(8) for k 2 5. The generator of this group is the suspension of the Hopf map v: S7+ S4 and is denoted by 3. To determine E 7k-3= 7r2k-3(S2k-7) for k 2 5 we need to look at the 4-stem, i.e., at the column El.*. The differentials affecting those groups are indicated on the chart. Hence we have E + 2= 0 so 7r7(S3)= E Y = Z/(2); the d, hitting E:3 means that the corresponding element dies (i.e., vanishes) when suspended to 7r9(Ss);since it first appears on S 3 we say it is born there. Similarly, the generator of E? corresponds to an element that is born on S4and dies on S6 and hence shows up in E t S 3= 7rg(S5).We leave it to the reader to determine the remaining groups shown in the chart, assuming the differentials are as shown. We now turn to the problem of computing differentials and group extensions in the EHPSS. For the moment we will concentrate on the prime 2. The fibration 1.5.2 can be looped n times to give

R"S"

+

Qfl+lSn+l+

~fl+lS2"+1

In Snaith [ 11 a map is constructed from R"S"to QRP"-’which is compatible RkZkX.) with the suspension map R"S"+ R"+'S"+'.(Here QX denotes Hence we get a commutative diagram 1.5.10

I - I - 1

n"s"

QRP"-’

nn+lSn+l

QRP"

,

fl"+lSZ"+l

QS",

where both rows are fibre sequences and the right-hand vertical map is the standard inclusion. The LES in homotopy for the bottom row leads to an exact couple and a spectral sequence as in 1.5.7. We call it the stable EHPSS. There is an odd primary analog of 1.5.10 in which RP" is replaced by an appropriate skeleton of BZ.,,the classifying space for the symmetric

34

1. Introduction to the Homotopy Groups of Spheres

group on p letters. Recall that its mod ( p ) homology is given by 1.5.11

Hi( B E , ;Z/ ( p ) ) = {:I(')

if i=Oor -1 mod ( 9 ) otherwise.

1.5.12 Proposition.

( a ) For p = 2 there is a spectral sequence converging to .rr;(RP") (stable for n 2 2 and d,: EF" + E!-'-"-' . Here homotopy of RP") with EFn = &,+, E Z k is the subquotient im rrz(RP"-')/im T;(RP"-~) of .rr;(RP"). There is a similar spectral sequence converging to .rr;(RPJ-') with EF" as above for n s j a n d E f ; . " = Of o r n > j . ( b ) For p > 2 there is a similar SS converging to rrz( BZ, ) with E ym+’= .rr: and E F2"' = r k + l - m q . There is a similar ss with EF" = 0for n >j converging to .rr:(BZp)J-') i f j is even and to .rr*(BZbg)(Jp')) i f j is odd. to these from the corresponding EHPSSs There are homomorphisms (c) of 1.5.7 induced by suspension on the El level, e.g., at p = 2 by the suspension map .rrk+,,(S2"-I)+r k s- , , + l . Hence the El-terms are isomorphic in the stable zone. We remark that this stable EHPSS is nothing but a reindexed form of the Atiyah-Hirzebruch SS (see Adams [4], section 7 ) for m Z ( B Z p ) . In the latter one has E;' = H,(BZ:,; T:) and this group is easily seen to be Ei+'3-r(s) in the EHPSS where f(s, =

{ s / +( p - l )p+ l (s 1 ) / (

-1 )

if if

s = 0 mod (2p - 2 ) s = - 1 mod (2p - 2).

Since everything in 1.5.12 is stable one can use stable homotopy theoretic methods, such as the ASS and K-theory, to compute differentials and group extensions. This is a major theme in Mahowald [ 13. Differentials originating in EFk+' for p = 2 correspond to attaching maps in the cellular structure of RP", and similarly for p > 2. For example, we have 1.5.13 Proposition.

multiplication by p

In the stable EHPSS (1.5.12), d , : E;"+ E:-'."-' is is even and trivial if k is odd.

if k

Another useful feature of this SS is James periodicity: for each r there is a finite i and an isomorphism EF" = E:+qp'3n+2p'which commutes with differentials (note that q = 2 when p = 2). This fact is a consequence of the vector field theorem and will be explained more fully below (1.5.18). For p = 2, the diagram 1.5.10 can be enlarged as follows. An element in the orthogonal group O( n ) gives a homeomorphism S"-' + S"-'. Suspension gives a basepoint-preserving map S" + S" and therefore an element in R"S".

35

5. Unstable Homotopy Groups and the EHP Spectral Sequence

Hence we have a map J: O ( n ) + R " S " (compare 1.1.12). We also have the reflection map I : RP"-' + O( n ) sending a line through the origin in R" to the orthogonal matrix corresponding to reflection through the orthogonal hyperplane. Combining these we get 1.5.14

RP"-’

I

-

RP"

I

-

S"

/I

I - I - I

QRP"-'

QRP"

QS".

Here the top row is a cofiber sequence while the others are fiber sequences. The right-hand vertical maps are all suspensions, as is the composite RP" -+ ORP".The second row leads to a spectral sequence (which we call the orthogonal S S ) converging to 7r,(O) which maps to the EHPSS. The map on E 7" = T k ( S " - ' ) is an isomorphism for k < 2n - 3 by the Freudenthal suspension theorem 1.1.10. Hence we have three spectral sequences corresponding to the three lower rows of 1.5.14 and converging to rr*(O), the 2-component of P:, and 7r:(RP"). In all three we have generators X k E EFk+' = Z and we need to determine the first nontrivial differential (if any exists) on it for k odd. We will see that this differential always lands in the zone where all three spectral sequences are isomorphic. In the orthogonal SS x k survives to E, iff the projection O( k + 1)/ O( k + 1 - r ) + S k admits a cross section. It is well known (and easy to prove) that such a cross section exists iff S k admits r - 1 linearly independent tangent vector fields. The question of how many such vector fields exist is the vector field problem, which was solved by Adams [ 161 (see 1.5.16). We can give equivalent formulations of the problem in terms of the other two spectral sequences. 1.5.15 Theorem (James [2,3]). equivalent:

7 3 e following three statements are

( a ) Sk-' admits r - 1 linearly independent tangent vector jields. ( b ) Let L be the generator of r 2 k - l ( ~ 2 k - 1 ) = Z. Then P ( L )G 7 r 2 k - 3 ( s k - ' ) (see 1.5.2) desuspends to 7r2k-r-2(Sk-r). ( c ) The stable map RPk-'/RPk-' +. Sk-' admits a cross section.

36

1. Introduction to the Homotopy Groups of Spheres

The largest possible r above depends on the largest powers of 2 dividing k + l . Let k = 2 J ( 2 s + l ) ,

1"

4(j)= 2j+l 2j+2 and p ( k ) = 4 W .

or 2 m o d ( 4 ) if j - 1 if j = O mod (4) if j - 3 m o d ( 4 )

1.5.16 Theorem (Adams [ 161). ( a ) With notation as above, Sk-' admits p ( k ) - 1 linearly independent tangent vector jields and no more. ( b ) Let (Yo = 2 E n," and for j > 0 let 5, denote the generator of im J in S T ~ ( ~ )(see - ~ 1.1.19). Then in the 2-primary EHPSS (1.5.7) d d ( J ) ( x k - is l ) the (nontrivial) image of in EiYf;k-J. We remark that the p ( k ) - 1 vector fields on Sk were constructed long ago by Hurwitz and Radon (see Eckmann [l]), Adams [16] showed that no more exist by using real I n - 1. Clearly the d,: E:" +

37

5. Unstable Homotopy Groups and the EHP Spectral Sequence

Ek-l,n-r in the stable EHPSS is the same as that in the SS for r*(RP:TiLI,), and similar statements can be made for p > 2, giving us 1.5.18 James Periodicity Theorem.

In the stable EHPSS (1.5.12) there is an isomorphism E:"+ E ~ + w L , n +commuting 2p' with d, where i = 3 r / 2 .

Note that 1.5.17 is simpler than its 2-primary analog 1.5.16(b). The same is true of the next question we shall consider, that of the general behavior of elements in im J in the EHPSS. It is ironic that most of the published work in this area, e.g., Mahowald [2,4], is concerned exclusively with the prime 2, where the problem appears to be more difficult. Theorem 1.5.17 describes the behavior of the elements X q k - l in the odd primary EHPSS and indicates the need to consider the behavior of imJ. The elements CYJl and their multiples occur in the stable EHPSS in the groups ~ q k - 2 . 2 m and Eqk-l,2m+l for all k > m. To get at this question we use the 1 spectrum J, which is the fibre of a certain map bu + Z2bu, where bu is the spectrum representing connective complex K-theory, i.e., the spectrum obtained by delooping the space Z x BU. There is a stable map So + J which maps im Jc ri isomorphically onto r.+.(J). The stable EHPSS, which maps to a similar ss converging to J,(B,,)= converges to ri(KLP), r*(J A B E p ) .This latter SS is completely understood and gives information about the former and about the EHPSS itself.

1.5.19 Theorem. ( a ) For each odd prime p there is a connective spectrum J and a map So-+Jsending thep-component of im J (1.1.19) isomorphicallyonto v * ( J ) , ie.,

rp'

r i ( J ) = Z/(pJ")

if i=O if i = q k - 1 , k > 0 , k = s p ' w i t h p ) s otherwise.

- 7Tk-mq(J)and ( b ) There is an SS converging to J*( B Z p ) with E ym+l Ek2m = r k + l - , q ( J ) ; the map So+J induces a map to this SS from the stable EHPSS of 1.5.12. ( c ) The d , in this SS is determined by 1.5.13. The resulting E,-term has the following nontrivial groups and no other:

for k > 0,

-Z/(p)

generated by

Xqk-l

Z / (p )

generated by

GJ for

k, j > 0,

-Z / (p )

generated by

aJ for

k, j > 0,

~ q k - l , 2 k-

~ q ' k + ~ ) - 2 . 2 k-

and

E 5' k+j 1- 1,2k+ 1 -

where aj is an element of order p in r q j - , ( J ) .

38

1. Introduction to the Homotopy Groups of Spheres

( d ) The higher diflerentials are determined by 1.5.17 and the fact that all group extensions in sight are nontrivial, i.e., with k and j as in 1.5.17, k-2,2(k-l-l) d2,+2(Xqk-1) = &]+I ~ 4wJ+2 and d2J+3is nontrivial on E$Li*2m+lfor j+2<m 0 Ji(BE,)

(,,(pJ)

for i=qspi-2 with p ) s otherwise. W

= .rr,(J)O

We will sketch the proof of this theorem. We have the fibration J + bu + X2bu for which the LES of homotopy groups is known; actually bu (when localized at the odd prime p ) splits into p - 1 summands each equivalent to an even suspension of BP( l), where T*(BP( 1)) = Z, ,)[ u , ] with dim vl = q. It is convenient to replace the above fibration by J + BP( 1)+ ZqBP(1). We also have a transfer map B Z , + S:,,, which is the map which Kahn and Priddy [ l ] show induces a surjection of homotopy groups in positive dimensions (see also Adams [ 151); the same holds for J-homology groups. Let R be the cofiber of this map. One can show that SPp)+ R induces a monomorphism in BP( 1)-homology (or equivalently in bu-homology) and that BP(1) A R -- V j z 0Z4'HZ,,,, i.e., a wedge of suspensions of intebral Eilenberg-Mac Lane spectra localized at p. Smashing these two fibrations together gives us a diagram 1.5.20

JAR

T JABX,

-

-

B P ( ~ )R A

T BP(~)ABZ,

-

-

X q B P ( l )R ~

T Z4BP(1)~BZ,

in which each row and column is a cofiber sequence. The known behavior of ~ * ( f determines ) that of T,( f A R ) and enables one to compute T*(J A B Z , ) = J,(BZ,). The answer, described in 1.5.19(c), essentially forces the SS of 1.5.19 to behave in the way it does. The E,-term [1.5.19(c)] is a filtered form of T*(BP(1) A BZ,)O T,(Zq-'BP(1) A B E p ) . Corresponding statements about the EHPSS are not yet known but can most likely be proven by using methods of Mahowald [4]. We surmise they can be derived from the following. 1.5.21 Conjecture.

( a ) The composite ?Tk(fi2n+lS2n+1)+ Tk(QBZr)+Jk(BZ.p4")is onto unless k = qsp' - 2 (wirh j > 0, sp' > p and p i s) and n = spJ - i for 1 5 i Ij .

5. Unstable Homotopy Groups and the EHP Spectral Sequence

39

( b ) The groups Eg,k-1.2"+'of 1.5.19(e) pull back to the Em-term of the EHPSS and correspond to the element a k / m (1.3.19) of order p m in im J E S r q k - l . Hence a k I m is born in SZm+'and has Hopf invariant a k - , , , except for al,which is born on with Hopf invariant one. (This was not suspected when the notation was invented !)

s2

We will give an example of an exception to 1.5.21(a) for p = 3. One has as E E:9*5,which should support a d3 hitting E9 E Ei8,', but E:*,' = 7r4,,( S 5 ) and a9 is only born on S', so the proposed d3 cannot exist (this problem ) and does not occur in the stable EHPSS). In fact, a l a s# 0 E T ~ ~ ( S='E:8,3 E:995. this element is hit by a d2 supported by the The other groups in 1.5.19(e), Jpql-2(BXp),are harder to analyze. EZ-2.q pulls back to the EHPSS and corresponds to PI E T&, (1.3.14), the first stable element in coker J (1.1.20), so PI is born on and has Hopf invariant al.Presumably the corresponding generators of E?q-2,2p'-2 for i > 1 each supports a nontrivial dq hitting a PI in the appropriate group. The behavior of the remaining elements of this sort is probably determined by, that of the generators of E~'q-2~wp'-2J for j z 2 , which we now denote by 6,. These appear to be closely related to the Arf invariant elements @, = P p ~ - l / p ~ - l (1.4.10) in E$p14of the ANSS. The latter are known not to survive (6.4.1), so presumably the do not survive either. In particular we know d2p2-6(I$) = P: in the appropriate group. There are similar elements at p = 2 as we shall see below. In that case the 6, are presumed but certainly not known (for j > 5) to exist in 7 ~ $ + 1 - ~ . Hence any program to prove their existence at p = 2 is doomed to fail if it would also lead to a proof for p > 2. We now consider the 2-primary analog of 1.5.19 and 1.5.21. The situation is more complicated for four reasons.

sq

6

(1) im J (1.1.19) is more complicated at p = 2 than at odd primes. (2) The homotopy of J (which is the fiber of a certain map bo + Z4bsp, where bo and bsp are the spectra representing connective real and symplectic K-theory, respectively) contains more than just im J. (3) Certain additional exceptions have to be made in the analog 1.5.21. (4) The groups corresponding to the JPlq-*(B Z , ) are more complicated I Mahowald [6] in addition to the and lead us to the elements 7, E ~ Z of hypothetical @, E T & + ~ L ~ . Our first job then is to describe a*(J)and how it differs from i m J as described in 1.1.19. We have r , ( b o ) = 7rE+,(O)and r , ( b s p )= V ~ + ~ (for O) i r O and r * ( O ) is described in 1.1.17, i.e., if i = 3 mod (4) if i = O o r l m o d ( 8 ) otherwise.

40

1. Introduction to the Homotopy Groups of Spheres

The map bo + Z4bsp used to define J is trivial on the torsion in r * (bo), so these groups pull back to r*(J).Hence rTgi+l(J) and rBi+z(J) for ir 1 contain summands of order 2 not coming from im J 1.5.22 Proposition. A t p = 2

if if if if if if

1z,

Z/(2)02/(2) (2J+l)

i=O

i = l or 2 i = 3 m o d ( 8 ) and i > O i = O or 2mod(8) and i r 8 i = 1 mod (8) and i 2 9 i = 8 m - l , m r l and 8m=2'(2s+l).

Here, im J c r*(J ) consists of cyclic summands in r,(J) for i > 0 and i = 7, 0, 1, or 3 mod (8).

Now we need to name certain elements in r * ( J ) . As in 1.5.16 let 6, denote the generator of im J in dimension +( j)- 1, where 2j-1

if j - 1 o r 2 m o d ( 4 ) if j = 0 mod (4) if j = 3 m o d ( 4 ) .

2j+l

We also define elements aj in r * (j ) of order 2 as follows. a1= 7 E r,(J) and a 4 k + l ~r 8 k + l ( J ) is a certain element not in i m J for k 2 1. ( ~ 4 k + 2 = 2 7 a 4 k + l , (Y4k+3 = 7 a 4 k + l = 4 6 4 k + 2 , and (Y4k E v & - 1 ( J )is an element of order 2 in that cyclic group. 1.5.23 Theorem (Mahowald [4]).

( a ) There is an SS converging ro JJRP") with EF" = r k - n + l ( . f ) ; the map So+ J induces a homomorphism to this SS from the stable EHPSS of 1.5.12. ( b ) The d , in this SS is determined by 1.5.13. The following is a complete list of nontrivial d2's and d3's. For k r 1 and t 2 0 , d2 sends X4k+,

a41+3+1 a41+l

-

~4k+1.4k+2

E 42 k +8+

I

+8 i,4k + 2

~4kf2+81,4k+2 2 ~4k+I+81+7,4k+l

a41+4

2

to L Y I to 64,+1 for i = O , 1 to a41+2 to 6 4 1 + 5

and a’%,+*~ 42k + 1 + 8 1 , 4 k + I to

(~4l+i+l

for i = 1,2.

41

5. Unstable Homotopy Groups and the EHP Spectral Sequence

For k > 1 and

t>

1 , d3 sends a ~

4 k 1+81,4k+3 + 2

41

to

a41+1

and 4 k + 8 l + 1,4k+1 Ly41+l E E 2

to Ly41+2.

See Fig. 1.5.24. ( c ) The resulting E4-term is a Z/(Z)-vectorspace on the following generators for k 2 1, t 2 0. LY2e E22;

x , E E:22; 81+1+5,2

'4l+,

for

E4

E

ff41+3

~4k+St+6,4k. 4 I

*41+2

~4k+8[+3,4k+2. 4

4k+4,4k+ZO 5 2 E E 4 9

~y+St+3,4k+3.

and

, ,

ff4f+ 1

a 4 r + 4 ~

4k+81+2,4k. 541+2€ E4 3

9

a4f+3E

,

(Y&+lE E4

~ik-L.4k. X4k-1

9

8 1+9,3. E4 ,

81+3,3.

i=3,4,5;

81+ 10.3.

Ly41+4EE4

i=l,2;

E:f+'+1*2 for

4k+81+3,4k+l. E4 9

4k+81+7,4k+l. ff41+4EE4

,

4k+Xf+10,4k+2. Ly41+5E E 4 9 4k+81+10,4k+3

Ly4r+4~E4

( d ) The higher differentials are determined by 1.5.15 and the fact that most group extensions in sight are nontrivial. The resulting Em-term has the following additive generators and no others for t 2 0. 61+9,3.

xi E Ek2; 8t i 3 . 3 . a4t+lEES 9

m

2J+ 1( 1 + I ) - 1 a 2 ~ 1 + 2 ~ - ~E - 2E m

(y2

,*

~2+4,41+2.

,

i = 1,2;

9

for i = l , 2 , 3 ;

2 3;

for j and

for

X ~ EL'; E

~Blf7.8-i rn

a4i+l

9

8t+r+l,?

Em

~ 4 1 + E4 m ~

i=3,4;

for

Em

E

81+11,5.

x ~ E E ,;

~81+15,9.

&41+4

a41+,

34

81+#+5,2

'4l+t

,

Em

a 4 1 + 4 ~

Ly,

EZf'(l+l)-2,*

for j

2 2.

( e ) Fori>O

Z/(2) J,(RPm)= r,(J)O Z / ( 2 ' ) [O

izOmod(4) i f i = 2-"'s - 2 for s odd otherwise. W if

Note that the portion of the Ex-term corresponding to the summand r , ( J ) in 1.5.23(e) [i.e., all but the last two families of elements listed in 1.5.23(d)] is near the line n = 0, while that corresponding to the second summand is near the line n = k The proof of 1.5.23 is similar to that of 1.5.19, although the details are messier. One has fibrations J + bo +X4bsp and RPm+Sy2)+R. We have

Figure 1.5.24. A portion of the &-term of the SS of Theorem 1.5.23 converging to J,(RP") and showing the d,'s and d3's listed in Theorem 1.5.23, part (c).

5. Unstable Homotopy Groups and the EHP Spectral Sequence

43

R A bo = ',Ijzo Z4JNZ(,,and we can get a description of R A bsp from the fibration Z4bo+ bsp+ H Z ( z , . The €?,-term in 1.5.22 is a filtered form of rr,(X3bsp A RP")O rr,(bo A RP"'); elements with Hopf invariants of the form Gj are in the first summand while the other generators make up the second summand. By studying the analog of 1.5.20 we can compute J,(RP") and again the answer [1.5.23(e)] forces the S S to behave the way it does. Now we come to the analog of 1.5.21. 1.5.25 Theorem (Mahowald [4]). ( a)

The composite rrk

(a

2 n + 1 S2"+ I

) + 'Tk(QRP2")-+Jk(RP2")

is onto unless k = 0 mod (4) and k I2n, or k = 6 mod ( 8 ) . It is also onto if k = 2 ' f o r j r 3 or ifk=2'-2mod(2'+') and k r 2 n + 8 + 2 j . When k 5 2 n is a multiple of 4 and not a power of 2 at least 8 , then the cokernel is Z / ( 2 ) ; when k 5 2n is 2 less than a multiple of 8 but not 2 less than a power of 2, then the cokernel is J k ( R P 2 " )= Jk(RP"). ( b ) All elements in the Em-term corresponding to elements in T*(J ) pull back to the EHPSS except some of the 'Y4r+i E E2+i+5,2for i = 3,4 and t 2 0. Hence H ( a , ) = H ( G 2 )= H(G,) = 1 , H ( C X , +=~ a,, ) and if 2'x = Jor X E im J then H ( x ) = a f P i .

Theorem 1.5.23 leads one to believe that H(L?~,+~) = C?4,+,-, for i = 4 , 5 and t 20, and that these elements are born on S2, but this cannot be true in all cases. If G4 were born on S2, its Hopf invariant would be in rlO(S3), but this group does not contain G3, which is born on S4. In fact we find H ( G 4 )= G 2 , H ( G 5 )= G:, and H(LY8) is an unstable element. 1.5.26 Remark. Theorem 1.5.25(b) shows that the portion of im J generi.e., the cyclic summands of order 2 8 in dimensions ated by G4r+2 and, , , , ,C? 4 k - 1 , are born on low-dimensional spheres, e.g., E 4 r + 2 is born on S'. However, simple calculations with 1.5.14 show that the generator of ' T 4 k - 1 ( 0 ) pulls back to ~ 4 k - I ( 0 ( 2 k + l )and ) no further. Hence G4r+2€ T ~ ~ S')+ is~not ( actually in the image of the unstable J-homomorphism until it is suspended to S4"3. Now we consider the second summand of J,(RP") of 1.5.23(e). The elements G 2 E E4,4k-2 for k 2 1 have no odd primary analog and we treat them first. The main result of Mahowald [ 6 ] says there are elements T ~ E r p ( S 0 ) for j 2 3 with Hopf invariant v = G2. This takes care of the case k = 2'-2 above. 1.5.27 Theorem. In the EHPSS the element v = G 2 e E:k4k-2 for k 2 2 behaves as follows (there is no such element for k = 1).

44

1. Introduction to the Homotopy Groups of Spheres

( a ) If k = 2'-2, j 2 3 then the element is a permanent cycle corresponding to 17,; this is proved by Mahowald [6]. ( b ) I f k = 2 s + l thend,(v)=v2. 1.5.28 Conjecture.

If k = (2s + 1)2'-' with s > 0 then d 2 , ~ 2v() = 77j.

The remaining elements in 1.5.23(e) appear to be related to the famous Kervaire invariant problem (Mahowald [ 7 ] ,Browder [ 11). 1.5.29 Conjecture. In the EHPSS the elements 6;E E:'*'('+')-27*for j

2 2,

t 2 0 behave as follows: ( a ) If there is a framed (2'+'-2)-man$old with Kervaire invariant one then a;. E E :'+'-'** is a nontrivial permanent cycle corresponding to an element 0, E I T ~ J + ~ - ~ S( o ) . ( These elements are known (Barratt, Jones, and Mahowald [ 2 ] ) to exist for j 5 5.) satisJies d,( 6,)= 0,, ( b ) If ( a ) is true then the element 6,E E:'+1(2s+')-23* where r = 2'+' - 1 -dim( Ci). The converse of 1.5.29(a) is proved by Mahowald [4] 7.11. Now we will describe the connection of the EHPSS with the Segal conjecture. For simplicity we will limit our remarks to the 2-primary case, although everything we say has an odd primary analog. As remarked above, the stable E H P S S (1.5.12) can be modified so as to converge to the stable homotopy of a stunted projective space. Let R q = RPm/R&, for n > 0; i.e., R q is the infinite-dimensional stunted projective space whose first cell is in dimension j . It is easily seen to be the Thom spectrum of the j-fold Whitney sum of the canonical line bundle over RP". This bundle can be defined stably for j 5 0, so we get Thom spectra RP, having one cell in each dimension r j for any integer j . 1.5.30 Proposition.

For each j

EZ

there is a spectral sequence converging to

?r,(RP,) with T~-,+~(SO)

E:"={o

if if

n-lrj n-l<j

and d,: EFn+ EF-ls"-'. For j = 1 this is the stable EHPSS of 1.5.12. If j < 1 this SS maps to the stable EHPSS, the map being an isomorphism on E for nr2.

:"

The Segal conjecture for Z / ( 2 ) ,first proved by Lin [ l ] ,has the following consequence. 1.5.31 Theorem. For each j < 0 there is a map S-'

S-’ + RP-,

=

+ RP,

such that the map

lim R q is a homotopy equivalence after 2-adic completion of

the source (the target is already 2-adically complete since RP, is for j odd).

5. Unstable Homotopy Groups and the EHP Spectral Sequence

45

Consequently the inverse limit over j of the spectral sequences of 1.5.30 converges to the 2-component of “ * ( S - ’ ) . W e will call this limit SS the superstable EHPSS. Nothing like this is stated in Lin [ 11 even though it is an easy consequence of his results. A proof and some generalizations are given in Ravenel [4]. Notice that H*(RP-,) # @ HJRP,); this is a spectacular example of the failure of homology to commute with inverse limits. Theorem 1.5.31 was first conjectured by Mahowald and was discussed by Adams [14]. Now consider the spectrum RP,. It is the Thom spectrum of the trivial bundle and is therefore So v RP,. Hence for each j < 0 there is a map RP, + So which is nontrivial in mod ( 2 ) homology. The cofiber of this map f o r j = -1 can be shown to be R, the cofiber of the map RPI+ Soof Kahn and Priddy [ 2 ] . The Kahn-Priddy theorem says this map is surjective in homotopy in positive dimensions. Using these facts we get

1.5.32 Theorem.

In the SS of 1 S.30 for j < 0,

( a ) no element in E$k supports a nontrivial diflerential; ( b ) no element in E:*k is the target of a nontrivial diflerential; ( c ) every element of E7k= “ k + 1 ( S o ) that is divisible by 2 is the target of a nontrivial d , and every element of E?k for k > -1 is the target of some d , for r r 2 ; and ( d ) every element in E : * k = ?rk( S o ) not of order 2 supports a nontrivial d , and every element of E$k supports a nontrivial d, for some r 1 2 .

Prooj Parts (a) and (b) follow from the existence of maps S-’ + RP, + So, (c) follows from the Kahn-Priddy theorem, and (d) follows from the fact that the map !imRP, + So is trivial. Now the SS converges to 7r*(S-’), yet 1.5.32(c) indicates that the map S-’+ R E , induces a trivial map of Em-terms, except for E:loo, where it is the projection of Z onto Z/(2). [Here we are using a suitably indexed, collapsing AHSS for n,( S-’).I This raises the following question: what element in E$-” (for some n > 0 ) corresponds to a given element X E “k(S-l)? The determination of n is equivalent to finding the smallest n such that the composite S k A S-’ +RP-,-I is nontrivial. The KahnPriddy theorem tells us this composite is trivial for n = 0 if k 2 0 or k = -1 and x is divisible by 2; and the Segal conjecture (via 1.5.31) says the map is nontrivial for some n > 0. Now consider the cofiber sequence S-”-’+ RP-,-I + R E , . The map from S k to R E , is trivial by assumption so we get a map from S k to Spl-”, defined modulo some indeterminacy. Hence x E “ k + ] ( S o ) gives us a coset ~ ( xc )“ k + l + n ( ~ o ) which does not contain zero. We call M ( x ) the Mahowald invariant of x, and note that n, as well

46

1. Introduction to the Homotopy Groups of Spheres

as the coset, depends on x. The invariant can be computed in some cases and appears to be very interesting. For example, we have 1.5.33 Theorem. Let L be a generator of 7ro( S o ) . Thenfor each j > 0, M ( 2'~) H contains aj,a preimage in a,(So) ofrhe ajE 7 r , ( J ) of 1.5.23. A similar result holds for odd primes. In 1.5.31 we replace the RP, by Thorn spectra of certain bundles over BXm and M ( p j ~3) aj for aj as in 1.5.19. We also have 1.5.34 Conjecture. M ( 0,) contains Oj+l for 0, as in 1.5.29.

1.5.35 Conjecture. Whenever the Greek letter elements (1.3.17) a:") and a ( " + l ) exist in hornotopy, ( Y ~ " + " E ~ ( c r j " ' ) . H

,

One can mimic the definition of the Mahowald invariant in terms of the ASS or ANSS E,-terms and in the latter case prove an analog of these conjectures. At p = 2 one can show (in homotopy) that M ( a , ) 3 G 2 , M ( C z )3 E3, and M(ci,)G:= 03. This suggests using the iterated Mahowald invariant to define (up to indeterminacy) Greek letter elements in homotopy, and that 0, is a special case (namely a y + ' ) )of this definition.

CHAPTER 2

SEITING UP THE ADAMS SPECFRAL SEQUENCE

In this chapter we introduce the spectral sequence that will be our main object of study. We do not intend to give a definitive account of the underlying theory, but merely to make the rest of the book intelligible. Nearly all of this material is due to Adams. The classical Adams spectral sequence [i.e., the one based on ordinary mod ( p ) cohomology] was first introduced in Adams [3] and a most enjoyable exposition of it can be found in Adams [7]. In Section 1 we give a fairly self-contained account of it, referring to Adams [4] only for standard facts about Moore spectra and inverse limits. We include a detailed discussion of how one extracts differentials from an exact couple and a proof of convergence. In Section 2 we describe the Adams spectral sequence based on a generalized homology theory E , satisfying certain assumptions (2.2.5). We rely heavily on Adams [4], referring to it for the more difficult proofs. The E,-Adams resolutions (2.2.1) and spectral sequences (2.2.4) are defined, the &-term is identified, and the convergence question is settled (2.2.3). We do not give the spectral sequence in its full generality; we are only concerned with computing n-*( Y), not [X,Y] for spectra X and Y. Most of the relevant algebraic theory, i.e., the study of Hopf algebroids, is developed in Appendix 1. In Section 3 we study the pairing of Adams spectral sequences induced by a map a: X' A X"+ X and the connecting homomorphism associated with a cofibration realizing a short exact sequence in E-homology. Our smash product result implies that for a ring spectrum the Adams spectral 47

48

2. Setting Up the Adams Spectral Sequence

sequence is one of differential algebras. To our knowledge these are the first published proofs of these results in such generality. Throughout this chapter and the rest of the book we assume a working knowledge of spectra and the stable homotopy category as described, for example, in the first few sections of Adams [4].

1. The Classical Adams Spectral Sequence In this section we will set up the Adams spectral sequence based on ordinary mod ( p ) cohomology for the homotopy groups of a spectrum X . Unless otherwise stated all homology and cohomology groups will have coe5cients in Z / ( p ) for a prime number p, and X will be a connective spectrum such that H * ( X ) (but not necessarily X itself) has finite type. Recall that H * ( X ) is a module over the mod ( p ) Steenrod algebra A, to be described explicitly in the next chapter. Our object is to prove 2.1.1 Theorem (Adams [3]).

Let X be a spectrum as above. There is a

spectral sequence E$*(X)

with d,: E:'+ Es+r,'+r-'

such that ( a ) E;'=Ext>'(H*(X),Z/(p)). ( b ) i f X is of finite type, E%* is the bigraded group associated with a certain filtration of n-*(X)OZ,, where Z , denotes the ring of p-adic integers. Let E = H Z / ( p ) , the mod ( p )Eilenberg-Mac Lane spectrum. We recall some of its elementary properties. 2.1.2 Proposition.

( a ) H , ( X ) = n,(E A X ) . ( b ) H * ( X )= [ X , El. (c) H*(E)=A. ( d ) If K is a locally finite wedge of suspensions of E, i.e., a generalized mod ( p ) Eilenberg- Mac Lane spectrum, then n*( K ) is a graded Z / ( p)-vector space with one generator for each wedge summand of K . More precisely, .rr,(K)= HomA(H*(K),Z l ( p ) ) . ( e ) A mapfrom X to K is equivalent to a locallyfinite collection of elements in H*( X ) in the appropriate dimensions. Conversely, any locally finite collection of elements in H * ( X ) determines a map to such a K.

49

1. The Classical Adam Spectral Sequence

(f) I f a locallyfinite collection of elements in H*(X) generate it as an Amodule, then the corresponding map f:X + K induces a surjection in cohomology. ( g ) E A X is a wedge of suspensions of E with one wedge summand for each Z / ( p ) generator of H * ( X ) . H * ( E A X ) = A @ H * ( X ) and the map f : X + E A X (obtained by smashing X with the map S o + E ) induces the A-module structure map A@ H * ( X ) + H * ( X ) in cohomology. In particular H*( f ) is a surjection. The idea behind the Adams spectral sequence is to use maps such as those of (f) or (g) and our knowledge of .rr,(K) or .rr,(E A X ) to get information about r * ( X ) .We enlist the aid of homological algebra to make the necessary calculations. More specifically, we have

2.1.3 Definition. A mod ( p ) Adams resolution ( X 9 ,g s )for X is a diagram

KO

Kl

K2

where each K , is a wedge of suspensions of E, H * ( f s ) is onto and X,,, is the jiber offs. W Proposition 2.1.2(f) and (g) enable us to construct such resolutions for any X , e.g., by setting K , = E A X , . Since H * ( f s ) is onto we have short exact sequences O+ H * ( X s )+ H * ( K s )+ H*(CX,+,)+ 0.

We can splice these together to obtain a long exact sequence 2.1.4

O+ H * ( X ) + H*( K O )+ H * ( Z K , )+ H*(C2Kz)+

-.

Since the maps are A-module homomorphisms and each H * ( K , ) is free over A p , 2.1.4 is a free A-resolution of H * ( X ) . Unfortunately, the relation of .rr*(Ks)to .rr*(X) is not as simple as that between the corresponding cohomology groups. Life would be very simple if we knew ..*(A) was onto, but in general it is not. We have instead long exact sequences

2.1.5

arising from the fibrations

50

2. Setting Up the A d a m Spectral Sequence

If we regard .rr*(X,) and .rr*(KS)for all s as bigraded abelian groups D , and El, respectively [i.e., D:'= T ' - ~ ( Xand ~ ) E ? ' = ~ T - ~ ( K ,then ) ] 2.1.5 becomes

2.1.6 ..

El

where i, = ~ ’ - ~ ( g ~Df+l.'+l+ ): D:', j i = T , - ~ ( F ~DS,'+ ) : ES,',

and E:'+ D;+l7'.

k, =

The exactness of 2.1.5 translates to ker i, = im k,,kerj, = im i,, and ker k, = i m j , . A diagram such as 2.1.6 is known as an exact couple. It is standard homological algebra that an exact couple leads one to a spectral sequence; accounts of this theory can be found in Cartan and Eilenberg [ 1, section XV.71, Mac Lane [ 1, section XIS], and Hilton and Stammbach [ 1, chapter 81 as well as Massey [2]. Briefly, d , =j,k,:E:'+ E;+l7' has ( d , ) * = j l k l j l k l = Oso ( E l ,d, ) is a complex and we define E2 = H(E l , d,). We get another exact couple, called the derived couple, 2.1.7

D

2

x?\/

A D,

E2

where 0:' = i, D:', i2 is induced by i, ,j,( i , d ) = j , d for d E D,, and k2(e) = k , ( e ) for e E ker d, c E l . Since 2.1.7 is also an exact couple (this is provable by a diagram chase), we can take its derived couple, and iterating the procedure gives a sequence of exact couples

Er

where Dr+,= i,D,, d, =j,k,, and E,,, = H ( E,, dr). The sequences of complexes {( E,, 4)) constitutes a spectral sequence. A close examination of the indices will reveal that d,: E:'+ Es+r,'+r-l. It follows that for s < r,

51

1. The Classical Adam Spectral Sequence

the image of d , in E:' is trivial so E::l can define

is a subgroup of E:', hence we

~ g n= E;'. r>s

This group will be identified (2.12) in certain cases with a subquotient of r l P s ( X ) namely, , im ~ , - ~ ( X , ~ )T,-,(X,+~). /im The subgroups im r.JXs) = F " r , ( X ) form a decreasing filtration of r , ( X ) and Em is the associated bigraded group.

The mod ( p ) Adams spectral sequence for X is the spectral sequence associated to the exact couple 2.1.6.

2.1.8 Definition.

We will verify that d,: E S.' + Es+r,'+r-lby chasing diagram 2.1.9, where we write r,(X,) and r,(K,) instead of D , and El, with u = t -s. 2.1.9

+

P Y

+

I I

ruu(fs+*)

dXs+2)-

( gs+ I )

.rr,(Ks+2)

JS+2,Ll

I

' ru-1(xs+3)

i

nu-1(gs+*)

.rr,(fs+l)

r,(Xs+1)-

T"-I(h+J A

~ u - I ( ~ s + ~ + )

Pu-l(fr+Z)

~,-l(Xs+*)

ru-,(Ks+z)

+

The long exact sequences 2.1.5 are embedded in this diagram; each consists of a vertical step n,(g,) followed by horizontal steps r,(J,)and d,, and so on. We have ES,' = r,(K , ) and d:'= (~s+,,u~,)(ru~,(~+l)). We have E:'= ker df.'/im df-'.'. Suppose an element in E:' is represented by X E r u ( K s ) .We will now explain how d 2 [ x ] (where [ x ] is the class represented by x ) is defined. x is a d , cycle, i.e., d , x = 0, so exactness in 2.1.4 implies that d , , x = (ru+l(gs+l))(y)for some y E T " - , ( X ~ + ~ )Then . (7~,-,(fS+~))(y) is a d , cycle which represents d 2 [ x ]E Ei+z3f-1. If d 2 [ x ]= 0 then [ X I represents an element in E;' which we also denote by [ X I . To define d 3 [ x ]it can be shown that y can be chosen so that y = (ru-l(gs+2))(y’) for some Y’E ~ T , - ~ ( X and , + ~that ) (~,-,(f~+~))(y’) is a d , cycle representing a d2 cycle which represents an element in Es+3,t+2which we define to be d 3 [ x ] .These assertions may be verified by drawing another diagram which is related to the derived couple 2.1.7 in the same way that 2.1.9 is related

52

2. Setting Up the A d a m Spectral Sequence

to the original exact couple 2.1.6. The higher differentials are defined in a similar fashion. In practice, even the calculation of d2 is a delicate business. Before identifying E&' we need to define the homotopy inverse limit of spectra. 2.1.10 Definition.

Given a sequence of spectra a maps

lim Xi is theJiber of the map

c

whose ith component is the diference between the projection p i :Il X, + X, and the composite

n xj

e

b

-

xi+,A+, xi. rn

For the existence of products in the stable category see 3.13 of Adams [4]. This !im is not a categorical inverse limit (Mac Lane [2, section 111.41) because a compatible collection of maps to the X, does not give a unique map to &I Xi. For this reason some authors (e.g., Bousfield and Kan [ 11) denote it instead by h a m . The same can be said of the direct limit, which can be defined as the cofiber of the appropriate self-map of the coproduct of the spectra in question. However this & I has most of the properties one would like, such as the following. 2.1.11 Lemma. Given spectra X,,jfor i, j 2 0 and maps f : X i . j+ Xi-l,jand g : Xi,j+ Xi,j-l such thatfg is homotopic to gJ

lirn lirn Xi.j= lim lirn Xi,j.

cc

i

j

j

i

Proof: We have for each i a cofibre sequence

Next we need to know that products preserve cofiber sequences. For this fact, recall that the product of spectra n Y , is defined via Brown's representability theorem (Adams [4], Theorem 3.12) as the spectrum representing the functor n[-, Yi]. Hence the statement follows from the fact that a product (although not the inverse limit) of exact sequences is again exact. Hence we get the following homotopy commutative diagram in which both rows and columns are cofiber sequences.

53

1. The Classical A d a m Spectral Sequence

I

I

I

Everything in sight is determined by the two self-maps of n, n, X,, and the homotopy that makes them commute. Since the product is categorical we have II, 11, X , =n, n, X , . It follows that II,&I, X,, =@, II,X,, because they are each the fiber of the same map. Similarly

n lim x,,=@n x,, 1

1

Z

so one gets an equivalent diagram with corner. 1

J

&Ij l h i X ,

in the upper left

Now we will show that for suitable X, E&' is a certain subquotient of %(X). Let X be a spectrum with a n Adams resolution ( X s ,g s ) such that !imX , = pt. Then EL' is the subquotient im 7ru(Xs)/im 7rU(Xstl)of = 0. im .rr*(x,) 7r,(X) and

2.1.12 Lemma.

Proof: For the triviality of the intersection we have lim X, = pt. Let G, = T * ( X , ~and )

!im 7r*(Xs)= 0 since

c

{

G' = Gs ' im G , < G ,

if if

s>t t>s.

We have injections GI + Gj-' and surjections Gf + Gj-l, so l k , Gf = 0, Gf and lir~, Gf = G,. We are trying to show l i ~Gi ~= , 0. l h , Gf maps onto c lim, Gip1, so I & l h t Gj maps onto & l GA. But Gf= lim, lim, Gf =&, G, = 0. For the identification of EL', let 0 # [XI E E:'. First we show aS,,(x)=O. Since d , [ x ] = O , a , , ( x ) can be lifted to ' T T , - ~ ( X ~for + ~each + ~ ) r. It follows that a,,(x) E im !im7r,-,(Xstr)= 0, so

m,b,

c c

as,&)

= 0.

54

2. Setting Up the Adams Spectral Sequence

Hence we have x = 7ru(fi)(y) for y t r U ( X x )It. suffices to show that y has a nontrivial image in a , ( X ) . If not, let r be the largest integer such that y has a nontrivial image Z E T , , ( X ~ - ~ +Then , ) . z = d , - , , , ( w ) for w E T ~ ( K ~and - ~d,[w] ) = [ X I , contradicting the nontriviality of [XI. W Now we prove 2.1.1(a), the identification of the E,-term. By 2.1.2(d), E:' = Horn,( HI-$( K s ) , Z / ( p ) ) . Hence Horn,(-, Z / ( p ) ) to 2.1.3 gives a complex

p s . ,E~.I&,E:.~+.

applying

...

The cohomology of this complex is by definition the indicated Ext group. It is straightforward to identify the coboundary 6 with the d , in the spectral sequence and 2.1.1(a) follows.

2.1.13 Corollary. Iff:X --* Y induces an isomorphism in mod ( p ) homology then it induces an isomorphism (from E, onward) in the mod ( p ) Adams spectral sequence. 2.1.14 Definition. Let G be an abelian group and X a spectrum. Then X G = X A SG, where SG is the Moore spectrum associated with G (Adams [ 4 ] , p. 200). Let 2 = X Z , , where Z , is the p-adic integers, and X" = XZ/(P"). 2.1.15 Lemma. (a) The map X +. 2 induces an isomorphism of mod ( p ) Adams spectral sequences. (b) ?,(*I = .rr*(X)@Z,. (c) X = !im X"' i f X has jinite type.

Proof. For (a) it suffices by 2.1.11 to show that the map induces an isomorphism in mod ( p ) homology. For this see Adams [ 4 ] , proposition 6.7, which also shows (b). Part (c) does not follow immediately from the fact that S Z , = lim S Z / ( p " ) because inverse limits do not in general commute with smash products. Indeed our assertion would be false for X = S Q , but we are assuming that X has finite type. By 2.1.10 there is a cofibration c

sz,+n SZ/(P")+rI W ( P r n ) , rn rn so it suffices to show that

x

A n

S Z / ( p r n ) = r IX Z / ( P " ) .

This is a special case (with X = E and R = Z ) of theorem 15.2 of Adams [ 4 ] . 4

55

1. The Classical A d a m Spectral Sequence

I f X is a connective spectrum with each r i ( X ) ajinitep-group, then for any mod ( p ) A d a m resolution ( X , , G , ) ofX, @ X, =pt.

2.1.16 Lemma.

Roo$

Construct a diagram

x=x;cx;cx;t.

* *

(not an Adams resolution) by letting Xt+l be the fiber in

x:+l-+ x:+ K , where the right-hand map corresponds [2.1.2(e)] to a basis for the bottom cohomology group of X,. Then the finiteness of .rri(X) implies that for each i, rj(X:) = 0 for large s. Moreover, T*(XL+~)+ r*(XL) is monomorphic so lim X : = pt. Now if (X,, g,) is an Adams resolution, the triviality of g, in cohomology enables us to construct compatible maps X, + X:. It follows that the map lim r * ( X , ) - +r J X ) is trivial. Each X, also satisfies 2.1.14, so we conclude that @ r*(X,) has trivial image in each r*(X,) and is therefore trivial. Since r i ( X , ) is finite for all i and s, @' r*(X,) = 0 so ~ !J I X, = pt. W

c

c

We are now ready to prove 2.1.l(!), i.e., to identify the &,-term. By 2.1.15(a) it suffices to replace X by X . Note that since SZ, A S Z / ( p " ) = SZ/( p"), X" = 2".It follows that given a mod ( p ) Adams resol$on (X,, g,) for X, smashing with S Z , and S Z / ( p " ) gives resolutions (X,, 2,) and (X,", g ; ) for 2 and X", respectively. Moreover, 5" satisfies 2.1.14 so @, Xy = pt. Applying 2.1.13(c) to each X, we get X, = @ , m XY, so lim X = !im lim X Y

c

5

m

=limlimX': cc

m

-

by2.1.11

s

pt.

Hence the result follows from 2.1.12. 2.1.17 Remark. The Em term only gives us a series of subquotients of r*(X ) 0 Z,, not the group itself. After computing Em one may have to use

other methods to solve the extension problem and recover the group. We close this section with some examples. Let X = HZ, the integral Eilenberg-Mac Lane spectrum. The fundamental cohomology class gives a map f : X + E with H * ( f ) surjective. The fiber o f f is also X, the inclusion map g : X + X having degree p. Hence we get an Adams resolution (2.1.3) with X, = X and K, = E

2.1.18 Example.

56

2. Setting Up the A d a m Spectral Sequence

for all s, the map X

=X , +X , =X

having degree p s . We have then

There is no room for nontrivial differentials so the spectral sequence collapses, i.e., Em=E l . We have E F = Z / ( p )=i m r,,(Xs)/im r 0 ( X r + , )In . this case 2= HZ,,the Eilenberg-Mac Lane spectrum for Z,. Let X = H Z / ( p ' ) with i > 1. It is known that H * ( X ) = H * ( Y ) @ X H * ( Y ) as A-modules, where Y = H Z . This splitting arises from the two right-hand maps in the cofiber sequence 2.1.19 Example.

Y-, Y + X + X Y , where the left-hand map has degree p'. Since the El-term of the Adams spectral sequence depends only on H * ( X ) as an A-module, the former will enjoy a similar splitting. In the previous example we effectively showed that if if

t=s

tZs.

It follows that in the spectral sequence for X we have if r--s=Oor 1 otherwise.

In order to give the correct answer we must have EL' = 0 if t - s = 1 and EZf = 0 if t = s for all but i values of s. Multiplicative properties of the spectral sequence to be discussed in Section 3 imply that the only way we can arrive at a suitable E,mterm is to have d,: E:,'+'+ Et+'s'+' nontrivial for all s 2 0. A similar conclusion can be drawn by chasing the relevant diagrams. Let X be the fiber in X + g + H Z , where the right-hand map is the fundamental integral cohomology class on So. Smashing the above fibration with X we get

2.1.20 Example.

x

A

x

~

O

-

X X ~A

HZ.

It is known that the integral homology of X has exponent p , so X A H Z is a wedge of E and H * ( f , ) is surjective. Similar statements hold after smashing with X any number of times, so we get an Adams resolution (2.1.3) with K , = X , A H Z and X , = X ( * + l ) the , (s+ l)-fold smash product of X with itself, i.e., one of the form

X-XAX-XAXAXt-."

XAHZ XAXAHZ.

2. A d a m Spectral Sequence Based on Homology Theory

57

Since X is (2p -4)-connected, X , is ( ( s + 1)(2p -3) - 1)-connected, so

lim X,s is contractible.

2. The Adams Spectral Sequence Based on a Generalized Homology Theory In this section we will define a spectral sequence similar to that of 2.1.1 (the classical Adams spectral sequence) in which the mod ( p ) EilenbergMac Lane spectrum is replaced by some more general spectrum E. The main example we have in mind is of course E = BP, the Brown-Peterson spectrum, to be defined in 4.1.12. The basic reference for this material is Adams [4] (especially section 15), which includes the requisite preliminaries on the stable homotopy category. Our spectral sequence should have the two essential properties of the classical one: it converges to a,(X) localized or completed at p and its E,-term is a functor of E * ( X ) (the generalized cohomology of X ) as a module over the algebra of cohomology operations E * ( E ) ; i.e., the E,-term should be computable in some homological way, as in 2.1.1. Experience has shown that with regard to the second property we should dualize and consider instead E , ( X ) (the generalized homology of X ) as a comodule over E,(E) (sometimes referred to as the coalgebra of cooperations). In the classical case, i.e., when E = H Z / ( p ) , E , ( E ) is the dual Steenrod algebra A , . Theorem 2.1.1(a) can be reformulated as E, = ExtA*(Z/(p ) , H , ( X ) ) using the definition of Ext in the category of comodules given in A1.2.3. In the case E = BP substantial technical problems can be avoided by using homology instead of cohomology. Further discussion of this point can be found in Adams [6], pp. 51-55. Let us assume for the moment that we have known enough about E and E , ( E ) to say that E , ( X ) is a comodule over E,( E ) and we have a suitable definition of ExtE*(E)( E , ( S o ) , E , ( X ) ) , which we abbreviate as Ext(E,(X)). Then we might proceed as follows.

-/,I

2.2.1 Definition. An E,-Adams resolution for X is a diagram

x = x, hi Kcl

gn

g1 x,+--. * . x,-

K,

f21 K2

such that for all s 2 0 the following conditions hold.

58

2. Setting Up the Adams Spectral Sequence

( a ) X,,, is thejber offs. ( b ) E A X , is a retract of E A K , , i.e., there is a map h,y:E A K,5+ E A X, such that ( E Af,)h, = E A X,. In particular E,(fs) is a monomorphism. (c) K , is a retract of E A K,.

(4 As we will see below, conditions (b) and (c) are necessary to insure that the spectral sequence is natural, while ( d ) is needed to give the desired &-term. As before it is convenient to consider a spectrum with the following properties. 2.2.2 Definition. A n E-completion

2 of X

is a spectrum such that

( a ) There is a map X + 2 inducing an isomorphism in E,-homology. with !&I gS= pt. ( b ) 2 has an E,-Adams resolution {2s} This is not necessarily the same as the 2 of 2.1.14, which will be denoted in this section by X , (2.2.12). Of course, the existence of such a spectrum (2.2.13) is not obvious and we will not give a proof here. Assuming it, we can state the main result of this section. 2.2.3 Theorem (Adams [4]). A n E,-Adams resolution for X (2.2.1 ) leads to a natural spectral sequence E X * ( X ) with d,: E:'+ ES+r.'fr-' such that

( a ) E:'= Ext(E,(X)). ( b ) EZ* is the bigraded group associated with a certainjltration ofn,(g). ( This Jiltration will be described in 2.2.14.) 2.2.4 Definition. 7he spectral sequence of 2.2.3 is the Adams spectral m sequence for X based on E-homology.

2.2.5 Assumptions. We now list the assumptions on E which will enable us to define Ext and 2. (a) E is a commutative associative ring spectrum. (b) E is connective, i.e., .rr,(E) = O for r 1 such that the following diagram commutes K , s + r + , A Kr,i+r+t

I

Ks,v+r A

K,r+r

Ks+r,s+i+r+~

- I

Ks+t,s+t+r

where the vertical maps come from ( a ) . ( e ) The following diagram commutes Ks,,+r A Kr,r+,

( X K + r . s + z r A Kr,r+r) v ( K s , s + r A

Ks + t,s+

X Ks + r +r,s + 1+2r

I r+

r

I

IKr+r,r+lr)

where the vertical maps are those of ( d ) and the horizontal maps come from ( a ) , the maps to and from the wedge being the sums of the maps to and from the summands.

Proof: Part (a) is elementary. For (b) we refer the reader to Cartan and Eilenberg [l], section XV.7, where a spectral sequence is derived from a set of abelian groups H ( p , q ) satisfying certain axioms. Their H ( p , q ) in this case is our T * ( & ~ ) ,and (a) guarantees that these groups have the appropriate properties. For (c) we use the fact that X i = X ' A E('), X:' = X " AI?('), and X,+, = X A E(S+'). For (d) we can assume the maps I?("+')+ E") are all inclusions with K,,,+, = l?(s)/l?(s+r). Hence we have

66

2. Setting Up the Adams Spectral Sequence

and this maps naturally to

For (e) if E(S+')+ E") are inclusions then so is Ks+,,s+2,+K s , s + 2 r SO we have K s , s + r = K s , s + 2 , / K s + r , r +and 2 r K r , t + r =K t , t + ~ r l K r + r , t + 2 , . With this in mind we get a commutative diagram

1 x(

Ks,s+r

Kt+r,r+2r

1

u K s + r , s + ~ rK,r+r) ~

- I

EKs+r+r,s+r+2r

where the horizontal maps come from (d) and the upper vertical maps are inclusions. The lower left-hand map factors through the wedge giving the H desired diagram. We are now ready to prove 2.3.3. In light of 2.3.6(b), the pairing is induced by the maps of 2.3.6(d). Part 2.3.3(a) then follows from 2.3.6(e) as the differential on ET*(X')@"*(X'') is induced by the top map of 2.3.6(e). Part 2.3.3(b) follows from the commutative diagram in 2.3.6(d). Part 2.3.3(c) follows from the compatibility of the maps in 2.3.6(c) and (d). Assuming 2.3.3(e), ( d ) is proved as follows. The pairing on Ext is functorial, so if E , ( X ) has a product which is associative or commutative, so will E ; * ( X ) . Now suppose inductively that the product on E T * ( X ) has the desired property. Since the product on Er+, is induced by that on E, the inductive step follows. It remains then to prove 2.3.3(e). We have E , ( X ' A Ks,s+l)= Ds((E,(X')) (A1.2.11) and similarly for X", so our pairing is induced by a map E*(X'

A Ks,s+,) @=*(E)

E,(X"A Kr,r+,)+ E * ( X

A

KS+1,S+l+l),

i.e., by a pairing of resolutions. Hence the pairing on E2 coincides with the specified algebraic pairing by the uniqueness of the latter (A1.2.14). We prove 2.3.4 by reducing it to the following special case.

2.3.7 Lemma. Theorem 2.3.4 holds when X is such that Ext"(E,(X)) = O for s > 0 and r * ( X )= Exto(E,(X)). H

67

3. Smash Product and Connecting Homomorphism

Proof of 2.3.4. Let W be the fiber of the composite W - fX + X A E . Since X f A h is trivial, h lifts to a map h': Y + X W'. Now consider the cofiber sequence W+X

A

E +X W'+ X W.

Lemma 2.3.7 applies here and gives maps 6,: E ; * ( X W ) + E:"3*(XW).

Composing this with the maps induced by h' gives the desired result.

W

Proof of 2.3.7. Disregarding the notation used in the above proof, let W = 2-l Y, X ' = 2-' Y A E, and Y' = Y A E. Then we have a commutative diagram in which both rows and columns are cofiber sequences X

I

- W - W '

XV(YAE)-

I -I I I

X

i

X'

Each row is the beginning of an Adams resolution (possibly noncanonical for W and X ) which we continue using the canonical resolutions (2.2.9) for W , X ' , and Y'. Thus we get a commutative diagram

2.3.8

w

-

w'AE(2)

W

W'AE

x’

X'AE

x'AE(2)

Y'AE

Y'AE(2'

* * *

I I-I - I I -I I I -

x

y

C-

y'

t--

t-

*..

' * *

in which each column is a cofiber sequence. The map Y Z W' induces maps 6,: E:*( Y )+ E:",*( W ) which clearly satisfy 2.3.4(a) and (b), so we need only to verify that is the connecting homomorphism. The resolutions displayed in 2.3.8 make this verification easy because they yield a short exact sequence of El -terms which is additively (though not differentially)

68

2. Setting Up the Adams Spectral Sequence

split. For s = 0 we have

EF*( W )= r , ( X ) ,

EF*(X)= r , ( X v ( Y A E ) ) , E:.*( W ) = T * ( Y A E ) ,

EF*( Y )= T * ( Y A E ) , E:,*(X)=T,(YAEAE)

-

and

E~s*(Y)=r,(2Y~E~E),

so the relevant diagram for the connecting homomorphism is

X

XV(YAE)

b

YAE

where a and b are splitting maps. The connecting homomorphism is induced by adb, which is the identity on Y A E, which alsoinduces tj2. For s > 0 we have

and

so the relevant diagram is

CEAE

I - I

CEAEAE

X'EA\(')

and again the connecting homomorphism is induced by the identity on

C'YAEAE'".

CHAPTER 3

THE CLASSICAL ADAMS SPECTRAL SEQUENCE

In Section 1 we make some simple calculations with the Adams spectral sequence (ASS) which will be useful later. In particular, we use it to compute T * ( M U )(3.1.5),which will be needed in the next chapter. The computations are described in some detail in order to acquaint the reader with the methods involved. In Sections 2 and 3 we describe the two best methods of computing the ASS for the sphere, i.e., the May spectral sequence and the lambda algebra. In both cases a table is given showing the result in low dimensions (3.2.9 and 3.3.11). Far more extensive charts are given in Tangora [ l , 41. The main table in the former is reproduced in Appendix 3. In Section 4 we survey some general properties of the ASS. We give E;* for s s 3 (3.4.1 and 3.4.2) and then say what is known about differentials on these elements (3.4.3 and 3.4.4). Then we outline the proof of the Adams vanishing and periodicity theorems (3.4.5 and 3.4.6). For p = 2 they say that E:' vanishes roughly for 0 < t - s < 2s and has a very regular structure for t - s < 5s. The &-term in this region is given in 3.4.16. An elementary proof of the nontriviality of most of these elements is given in 3.4.21. In Section 5 we survey some other past and current research and suggest further reading.

1. The Steenrod Algebra and Some Easy Calculations In this section we begin calculating with the classical mod ( p ) Adams spectral sequence of 2.1.1. We start by describing the dual Steenrod algebra 69

70

3. The Classical A d a m Spectral Sequence

A,, referring the reader to Milnor [ 2 ] or Steenrod and Epstein [ I ] for the proof. A, is a graded commutative, noncocommuta-

3.1.1 Theorem (Milnor [2]).

five Hopf algebra. ( a ) For p = 2 A, = P(t,,t 2 ,... ) as an algebra where P( ) denotes a polynomial algebra over Z / ( p ) on the indicated generators, and I&,l= 2” - 1 . The coproduct A:A,+A,OA, is given by Atn = & 5 1 = n t2n’-IOt,,where

eo=1.

( b ) F o r p > 2 , A , = P ( t , , t 2 ,... ) @ E ( T ~T ,, , . . . ) as an algebra, where E ( ) denotes the exterior algebra on the given generators, )(,,I = 2( p“ - I), and [ T I , = 2 p n - 1 . The coproduct A:A,+A,@A, is given by A& = 20,,,,&10t,, where t O = l and A T , = T , , @ ~ + X ~ ~,P ~, ‘ , ~@ T~, . ( c ) For all p there is a unit v: Z/( p ) -+ A, counit F : A, + Z/( p ) (both of which are isomorphisms in dimension 0, and a conjugation (canonical antiautomorphism) c: A, + A, which is an algebra map given recursively by ~ o ~ l ~ n c~ ( &P ) ,= O ~ forn>O l and7,+XOSlcn, $ , ‘ , ~ ( ~ , ) = O f o r n r OA, . will denote ker E ; i.e., A, is isomorphic to A, in positive dimensions, and is trivial in dimension 0.

A, is a commutative Hopf algebra and hence a Hopf algebroid. The homological properties of such objects are discussed in Appendix 1 . We will consider the classical Adams spectral sequence formulated in terms of homology (2.2.3) rather than cohomology (2.1.1). The most obvious way of computing the E2-term is the cobar complex. The following description of it is a special case of 2.2.10 and A1.2.11. 3.1.2 Proposition.

The E2-termfor the classical Adams spectral sequence for P + ( X ) is the cohomology of the cobar complex C:,( H , ( X ) ) dejined by

C>*(H,( X ) ) = A,@

*

. *@A,@ H,( X )

(with s censor factors of A,). For a, E A, and X E H , ( X ) , the element a,@. . . a, O x will be denoted by [a,I a2I * * . I a,]x. The coboundary operator d,: C>*(H,(X))+ C”,+*’(H,(X))is given by d,[a, I

I a,]x = [ l I a, 1. . . I a s ] x E

+ 1 ( - l ) ‘ [ a , I . . . a~-,la~la~Ia,+,l.. .la,Ix ,=I

+

(-1),+l[

a , 1.

*

I a, I X ’ I X ” ,

where ha, = a : O a l and +(x) = X‘@X”EA,@H,(X). [A priori this expression lies in A:”“@ H*( X ) . The diligent reader can verifv that it actually lies in A:’+’@ H,(x).]

1. The Steenrod Algebra and Some Easy Calculations

71

This E,-term will be abbreviated by Ext(H,(X)). Whenever possible we will omit the subscript A,. The following result will be helpful in solving group extension problems in the ASS. For p > 2 let a,€ Exti:(Z/(p), Z / ( p ) )be the class represented by [ rO]E C (Z/ ( p ) ) .The analogous element for p = 2 is represented by [El] and is denoted b y ao, hl,o, or ha. 3.1.3 Lemma. ( a ) For s20, Ext’*”(H,(So)) is generated by a;. ~ is a permanent cycle in the ASS represented by ( b ) I f x Ext(H,(X)) a E rr,(X), then a,x is a permanent cycle represented by pa. [The pairing Ext(H,(S’))@Ext(H,(X)) + Ext(H,(X)) is given by 2.3.3.1 ProoJ Part (a) follows from inspection of C * ( Z / ( p ) ) ;there are no other elements in the indicated bidegrees. For (b) the naturality of the smash product pairing (2.3.3)reduces the problem to the case x = 1 E Ext( H,(So)), where it follows from the fact that r o ( S o )= Z. The cobar complex is so large that one wants to avoid using it directly at all costs. In this section we will consider four spectra ( M O , MU, bo, and bu) in which the change of rings isomorphism of A1.1.18 can be used to great advantage. The most important of these for our purposes is MU, so we treat it first. The others are not used in the sequel. Much of this material is covered in chapter 20 of Switzer [l]. The computation of rr,(MU) is due independently to Milnor [4] and Novikov [2,3]. For the definition and basic properties of MU, including the following lemma, we refer the reader to Milnor [4] or Stong [ l ] or to Section 4.1. 3.1.4 Lemma. ( a ) H,(MU; Z ) = Z[b,, b,, . . . 1, where bi E H2,. ( 6 ) Let H/(p ) denote the mod ( p ) Eilenberg- Mac Lane spectrum for a prime p and let u : M U - , H / ( p ) be the Thom class, i.e., the generator of H o ( M U ;Z / ( p ) ) . Then H,(u) is an algebra map and its image in H , ( H / ( p ) ) = A, is P ( E : , E;, . . . ) for p = 2 and P ( & , E,,. . . ) for p>2. The main result concerning MU is the following. 3.1.5 Theorem (Milnor [4], Novikov [2,3]). ( a ) rr,(MU)=Z[x,,x,,. . . I with x i € r T T Z i ( M U ) . ( 6 ) Let h : 7r,(MU)+ H,(MU; Z ) be the Hurewicz map. Then modulo decomposables in H,(M U ; Z), -pb, if i = p k - 1 for some prime p h ( x i )= -bi otherwise.

72

3. The Classical A d a m Spectral Sequence

We will prove this in essentially the same way that Milnor and Novikov did. After some preliminaries on the Steenrod algebra we will use the change of rings isomorphisms A1.1.18 and A1.3.13 to compute the &-term (3.1.10). It will follow easily that the spectral sequence collapses; i.e., it has no nontrivial differentials. To compute the &-term we need to know H , ( M U ; Z / ( p ) )as an A,comodule algebra. Since it is concentrated in even dimensions, the following result is useful. 3.1.6 Lemma. Let M be a left A,-comodule which is concentrated in even dimensions. Then M is a comodule over P, c A , dejned as follows. For p > 2, P * = P ( ( , , &, . . ) and f o r p = 2 , P, = P ( ( : , t:,. . . ).

Proof: For m E M , let +( m ) = Z m ' O m". Then each m' E A, must be evendimensional and by coassociativity its coproduct expansion must consist entirely of even-dimensional factors, which means it must lie in f,. 3.1.7 Lemma. A s a left A,-comodule, H,(MU) = P*@C, where C = P( u, , u2,. . . ) with dim ui = 2i and i is any positive integer not of the form p k -1.

Proof: H , ( M U ; Z / ( p ) ) is a P,-comodule algebra by 3.1.4 and 3.1.6. It maps onto P, by 3.1.4(b), so by A1.1.18 it is f,OC, where C = Z/(p)O,,H,(MU). An easy counting argument shows that C must have the indicated form. 3.1.8 Lemma. Let M be a comodule algebra over A , having the form P, 0 N for some A,-comodule algebra N. Then E x t ~ * ( Z / ( p )M , ) =EXtE(Z/(P). Jf’)

where

In particular, ExtA*(Z/( P ) , H,(MU))=ExtEfZ/(p), Z / ( p f ) @ C . Roo$ The statement about H , ( M U ) follows from the general one by 3.1.7. For the latter we claim that M = A , O FN. We have A , = P,@ E as vector spaces and hence as E-comodules by A1.1.20, so

A,

O E

N

= f,O

E 0E N

and the result follows from A1.3.13.

= P,O

N

= M,

73

1. The Steeorod Algebra and Some Easy Calculations

Hence we have reduced the problem of computing the Adams &-term for MU to that of computing ExtE(Z / ( p ) , Z/( p ) ) . This is quite easy since E is dual to an exterior algebra of finite type. 3.1.9 Lemma. Let r be a commutative, graded connected Hopf algebra of finite type over a field K which is an exterior algebra on primitive generators x , , x2,. . . , each having odd degree i f K has characteristic other than 2 (e.g., let r =E ) . Then Extl(K,K)=P(yi,y2,...), where y , E Ext13'"i'is represented by [ x , ] in C , ( K ) (the cobar complex of A1.2.11).

Roo$ Let I ' , c T be the exterior algebra on x , . Then an injective resolution of K is given by

o+

r,-

d

K + r, + r n +r,-+

where d ( x , ) = 1 and d(1) = 0 applying Hom,,(K, ) gives a complex with trivial boundary operator and shows Ext,,(K, K ) = L[ Y,].Tensoring all the R, together gives an injective r-resolution of K and the result follows from the Kunneth theorem. Combining the last three lemmas gives

3.1.10 Corollary. Ext,*(Z/(p), H , ( M U ) ) = c O P ( a , , a , , . . . 1, where C is as in 3.1.9 and a , E Ext'*2p'-1is represented by 9 C,*(ff*(MW.

[T,]f o r p > 2

and

[&I for P = 2 in

Thus we have computed the &-term of the classical Adams spectral sequence for n-,(MU). Since it is generated by even dimensional classes, i.e., elements in E:' with f - s even, there can be no nontrivial differentials, i.e., E2 = Em. The group extension problems are solved by 3.1.3; i.e., all multiples of a: are represented in n-,(MU) by multiples of p s . It follows that n-,(MU)OZ(,, is as claimed for each p ; i.e., 3.1.5(a) is true locally. Since n , ( M U ) is finitely generated for each i, we can conclude that it is a free abelian group of the appropriate rank. To get at the global ring structure note that the mod ( p ) indecomposable quotient in dimension 2i, Q 2 , n - , ( M U ) C 3 Z / ( p ) ,is Z / ( p ) for each i > O so Q2,n,(MU) = Z. Pick a generator x, in each even dimension and let R = Z[x, ,x 2 , . . . 1. The map R + n,( M U ) gives an isomorphism after tensoring with Z(p)for each prime p, so it is an isomorphism globally.

74

3. The Classical A d a m Spectral Sequence

To study the Hurewicz map h: T,( M U ) H,( M U ; Z) recall H,( X ;Z ) = A H), where H is the integral Eilenberg-Mac Lane spectrum. We will prove 3.1.5(b) by determining the map of Adams spectral sequences induced by i : M U + M U A H. We will assume p > 2, leaving the obvious changes for p = 2 to the reader. The following result on H , ( H ) is standard. -f

P,@

3.1.11 Lemma. spectrum

The mod ( p ) homology ofthe integer Eilenberg- Mac Lane

H*( H ) = P* 0E ( 6,,? 2 , . . . ) as an A, comodule, where Ti denotes the conjugate of the conjugation c. H

T,,

i.e., its image under

Hence we have

H * ( W = A, OWso)Z / ( P ) and an argument similar to that of 3.1.8 shows

3-1.12

EXtA*(Z/(p), H*(X

A

H ) ) = ExtE(To,(Z/(P),H * ( X ) ) .

In the case X = M U the comodule structure is trivial, so by 3.1.11, ExtA*(Z/(p), H,(MU

A

H ) )= H , ( M U ) O P ( a d .

To determine the map of Ext groups induced by i, we consider three cobar complexes, CA*(H,(MU)), C,( C),and CE(ro)( H,( M U ) ) .The cohomologies of the first two are both ExtA,(Z/(p), H , ( M U ) ) , by 3.1.2 and 3.1.8, respectively, while that of the third is ExtA*(Z/(p),H,(MU A H)) by 3.1.12. There are maps from CA ( H , ( M U ) ) to each of the other two. The class a, E Extf;zI-'(Z/( p ) , H , ( M U ) ) is represented by [T,] E C,( C). The element - X i [ T i ] 5 < L i ~ CA*(H , ( M U ) ) [using the decomposition of H , ( M U ) given by 3.1.71 is a cycle which maps to [T,] and therefore it also , , we have i*(a,)= represents a,. Its image in C , , , , ( H , ( M U ) ) is [ ~ ~ ] f ,so a,&,. Since 5, E H , ( M U ) is a generator it is congruent modulo decomposables to a nonzero scalar multiple of b , n - , , while ui (3.1.9) can be chosen to be congruent to bi. It follows that the x i € 7rZi(MU)can be chosen to satisfy 3.1.5(b). We now turn to the other spectra in our list, MO, bu, and bo. The Adams spectral sequence was not used originally to compute the homotopy of these spectra, but we feel these calculations are instructive examples. In each case we will quote without proof a standard theorem on the spectrum's homology as an A,-comodule and proceed from there. For similar treatments of MSO, MSU, and MSp see, respectively, Pengelley [2], Pengelley [l], and Kochman [l]. The following result on MO was first proved by Thom [l]. Proofs can also be found in Liulevicius [ 11 and Stong [ 11, p. 95.

75

1. The Steenrod Algebra and Some Easy Calculations

3.1.13 Theorem. For p = 2, H,( MO) = A,@ N, where N is a polynomial algebra with one generator in each degree not of the form 2k - 1. For p > 2, H,(MO)=O. H

It follows immediately that 3.1.14

the spectral sequence collapses and For bu we have

T,(

MO) = N.

3.1.15 Theorem (Adams [8]).

@

H,(bu)=

C2’M

Osi 2

M = P , o E ( ~ , ,...I ,~~

for p = 2

H

where E for a E A, is the conjugate c ( a ) . Using 3.1.8 we get

ExtA*(Z/ ( p ) , M , = ExtE (z/( p),

( 7Z 3

T3

9

. )) * *

(again we assume for convenience that p > 2) and by an easy calculation A1.3.13 gives E X ~ E ( ~ / ( E(72,73,. P), . ))=ExtE(,,,,,(Z/(p), z / ( ~ ) ) = P ( a oai) , by 3.1.11, so we have 3.1.16 Theorem.

Ext4(Z/(p), H , ( b u ) ) = G Z ” P ( a , , a , ) i=O

where aoE Ext’,‘ and a , E Ext’s2P-’.

H

As in the MU case the spectral sequence collapses because the E2-term is concentrated in even dimensions. The extensions can be handled in the same way, so we recover the fact that

n;( bu) =

Z 0

if i r O a n d iiseven otherwise.

The bo spectrum is of interest only at the prime 2 because at odd primes it is a summand of bu (see Adams [S]). For p = 2 we have

76

3. The Classical Adam Spectral Sequence

3.1.17 Theorem (Stong [ 2 ] ) . For p = 2 , H , ( b o ) = P ( g , z , & , ( , ,,... ),

where S;: = c ( t i ) .

w

Let A(1), = A,/(,$, E , &,&, . . . ). We leave it as an exercise for the reader to show that A( l), is dual to the subalgebra A( 1) of A generated by Sq' and Sq2, and that H*(bO)=A*OA(l), z / ( 2 ) ,

so by A1.3.13, 3.1.18

ExtA*(Z/(2), H * ( b o ) )= EXfA(I)*(Z/(2), A/(2))-

A ( l ) is not an exterior algebra, so 3.1.9 does not apply. We have to use the Cartan-Eilenberg spectral sequence (CESS) A1.3.15. The reader can verify that the following is an extension (A1.1.15)

a)

3.1.19

A ( 1 )*

-+

(f 2 )

-+

9

where @ = P(tI)/(5;'). @ is isomorphic as a coalgebra to an exterior algebra on elements corresponding to 6, and .$, so by 3.1.9 Ext*(Z/(Z), Z/(2)) = P(h10, h , ) and E X ~ E , F ~ ) ( ZZ/(2)) / ( ~ ) , = P(h20), where h,, is represented by [if'] in the appropriate cobar complex. Since P(h2,) has only one basis element in each degree, the coaction of @ on it is trivial, so by A1.3.15 we have a CESS converging to E X ~ ~ ( ~ ) + (Z/(2)) Z/(~), with

E2= P(hI0, hll, h2o)

3.1.20

where hIiE E>O and hzoc E:.'. We claim

d2(h20) = ~

3.1.21

I O h I l ~

This follows from the fact that d(52)= &or: in cA(l)*(z/(2)).It follows that

E,= P ( u , h10, h

3.1.22

where

U E

I ) / ( ~ l O h l )

E Y corresponds to h:o. Next we claim d3( u ) = h:, .

3.1.23

We have in cA(l)*(z/(2))?

slOs:o(,.

d(5;Of;) = f20&0t:+

77

1. The Steeorod Algebra and Some Easy Calculations

In this E2 this gives d2h?o=

=o

h , 0 ~ , 1 ~ 2 0 + ~ 2 0 ~ 1 0 ~ 1 1

since E2 is commutative. However, the cobar complex is not commutative and when we add correcting terms to f2C3f2in the hope of getting a cycle, we get instead d ( F 2 0 F2

+ 510&2+

tli2C3t3

=

m s:o t:,

which implies 3.1.20. It follows that 3.1.24

E4=P(h10,

hll,

u, w ) / ( h l O h l l ~

h?l,

u2+h:Ow,

uhll),

where v E E:,, and w E E!*4correspond to h,,h?, and h;o, respectively. Finally, we claim that E4= E m ; inspection of E4 shows that there cannot be any higher diff erentihls because there is no E S.‘ for r 2 4 which is nontrivial and for which Es+r7t-rc1IS . also nontrivial. There is also no room for any nontrivial extensions in the multiplicative structure. Thus we have proved 3.1.25 Theorem. The E,-term for the mod ( 2 ) A S S f o r g*(bo),

EXtA,( z/ (21, H*( b o ) )= EXtA(1)*W/(2), z/ (2)) is

P(h11, h l l , u, w ) / ( ~ 1 0 ~ 1h:, 1 ,, u2+ h:oW uh,,), where

h , , Ext’.’, ~ h , , E Ext’.’,

u E Ext3,’,

and

w E Ext4*I2.

H

This E2-term is displayed in the accompanying figure. A vertical arrow over an element indicates that hiox is also present and nontrivial for all s > 0.

t

V

0

1 0

1

2

3

4

6

5 t-s

7

8

9

10

78

3. The Classical A d a m Spectral Sequence

Now we claim that this ASS also collapses,si.e., E 2 = Em. Inspection ) ~"h;:~. shows that the only possible nontrivial differential is d r ( w " h , , = However, 60 is a ring spectrum so by 2.3.3 the differentials are derivations and we cannot have d , ( h , , ) = hi," because it contradicts the relation h,,h,, =O. The extension problem is solved by 3.1.3, giving 3.1.26 Theorem (Bott [ l ] ) .

.rr*(bo) = a77, a,P1/(277, T 3 ,w ,a 2 - 4 P )

r:

w i t h q E . r r 1 , a E . r r . , , P E . r r g , i.e., f o r i 2 0

if i = 0 mod 4 if i ~ l o r 2 r n o d 8 otherwise.

.rri(60)= Z/2

For future reference we will compute E x ~ ~ ( ~ ) ( ZM) / ( for ~ )M , = A(O), = E ( & ) and M = Y = P ( ( , ) / ( t ; ) .Topologically these are the ASS E2-terms for the mod (2)-Moore spectrum smashed with 60 and bu, respectively. We use the CESS as above and our E2-term is Exta(Z/(2), ExtE(c2)(Z/(2), M)). An easy calculation shows that Ez=P(h1,,hzo)

for

M=A(O),

and E 2 =P ( h 2 , )

for

M = Y.

In the latter case the CESS collapses. In the former case the differentials are not derivations since A(O), is not a comodule algebra. From 3.1.23 we get d,(h:,) = h:, , so Em=Eq= P(w)O{l, h i i , h ? i , h20, h2oh11, h20h:iI. This Ext is not an algebra but it is a module over ExtA(,)*(Z/(2),Z/(2)). We will show that there is a nontrivial extension in this structure, namely h10h2,= h?,. We d o this by computing in the cobar complex CA(l)*(A(O).+). There the class h,, is represented by [5z]+[5f]51, so hI0h2,is represented by [511521+[t1[5~1&. The sum of this and [tj:l&:] (which represents hi.,) is the coboundary of [ t15,] + [ 6: + t2]5,. From these considerations we get 3.1.27 Theorem. As a module ouer E x ~ ~ ( ~ ) * ( ZZ/(2)) / ( ~ ) ,(3.1.25) we have ( a ) ExtA(,)*(Z/(2),A(O),) is generated by 1 E Exto.' and h2,e ExtIs3with h,, 1 =0, hlohzO=h:, . 1, u 1 =0, and ~ h ~ ~ = 0 . (6) ExtA(I)*(Z/(2), Y ) is generated by { h i 0 : 0 5 i 5 3 } with hioh;"= h,,hi,= vhi,=O.

-

-

79

2. The May Spectral Sequence

We will also need an odd primary analog of 3.1.27(a). A ( l ) = E ( T,, T ~0 ) P( tl)/(ef) is the dual to the subalgebra of A generated by the Bockstein p and the Steenrod reduced power PI. Instead of generalizing the extension 3.1.19 we use P(O)*+A(O),+~(1)*, where P(O), = P ( ( , ) / ( & f )and E ( l ) , = E ( T , , 7,).The CESS &-term is therefore ExtP(,,*(Zl(P), EXtE(I)*(Z/(P),N O ) , ) ) , where A(O),

= E ( T ~ )An .

easy calculation gives

3.1.28 Theorem. For p > 2 EXfA(,)*(Z/(P),N O ) , )

= E ( h o ) O P ( a , ,bo),

and b, E Ext2,p4 are represented by [ where hoE Ext'*4, a, E Ext H Z,,icp p - ' ( y ) [ f ; I [:-'I, respectively.

el],

[ ~ , ] T ~ + [ T , ]and ,

2. The May Spectral Sequence In this section we discuss a method for computing the classical Adams E,-term, ExtA*(Z/(p),Z/( p ) ) , which we will refer to simply as Ext. For the reader hoping to understand the classical Adams spectral sequence we offer two pieces of advice. First, do as many explicit calculations as possible yourself. Seeing someone else do it is no substitute for the insight gained by firsthand experience. The computations sketched below should be reproduced in detail and, if possible, extended by the reader. Second, the ,!?,-term and the various patterns within it should be examined and analyzed from as many viewpoints as possible. For this reason we will describe several methods for computing Ext. For reasons to be given in Section 4.4, we will limit our attention here to the prime 2. The most successful method for computing Ext through a range of dimensions is the spectral sequence of May [ 11. Unfortunately, crucial parts of this material have never been published. The general method for computing Ext over a Hopf algebra is described in May [2], and the computation of the differentials in the May spectral sequence for the Steenrod algebra through dimension 70 is described by Tangora [l]. A revised account of the May E2-term is given in May [4]. In our language May's approach is to filter A, by copowers of the unit coideal (A1.3.10) and to study the resulting spectral sequence. Its E2-term

80

3. The Classical A d a m Spectral Sequence

is the Ext over the associated graded Hopf algebra E'A,. The structure of this Hopf algebra is as follows. 3.2.1 Theorem (May [l]).

( a ) Forp=2, EoA,=E(&,:i>0,j20) with coproduct

1

' ( t I , J ) =

61-k,J+k@&,J,

Oiksr

where ifo,, = 1 and ( b ) Forp>2,

E

EYA, is the projection of = E ( T , : i 5 O)@

EOA,

sf'

T ( & J: i > 0 , j 2 0 )

with coproduct given by

1

tr-k,j+k@6k,]

01kit

and

A ( T , ) = T ~ @ ~ +1

tz-kt@~,),

Osksr

where T ( ) denotes the truncated polynomial algebra of height p on the indicated generators, T~ E Ey+,A, is the projection of E A , , and & J E E:A,* is the projection of 5:’. H May actually filters the Steenrod algebra A rather than its dual, and proves that the associated bigraded Hopf algebra EoA is primitively generated, which is dual to the statement that each primitive in EoA,* is a generator. A theorem of Milnor and Moore [3] says that every graded primitively generated Hopf algebra is isomorphic to the universal enveloping algebra of a restricted Lie algebra. For p = 2 let x,,, E EoA be the primitive dual to & J . These form the basis of a Lie algebra under commutation, i.e., LXt,J?

xk,TVtl

Xt,JXk,m

-Xk,rnX,,J

= '!kXl,rn

rn -'t

xk,J

where 8: Is the Kronecker 6. A restriction in a graded Lie algebra L is an endomorphism 6 which increases the grading by a factor of p. In the case at hand this restriction is trivial. The universal enveloping algebra V( L ) of a restricted Lie algebra L (often referred to as the restricted enveloping algebra) is the associative algebra generated by the elements of L subject to the relations xy - yx = [x, y ] and x p = (( p ) for x, y E L. May [ l ] constructs an efficient complex (i.e., one which is much smaller than the cobar complex) for computing Ext over such Hopf algebras. In particular, he proves

81

2. The May Spectral Sequence

3.2.2 Theorem (May [I]). For p =2, E ~ t ~ ~ ~ ~ ( Z/(2)) z / ( 2 )(the , third grading being the M a y jiltration) is the cohomology of the complex V*** = P ( h,,, : I > 0, j with

dh,,, = E A;.

el,,

hk,h,-kk+,, where

2 0)

h,,, E V’,2J(2‘-1’,1 corresponds to

Our h,,, is written R: by May [ l ] and R,, by Tangora [l], but as h,, (in a slightly different context) by Adams [3]. We will omit the comma in the subscript whenever practical. Notice that in C*(Z/(2)) one has d[lf’] = & 0 , j 2 0)O P ( b,, : i > 0,j 2O)O P ( a ,: i z O ) ,

where

, h,, E~s2(P'-l)P',2~-l b,, E:.2(Pt-l)P'+'.P(2'-l) 9

and

a, Ei,2P'p1.21+1 (h,,, and a, correspond respectively to (f' and T,).One has d, : E S,"" + E:-l.'.U-r, and i f x ES,'," ~ then d,(xy) = d,(x)y+(-l)'xd,(y). d , is given by hk,,hr-k,k+J,

dI(hI,J)=-

O:(Z/(2), Z/(2)) for t - s s 13 and s s 7 is generated as a vector space by the elements listed in the accompanying table. (There are no generators for t - s = 12 and 13, and the only generators in this range with s > 7 are powers of h,.) In the table c, corresponds to h,x,, while Rx corresponds to b:,o~.There are relations h:= hih,, h l = hfh,, and Phi=Phih2= hiPh,.

3. The Classical Adams Spectral Sequence

Ph:

,Ph:

Ph2

Ph1

h l =o

hih3

h:

h:

hl

h:h3

h'2

hoh2

h2

hOh3

Phoh2

co

h:h3

hlh3

h3

I 0

2

1

3

4

5

6

7

8

9

10

11

Inspecting this table one sees that there are no differentials in the Adams spectral sequence in this range, and all of the group extensions are solved by 3.1.3 and we get

3.2.12 Corollary. For n 5 13 the two component of.rr,(So) are given by the foilowing table.

In general the computation of higher May differentials is greatly simplified by the use of algebraic Steenrod operations (see Section A1.5). For details see Nakamura [ 11. Now we will use the May SS to compute ExtA(*)*(Z/(2), A(O),), where A( n)* = P ( [ , , t 2 ,. . . , S.+,)/(S:""-') is dual to the subalgebra A( n ) c A generated by Sql, S q 2 , .. . , Sq". We filter A ( 2 ) , just as we filter A,. The ~ 0, ~=)d1(h20), resulting May E,-term is P ( h , , , h , , , h,,, h,, , h30)with d , ( l ~ = dl(h2]= ) h l l h I 2and , d,(h,,) = h2,hI,. This gives 3.2.13

E2=

P(b21, b 3 0 ) 0 ( ( P ( h l l ,h 2 0 ) 0 E ( X , ) ) 0 ( h l 2 : i > O } ) ,

87

2. Tbe May Spectral Sequence

where bZ1= h:, , b30= hio, and x, = hllh30+ hzoh2,.The d2's are trivial except for 3.2.14

d2(hi0)= h i , ,

d z ( b , , )= h i 2 ,

and

dZ(b30)= hllb2l

Since A(O), is not a comodule algebra this is not an S S of algebras, but there is a suitable pairing with the May S S for ExtA(,,*(Z/(2), Z/(2)). Finding the resulting ,!?,-term requires a little more ingenuity. In the first place we can factor out P( b:o), i.e., E2 = Ez/(b:o)OP(b:o)'as complexes. We denote E2/(b:o)by l?, and give it an increasing filtration as a differential algebra by letting F o = P ( h l l ,h20)OE(x,)O{hf2: i > O } and letting 621, b30 E Fl . The cohomology of the subcomplex Fo is essentially determined by 3.1.27(a), which gives E x ~ ~ ( ~ ) * ( ZA(O),). / ( ~ ) , Let B denote this object suitably regraded for the present purpose. Then we have 3.2.15

H * ( F o ) = B O E ( x 7 ) O { h f 2i >: 0).

For k>Owehave F k / F k - l = ( b : l , b ~ ; ' b 3 0 } O F O ~ i t h d Z ( b ~ ~ 1 bb2klhll. 30)= Its cohomology is essentially determined by 3.1.27(b), which describes ExtA(,)*(Z/(2),Y ) . Let C denote this object suitably regraded, i.e., C = P(hZ0).Then we have for k > 0

This filtration leads to an S S converging to differential sends

in which the only nontrivial

for E = 0, 1, k > 0, and i 2 1. This is illustrated in Fig. 3.2.17(a), where a square indicates a copy of B and a large circle indicates a copy of C. Arrows pointing to the left indicate further multiplication by hI2,and diagonal lines indicate differentials. Now bzl supports a copy of C and a differential. This leads to a copy of C in E3 supported by h20b21shown in 3.2.17(b). There is a nontrivial multiplicative extension h20h12b30 = x7bZ1which we indicate by a copy of C in place of hI7b30 in (b). Fig. 3.2.17(b) also shows the relation h, lb&= hiZb30. The differentials in E3 are generated by d3(b&) = hI2b:,and are shown in 3.2.17(c). The resulting E4= Em is shown in 3.2.17(d), where the symbol in place of b:o indicates a copy of B with the first element missing.

h

R X

1

0

0

0

0 0

0

0

-y

0 0

0

0

0

90

3. The Classical Adnms Spectral Sequence

3. The Lambda Algebra In this section we describe the lambda algebra of Bousfield el al. [2] at the prime 2 and the algorithm suggested by it for computing Ext. For more details, including references, see Tangora [2,3]. For most of this material we are indebted to private conversations with E. B. Curtis. It is closely related to that of Section 1.5. The lambda algebra A is an associative differential bigraded algebra whose cohomology, like that of the cobar complex, is Ext. It is much smaller than the cobar complex; it is probably the smallest such algebra generated by elements of cohomological degree one with cohomology isomorphic to Ext. Its greatest attraction, which will not be exploited here, is that it contains for each n > 0 a subcomplex A( n ) whose cohomology is the E2-term of a spectral sequence converging to the 2-component of the unstable homotopy groups of S". In other words A ( n ) is the &-term of an unstable Adams spectral sequence. More precisely, A is a bigraded Z/(2)-algebra with generators A, E A',"+' ( n 2 0 ) and relations n-j-1 3.3.1 A i A 2 i + 1 + + n = jr0 C )Al+n-jA2i+l+, for i, n r O

(

with differential 3.3.2 Note that d behaves formally like left multiplication by A _ , . 3.3.3 Definition. A monomial A,,A, . . A,y E A is admissible if 2i, 2 ir+' for 1Ir < s. A( n ) c A is the subcomplex spanned by the admissible monomials with il < n. The following is an easy consequence of 3.3.1 and 3.3.2. 3.3.4 Proposition.

The admissible monomials constitute an additive basis for A. There are short exact sequences of complexes (b) (a)

O + A ( n )+ A ( n + 1 ) + X n A ( 2 n + 1) +O.

The significant property of A is the following. 3.3.5 Theorem (Bousfield et al. [2]). ( a ) H ( A )= ExtA*(Z/(2),Z/(2)), the classical A d a m &-term for the sphere.

91

3. The Lambda Algebra

( b ) H ( h (n ) ) is the E,-term of a spectral sequence converging to T*(S"). ( c ) The long exact sequence. in cohomology (3.3.6) given by 3.3.4(b) corresponds to the EHPsequence, i.e., 10 the long exact sequence of homotopy groups of the fiber sequence ( a t the prime 2 )

S" -3 as"+'+ Q p + '

(see 1.5.1).

H

The long exact sequence in (c) above is

3.3.6

2 H s * ' ( A ( +n 1))

+H"'(A(n))

Hs-':'-n-l ( A ( 2 n+ 1 ) )

The letters E, H, and P stand respectively for suspension (Einhungung in German), Hopf invariant, and Whitehead product. The map H is obtained by dropping the first factor of each monomial. This sequence leads to an inductive method for calculating H " . ' ( h ( n ) )which we will refer to as the Curtis algorithm. Calculations with this algorithm up to t = 51 (which means up to t - s = 3 3 ) are recorded in an unpublished table prepared by G. W. Whitehead. Recently, Tangora [4] has programmed a computer to find H" ' ( A ) at p = 2 for t 5 48 and p = 3 for t 5 9 9 . Some related machine calculations are described by Wellington [13. For the Curtis algorithm, note that the long exact sequences of 3.3.6 for all n constitute an exact couple (see Section 2.1) which leads to the following spectral sequence, similar to that of 1.5.7.

3.3.7 Proposition (algebraic EHP spectral sequence).

( a ) There is a trigraded spectral sequence converging to H"'(A) with Ef.'" = Hs-'7'-"(A(2n- 1)) for s > 0 and =

{

:/(*)

for t = n = O otherwise,

and d, ES,'," Es-1.l.n-r ( b ) For each m > 0 there is a similar spectral sequence converging to H " " ( A ( m ) )with as above for n 5 m Es,',n = 0 for n < m. ~

{

The EHP sequence in homotopy leads to a similar spectral sequence converging to stable homotopy filtered by sphere of origin which is described in Section 1.5. At first glance the spectral sequence of 3.3.7 appears to be circular in that the &-term consists of the same groups one is trying to compute.

92

3. The Classical A d a m Spectral Sequence

However, for n > 1 the groups in E;"" are from the ( t - s - n + 1)-stem, which is known by induction on t-s. Hence 3.3.7(b) for odd values of m can be used to compute the El-terms. For n - 1, we need to know H*(A(l)) at the outset, but it is easy to compute. A(1) is generated simply by the powers of A. and it has trivial differential. This corresponds to the homotopy of s'. Hence the EHP spectral sequence has the following properties. 3.3.8 Lemma. In the spectral sequence of 3.3.7(a), ( a ) E:'" = 0 for t - s < n - 1 (vanishing line); ( b ) ES,'," = Z / ( 2 )for t - s = n - 1 and all s 2 0 and if in addition n - 1 is is nontrivial for all s L 0 (diagonal even and positive, d , :E;*,"+ E;+17f7n-1 groups ) ; ( c ) Eft = H s - l . 1 - n ( A )for t - s < 3n (stable zone); and ( d ) E:','=O for t > s .

ProoJ The groups in (a) vanish because they come from negative stems in A ( 2 n - 1 ) . The groups in ( b ) are in the zero-stem of A ( 2 n - 1 ) and correspond to An-lAi-l E A. If n - 1 is even and positive, 3.3.2 gives d (A,- ,A:-')

= A n - 2 A ; ) mod A( n - 2),

which means d , behaves as claimed. The groups in (c) are independent of n by 3.3.6. The groups in ( d ) are in A(1) in positive stems. The above result leaves undecided the fate of the generators of ECl)."-'*" for n - 1 odd, which correspond to the A,_I. We use 3.3.2 to compute the differentials on these elements. (See Tangora [ 2 ] for some helpful advice on dealing with these binomial coefficients.) We find that if n is a power of 2, AnPl is a cycle, and if n = k * 2, for odd k > 1 then d ( A n - l ) = A , _ , - ~ J A ~ I -m , odA(n-1-2').

This equation remains valid after multiplying on the right by any cycle id A, so we get 3.3.9 Proposition. In the S S o f 3.3.7( a ) every element in E:'"' is apermanent cycle. For n = k2' for k > 1 odd, then every element in E : f , k 2 J is a d,-cycle for r 1. For t - s = 1 we have E:32.2=Z/(2), corresponding to A , or h l , while E;l+s*n = 0 for all other s and n. For this and 3.3.8(c) we get E:,n+2*n =2/(2) generated by A,_,Al for all n 2 2, while E:"'-' = 0 for all other s, t The element 11 cannot be hit by a differential because 3 is a cycle, so it survives to a generator of the 2-stem, and it gives generators of E:,n+4,n(correspond= 0 for all ing to elements with Hopf invariant 11) for n ? 2, while Es,r3r-sp1 other s and t. This brings us to t - s - 3 . In addition to the diagonal groups Es,s+3.4 given by 3.3.8(b) we have E:35.3generated by 21 and E:36.2generated by 111, with n o other generators in this stem. These two elements are easily seen to be nontrivial permanent cycles, so H s , s + 3 ( Ahas ) three generators: 3,21, and 111. Using 3.3.1 one sees that they are connected by left muitiplication by 0 (i.e., by A,,). Thus for t - s 5 3 we have produced the same value of Ext as given by the May spectral sequence in 3.2.1 1. The relation h:h2 = h: corresponds to the relation 003 = 111 in A, the latter being easier to derive. So far no differentials have occurred other than those of 3.3.8(b). These and subsequent calculations are indicated in Fig. 3.3.10, which we now describe. The gradings t - s and n are displayed; we find this more illuminating than the usual practice of displaying I- s and s. All elements in the spectral sequence in the indicated range are displayed except the infinite towers along the diagonal described in 3.3.8(b). Each element (except the diagonal generators) is referred to by listing the leading term of its Hopf invariant with respect to the left lexicographic ordering; e.g., the cycle 4111 +221+ 1123 is listed in the fifth row as 111. An important feature of the Curtis algorithm is that it sufices to record the leading term of each element. We will illustrate this principle with some examples. For more discussion see Tangora [3]. The arrows in the figure indicate differentials in the spectral sequence. Nontrivial cycles in A for 0 < t - s < 14 are listed at the bottom. We d o not list them for t - s = 14 because the table does not indicate which cycles in the 14th column are hit by differentials coming from the 15th column.

94

3. The Classical A d a m Spectral Sequence

Figure 3.3.10. The EHP spectral sequence (3.3.7) for t - s c 14.

3.3.11 Example. Suppose we are given the leading term 41 1 1 of the cycle above. We can find the other terms as follows. Using 3.3.1 and 3.3.2 we find d(4111)=21111. Referring to Fig. 3.3.10, we find 1111 is hit by the differential from 221, so we add 2221 to 4111 and find that d(4111+2221) =

95

3. The Lambda Algebra

11121. The figure shows that 121 is killed by 23, so we add 1123 to our expression and find that d (4111+ 2221 + 1123) = 4; i.e., we have found all of the terms in the cycle. Now suppose the figure has been completed for t - s < k. We wish to fill in the column t - s = k. The box for n = 1 is trivial by 3.3.8(d) and the boxes for n 2 3 can be filled in on the basis of previous calculations. (See 3.3.12.) The elements in the box for n = 2 will come from the cycles in the box for n = 3, t - s = k - 1, and the elements in the box for n = 2 , t - s = k- 1 which are not hit by d,'s. Hence before we can fill in the box for b = 2, t - s = k, we must find the d,'s originating in the box for n = 3. The procedure for computing differentials will be described below. Once the column t - s = k has been filled in, one computes the differentials for successively larger values of n. The above method is adequate for the limited range we will consider, but for more extensive calculations it has a serious flaw. One could work very hard to show that some element is a cycle only to find at the next stage that it is hit by an easily computed differential. In order to avoid such redundant work one should work by induction on t, then on s and then on n ; i.e., one should compute differentials originating in ES,*,"only after one has done so for all E S'""'' with t' < t, with t' = t and s' < s, and with s' = s, t' = t, and n’ < n. This triple induction is awkward to display on a sheet of paper but easy to write into a computer program. The procedure for finding differentials in the EHP spectral sequence (3.3.7) is the following. We start with some sequence (Y in the ( n + 1)th row. Suppose inductively that some correcting terms have already been added to hna, in the manner about to be described, to give an expression x. We use 3.3.1 and 3.3.2 to find the leading term i l i z . . . is+l of d ( x ) . If d ( x ) = 0 , then our (Y is a permanent cycle in the spectral sequence. If not, then beginning with u = 0 we look in the table for the sequence is-u+, is-u+2 . . is+l in the ( + 1)th row until we find one that is hit by a differential from some sequence p in the ( m+ 1)th row or until u = s - 1. In the former event we add Ail * * Ais-u-,Amp to x and repeat the process. The coboundary of the new expression will have a smaller leading term since we have added a correcting term to cancel out the original leading coboundary term. If we get up to u = s - 1 without finding a target of a differential, then it follows that our original (Y supports a dnpi1whose target is i2 * is+, . It is not necessary to add all of the correcting terms to x to show that our (Y is a permanent cycle. The figure will provide a finite list of possible targets for the differential in question. As soon as the leading term of d ( x ) is smaller (in the left lexicographic ordering) than any of these candidates, then we are done.

-

-

96

3. The Classical A d a m Spectral Sequence

* . . is+l in the In practice it may happen that one of the sequences + 1)th row supports a nontrivial differential. This would be a contradiction indicating the presence of an error, which should be found and corrected before proceeding further. Inductive calculations of this sort have the advantage that mistakes usually reveal themselves by producing contradictions a few stems later. Thus one can be fairly certain that a calculation through some range that is free of contradictions is also correct through most of that range. In publishing such computations it is prudent to compute a little beyond the stated range to ensure the accuracy of one's results. We now describe some sample calculations in 3.2.11.

(is-,,

3.3.12 Example. Elling in the table. Consider the boxes with t-S-

( n - 1 ) = 8.

To fill them in we need to know the 8-stem of H ( h ( 2 n - 1)). For convenience the values of 2n - 1 are listed at the extreme left. The first element in the 8-stem is 233, which originates on S 3 and hence appears in all boxes for n 2 2 . Next we have the elements 53, 521, and 5111 originating on S6. The latter two are trivial on S7 and so d o not appear in any of our boxes, while 53 appears in all boxes with n 2 4. The element 61 1 is born on S7 and dies on S9 and hence appears only in the box for n = 4. Similarly, 71 appears only in the box for n = 5.

3.3.13 Example. Computing diflerentials. We will compute the differentials originating in the box for t - S = 1 1 , n = 1 1 . To begin we have d ( 101) = (90+72+63+54)1=721+631+541. The table shows that 721 is hit by 83 and we find d(83) = (70+61+43)3

= 721 +433.

Hence

d ( 101 + 83) = 631 + 541 + 433. The figure shows that 31 is hit by 5 so we compute d(65) = 631 + (50+32)5 = 631 + 541,

so d(101+83+65)=433, which is the desired result. Even in this limited range one can see the beginnings of several systematic phenomena worth commenting on.

3.3.14 Remark. James periodicity. (Compare 1.5.18.) In a neighborhood of the diagonal one sees a certain periodicity in the differentials in addition

97

3. The Lambda Algebra

to that of 3.3.9. For example, the leading term of d ( A , h l ) is hn-2h,Al if n = 0 or 1 mod (4) and n 2 4, giving a periodic family of d,'s in the spectral sequence. The differential computed in 3.3.13 can be shown to recur every 8 stems; add any positive multiple of 8 to the first integer in each sequence appearing in the calculation and the equation remains valid modulo terms which will not affect the outcome. More generally, one can show that A(n) is isomorphic to

ZP2"'A(n +2")/A(2") through some range depending on n and m,and a general result on the periodicity of differentials follows. It can be shown that H * ( A ( n + k ) / A ( n ) )is isomorphic in the stable zone [3.3.8(c)] to the Ext for H * ( R P " + " - ' / R P " ~ ' ) and that this periodicity of differentials corresponds to James periodicity. The latter is the fact that the stable homotopy type of R P " + k / R P " depends (up to suspension) only on the congruence class of n modulo a suitable power of 2. For more on this subject see Mahowald [I-41. 3.3.15 Remark. The Adams oanishing line. Define a collection admissible sequences (3.3.3) a, for i > 0 as follows. a,=l,

a,=111,

a,=11,

a,=24111,

a 6 = 124111,

of

a4=4111, a7= 1124111,

a8=41124111,etc.

That is, for i > 1 a, =

i

(1, a t - l ) for i = 2,3 mod (4) ( 2 , ~ , - ~ ) for i = 1 mod (4) (4, for i = O mod(4)

It can be shown that all of these are nontrivial permanent cycles in the EHP spectral sequence and that they correspond to the elements on the Adams vanishing line (3.4.5). Note that H ( a , + , )= a,. All of these elements have order 2 (i.e., are killed by ho multiplication) and half of them, the a, for i = 3 and 0 mod (4), are divisible by 2. The a4,+3are divisible by 4 but not by 8; the sequences obtained are (2, a4r+Z) and (4, ~ 4 , + 1 )except for i = 1, when the latter sequence is 3. These little towers correspond to cyclic summands of order 8 in T ; , + ~(see 5.3.7). The a41are the tops of longer towers whose length depends on i. The sequences in the tower are obtained in a similar manner; i.e., sequences are contracted by adding the first two integers; e.g., in the 7-stem we have 4111, 511, 61, and 7. Whenever i is a power of 2 the tower goes all the way down to filtration 1; i.e., it has 4i elements, of which the bottom one is 8 i - 1. The table of Tangora [ I ] shows that the towers in the 23-, 39-, and 55-stems have length 6, while that in

98

3. The Classical Adams Spectral Sequence

the 47-stem has length 12. Presumably this result generalizes in a straightforward manner. These towers are also discussed in 3.4.21 and following 4.4.47.

3.3.16 Remark. dl’s. It follows from 3.3.9 that all d,’s originate in rows with n odd and that they can be computed by left multiplication by h a . In particular, the towers discussed in the above remark will appear repeatedly in the &-term and be almost completely cancelled by d,’s, as one can see in Fig. 3.3.11. The elements cancelled by d,’s d o not appear in any H*(A(2n - l ) ) , so if one is not interested in H*(A(2n)) they can be ignored. This indicates that a lot of repetition could be avoided if one had an algorithm for computing the spectral sequence starting from E , instead of El. 3.3.17 Remark. S3. As indicated in 3.3.5, A gives unstable as well as stable Ext groups. From a figure such as 3.3.11 one can extract unstable Adams E,-terms for each sphere. For the readers’ amusement we do this for S 3 for t - s s 2 8 in Fig. 3.3.18. One can show that if we remove the infinite tower in the O-stem, what remains is isomorphic above a certain line of slope f to the stable Ext for the mod (2) Moore spectrum. This is no accident but part of a general phenomenon described by Mahowald [3]. It is only necessary to label a few of the elements in Fig. 3.3.18 because most of them are part of certain patterns which we now describe. There are clusters of six elements known as lightning flashes, the first of which consists of 1,11,111,21,211,2111.Vertical lines as usual represent left multiplication by h o , i.e., by ho. Diagonal lines represent left multiplication by A , in the left-hand half of the flash but not in the right-hand half. The three right-hand elements each have the form 2y, which represents the Massey product (A1.4.1) (l,O, y ) . If the first element of a lightning flash is x, the others are lx, I l x , 2x, 21x, and 21 lx. In the clusters containing 23577 and 233577, the first elements are missing, but the others behave as if the first ones were 4577 and 43577, respectively. For example, the generator of E:,30is 24577. In these two cases the sequences l x and l l x are not admissible, but since 14= 23 by 3.3.1, we get the indicated values for lx. If X E E;‘ is the first element of a lightning flash, there is another one . The sequence for P x is obtained from that beginning with Px E E;+4,‘+12 for x by adding 1 to the last integer and then adjoining 41 11 on the right, e.g., P(233) = 2344111. This operator P can be iterated any number of times, is related to Bott periodicity, and will be discussed more in the next section. There are other configurations which we will call rays beginning with 245333 and 235733. Successive elements in a ray are obtained by left multiplication by A 2 . This operation is related to complex Bott periodicity.

-12 -1 1

-9

t - s

Figure 3.3.18.

T h e unstable Adam &-term for S3.

100

3. The Classical A d a m Spectral !+queoce

In the range of this figure the only elements in positive stems not part of a ray or lightning flash are 23333 and 2335733. This indicates that the Curtis algorithm would be much faster if it could be modified in some way to incorporate this structure. Finally, the figure includes Tangora’s labels for the stable images of certain elements. This unstable Adams spectral sequence for 7rTT*(S3) is known to have nontrivial d2’s originating on 245333, 222245333, and 2222245333, and d,’s on 2235733 and 22235733. Related to these are some exotic additive and multiplicative extensions: the homotopy element corresponding to Ph,d, = 2433441 11 is twice any representative of hoh2g= 235733 and (the generator of the 1-stem) times a representative of 2245333. Hence the permanent cycles 2245333,24334111,235733,22245333,2243341 11, and the missing element 35733 in some sense constitute an exotic lightning flash.

4. Some General Properties of Ext In this section we abbreviate ExtA,(Z/(p), Z / ( p ) ) by Ext. First we describe Ext” for small values of s. Then we comment of the status of its generators in homotopy. Next we give a vanishing line, i.e., a function f(s) such that Ext’” = 0 for 0 < t - s < f ( s ) . Then we give some results describing Ext”‘ for t near f ( s). 3.4.1 Theorem.

(a) (b) (c) (d)

For p

=2

Exto= 2 / ( 2 ) generated by 1 E Exto*’. Ext’ is spanned by {hi: i 2 0 ) with h, E Ext’.2’ represented by [St‘]. (Adams [12]) Ext’ is spanned by { h i h j : O si s j , j # i + l } . (Wang [ 13) Ext’ is spanned by h,hJhksubject co the relations

h.h. I J = h.hJ ”

hihi+’ = 0,

h,hf+, = 0,

and

hfh,,,

= h:+l,

along with the elements ci = ( h l + l , hi, h : + , ) ~Ext”.””’.

3.4.2 Theorem.

For p > 2

( a ) Ext’ = Z/( p) generated by 1 E Exto*’. ( b ) Extl is spanned by a , and {hi: i 2 0 ) where a , E Ext’.’ is represenfed by [ T ~ and ] hiE Ext’*qP’is represented by [[:‘I. ( c ) (Liulevicius [2]) Ext’ is spanned by a ; , {aohi: i > 01, { g i : i 2 01, { ki : i 2 0 } , { b, : i 2 0}, and IIoho, where

101

4. Some General Properties of Ext

g, = ( h # h,, , h,+,)E Ext2*'2i"'p'q9

6, =(A,, h,, . . . ,h,)E Ext2-w'+'

kt

= (ht, hi+,,

k + i l E Ext 212p+2)p'q ,

(with p factors h,),

and noh0= (h,, h,, a,) E Ext2,1+2q.

W

Ext3 for p > 2 has recently been computed by Aikawa [ 11. The behavior of the elements in Ext' in the ASS is described in Theorems 1.2.11-1.2.14. We know that most of the elements in Ext' cannot be permanent cycles, i.e., 3.4.3 Theorem. ( a ) (Mahowald and Tangora [S]). With the exceptions of h0h2, hoh3, and h2h, the only elements in Ext2for p = 2 which can possibly be permanent cycles are h: and h,h,. ( b ) ([Miller, Ravenel, and Wilson [ 13). For p > 2 the only elements in Ext2 which can be permanent cycles are a:, IIoho, lq,, hob,, and b,. W

Part (b) was proved by showing that the elements in question are the only ones with preimages in the ANSS E2-term. A similar proof for p = 2 is possible using the computation of Shimomura [ 11. The list in Mahowald and Tangora [S] includes h2h5and h,h,; the latter is known not to come from the ANSS and the former is known to support a differential. The cases hoh, and b, for p > 3 and h, h, for p = 2 are now settled. 3.4.4 Theorem.

( a ) (Browder [l]). For p = 2 hf is a permanent cycle fz there is a framed manifold of dimension 2'+' - 2 with Kervaire invariant one. Such are known to exist for j s 5. For more discussion see 1.5.29 and 1.5.35. ( b ) (Mahowald [6]). For p = 2 h,h, is a permanent cycle for all j 2 3. ( c ) (Ravenel [7]). For p > 3 and ir 1, b, is not a permanent cycle. ( A t p = 3 b, is not permanent but b2 is; bo is permanent for all odd primes.) ( d ) (R. L. Cohen and Goerss [ 11). For p > 3 hoh, is a permanent cycle corresponding to an element of order p for all i 2 2. ( e ) (R. L. Cohen [2]). Forp> 2 hob, is a permanent cycle corresponding to an element of order p for all i 2 0. The proof of (c) will be given in Section 6.4. Now we describe a vanishing line. The main result is 3.4.5 Vanishing Theorem (Adams [ 171).

( a ) Forp = 2 ExtsSf=Ofor O < t - s < f ( s ) , where f ( s ) = 2 s for,0’s 1 mod (4), E = 2 for s = 2 and ~ = for 3 s=3.

E

and

E

=1

102

3. The Classical A d a m Spectral Sequence

Hence in the usual picture of the ASS, where the x and y coordinates are t - s and s, the E2-term vanishes above a certain line of slope l / q (e.g., 1 3 for p = 2). Below this line there are certain periodicity operators IIn which raise the bigrading so as to move elements in a direction parallel to the vanishing line. In a certain region these operators induce isomorphisms.

3.4.6 Periodicity Theorem (Adams [17], May [6]).

S

1

2

3

4

5

6

7

8

29

Ms)

1

1

7

10

17

22

25

32

5s-7

These two theorems are also discussed in Adams [7]. For p = 2 these isomorphisms are induced by Massey products (A1.4) sending x to (x, h r + ' , hn+z).For n = 1 this operator is denoted in Tangora [ l ] and elsewhere in this book by P. The elements x are such that hi'+lx is above the vanishing line of 3.4.5, so the Massey product is always defined. The indeterminacy of the product has the form ~ y + h , , + ~with z ~ ~ ~ 2 " ' ' , 3 . 2and " + ~ EXt9-1+2"+'.r+2"+' . The group containing y is just below the vanishing line and we will see below that it is always trivial. The group containing z is above the vanishing line so the indeterminacy is zero. Hence the theorem says that any group close enough to the vanishing line [i.e., satisfying t - s < 2"+2+g(s)]and above a certain line with slope 4 [ t - s < h ( s ) ] is acted on isomorphically by the periodicity operator. In Adams [17] this line had slope f. It is known that f is the best possible slope, but the intercept could probably be improved by pushing the same methods further. The odd primary case is due entirely to May [6]. We are grateful to him for permission to include this unpublished material here. Hence for p = 2 Ext'.' has a fairly regular structure in the wedge-shaped region described roughly by 2s < t - s < 5s. Some of this (partially below

4. Some General Properties of

103

Ed

the line of slope 4 given above) is described by Mahowald and Tangora [ 141 and an attempt to describe the entire structure for p = 2 is made by Mahowald [ 131. However, this structure is of limited interest because we know that almost all of it is wiped out by differentials. All that is left in the Em-term are certain few elements near the vanishing line related to the J-homomorphism (1.1.12). We will not formulate a precise statement or proof of this fact, but offer the following explanation. In the language of Section 1.4, the periodicity operators H, in the ASS correspond to ul-periodicity in the ANSS. More precisely, H, corresponds to multiplication by up'. The behavior of the ul-periodic part of the ANSS is analyzed completely in Section 5.3. The u,-periodic part of the ANSS &-term must correspond to the portion of the ASS &-term lying above (for p = 2) a suitable line of slope 5. Once the ANSS calculation has been made it is not difficult to identify the corresponding elements in the ASS. The elements in the ANSS all have low filtrations, so it is easy to establish that they cannot be hit by differentials. The elements in the ASS are up near the vanishing line so it is easy to show that they cannot support a nontrivial differential. We list these elements in 3.4.16 and in 3.4.21 give an easy direct proof (i.e., one that does not use BP-theory or K-theory) that most (all for p > 2) of them cannot be hit by differentials. The proof of 3.4.5 involves the comodule M given by the SES 3.4.7

o* z/(p)

+

A* m A ( O ) * z/(P)

+-

+

O,

where A(O),= E ( T J for p > 2 and E f t , ) for p = 2 . M is the homology of the cofiber of the map from So to H, the integral Eilenberg-Mac Lane spectrum. The E2-term for H was computed in 2.1.18 and it gives us the tower in the 0-stem. Hence the connecting homomorphism of 3.4.7 gives an isomorphism Exts-l.f 3.4.8 A* (Z/(P), M ) EXtS.'

for t - s > O . We will

consider

the

subalgebras

A(n)cA

generated

by

{Sq', Sq2, . . . ,Sq2'} for p = 2 and { p, P', Pp, . . . , Pp'-'} for p > 2. Their for p = 2 and duals A ( n ) , are P { & , t 2 ,... , &+1)/(t5'+2-')

. . , 7 " ) 0 P ( Z l , .. . ,Zn)/(E"+'-I

E(70,.

for p > 2. We will be considering A,-comodules N which are free over A(O), and (-1)-connected. Z-'M is an example. Unless stated otherwise N will be assumed to have these properties for the rest of the section.

104

3. The Classical Adams Spectral Sequence

Closely related to the questions of vanishing and periodicity is that of approximation. For what (s, t ) does Ext>:(Z/( p), N ) = Exti&,*(Z/(p), N ) ? This relation is illustrated by 3.4.9 Approximation Lemma. Suppose that there is a nondecreasingfunction fn(s) dejined such that for any N as aboue, Ext>:n,,(Z/(p), N ) = O for t -s ;,,*(Z/(p), N ) for t-s5l ~ d(( ct , )~= ~ ~ [(,I- . -It,lt;"+'] with 2"factors 5,;a n d f o r p > 2 d ( c , ) = - [ ~ ~ 1.)T,)~;"I. * ( c ) c, is uniquely determined up to a coboundsry by ( a ) and ( b ) . ( d ) For n 2 1 ( p > 2 ) or n 2 2 ( p = 2) c, projects to a cocycle in CA(,)* representing a nontrivial element w , E E x t ~ ~ ~ : l ) p n (( pZ)/, Z/( p ) ) . ( e ) Forp = 2, w 2 maps to w as in 3.1.27, and in general w,+, maps to w",

- IT,]

Boo$ We will rely on the algebraic Steenrod operations in Ext described in Section A1.5. We treat only the case p = 2. By A1.5.2 there are operations Sq': ExtS"+ ExtS+'*2'satisfying a Cartan formula with S q o ( h , )= h,,, (A1.5.3) and Sq'(h,) = h f . Applying Sq' to the relation hoh, = 0 we have 0 = Sq'(hoh,) = Sq0(h0)Sq'(h,)+ Sq'(h0)Sq0(h,) = h:+

hzh2.

Applying Sq' to this gives h t h 2 + h i h 3 = 0 . Since h,h,=O this implies hih3 = 0. Applying Sq4 to this gives h:h, = 0. Similarly, we get h:'h,+, = 0 for all i 2 2. Hence there must be cochains c, satisfying (b) above. To show that these cochains can be chosen to satisfy (a) we will use the Kudo transgression theorem A1 S.7. Consider the cocentral extension of Hopf algebras (A1.1.15)

P ( S , ) + P ( E , , t 2 ) + P(52). In the CESS (A1.3.14 and A1.3.17) for Extp(cl,E2)(Z/(2), Z / ( 2 ) ) one has E2 = P(h,,, h , :j 2 0) with h,, E E:.' and h , E E:,'. By direct calculation one has d2(hZ0) = h,,h,,. Applying Sq2Sq' onegets d,(h:,) = h ~ , h , , + h ~ , h , , . T h e second term was killed by d2(h:lh21)so we have d,(h:,) = h:,h,,. Applying appropriate Steenrod operations gives d2n+,(h:i)= h f & l , + l . Hence our cochain c, can be chosen in Cp(5,,52) so that its image in cp(52) is [t2 1 * * * 1 t2] representing h:;, so (a) is verified. For (c), note that (b) determines c, up to a cocycle, so it suffices to show that each cocycle in that bidegree is a coboundary, i.e., that 2" = 0. This group is very close to the vanishing line and can be computed directly by what we already know. For (d), (a) implies that c, projects to a cocycle in CA(,)*which is nontrivial by (b); (e) follows easily from the above considerations. For p = 2 suppose x E Ext satisfies h;"x = 0. Let 2 E C,. be a cocycle representing x and let y be a cochain with d ( y ) = 2[& I * . It,]with 2" factors. Then $c, + y [ t:""] is a cocycle representing the Massey product 9

~l

106

3. The Classical Adam Spectral Sequence

(x, hi", h,+,), which we define to be the nth periodicity operator II,. This cocycle maps to ic,, in cA(,,)*,so II, corresponds to multiplication by w , as claimed. The argument for p > 2 is similar. Now we need to examine w,' multiplication in ExtA(I)*(Z/(p),A(O),) for p > 2 using 3.1.30 and w2 multiplication in E x ~ ~ ( ~ ) * ( ZA(O),) / ( ~ ) , using 3.2.17. The result is 3.4.1 1 Lemma.

( a ) For p =2, multiplication by w2 in Ext>i2,,(Z/(2), A(O),) is an isomorphism for t - s < v ( s ) and an epimorphism for t - s < w ( s ) , where v ( s ) and w ( s ) are given in the following table.

( b ) For p > 2 multiplication by o1 in Ext>;,,*(Z/(p), A(O),) is a monomorphism for all s 2 0 and an epimorphism for t - s < w ( s ) where w ( s )=

{

( p 2 - p - 1)s - 1 (p2-p-1)s+p2-3p

for seven for sodd.

Next we need an analogous result where A(O), is replaced by a (-1)connected comodule N free over A(O), . Let N o c N be the smallest free A(O),-subcomodule such that N / N o is 1-connected. Then O + No+ N + N / N o + O

is an SES of A(O),-free comodules inducing an LES of A( n ) - Ext groups on which w, acts. Hence one can use induction and the 5-lemma to get 3.4.12 Lemma.

Let N be a connective A(n),-comodule free over A(O),.

( a ) For p = 2 multiplication by w2 in Ext>i2)*(Z/(2),M ) is an isomorphism for t - s < G(s) and an epimorphism for t - s < 6(s), where these functions are given by the following table

t

i

&(s)

1

7

10

18

22

25

33

5s+3

107

4. Some General Properties of Ext

( b ) For p > 2 a similar result holds for w1-multiplication where

q s )= and $(s) =

{

I

(p2-p-1)s-2p+1 (p2-p-1)s-p2+p

(p2-p-l)s-l (pZ-p-l)s-p2+2p-1

for seven for s o d d for seven for sodd.

H

3.4.13 Remark. If N / N o is (q-1)-connected, as it is when N = V M (3.4.7), then the function 3 ( s ) can be improved slightly. This is reflected in 3.4.6 and we leave the details to the reader. The next step is to prove an analogous result for on-multiplication.We sketch the proof forp = 2. Let N be as above and define N = A( n ) , OA(2)* N, and let C = N/N. Then C is 7-connected if N i s (-1)-connected, and ExtA,,,*(Z/(2), A) = ExtA,,,*(Z/(2),N ) . Hence in this group w , = w:"-* and we know its behavior by 3.4.12. We know the behavior of w , on C by induction, since C is highly connected, so we can argue in the usual way by the 5-lemma on the LES of Ext groups. If N satisfies the condition of 3.4.13, so will 15 and so we can use the improved form of 3.4.12 to start the induction. The result is

c,

3.4.14 Lemma. Let N be as above and satisfy the condition of 3.4.13. 7 I e n multiplication by w, (3.4.10) in Ext>;,),(Z/(p), N ) is an isomorphism for t - s < h ( s + 1)- 1 and an epimorphism for t - s < h ( s ) - 1, where h ( s ) is as in 3.4.6. Now the periodicity operators II,, defined above as Massey products, can be described in terms of the cochains c, of 3.4.10 as follows. Let x represent a class in Ext (also denoted by x ) which is annihilated by hi" and let y be a cochain whose coboundary is x[ll I t11. 11]with 2" factors tl.Then y [ [ : " + ' ] +xc, is a cochain representing I I , ( x ) . Hence it is evident that the action of II, in Ext corresponds to multiplication b y w , in A(n),-cohomology. Hence 3.4.14 gives a result about the behavior of II, in ExtA*(Z/(p),M ) with M as in 3.4.7, so 3.4.6 follows from the isomorphism 3.4.8. Having proved 3.4.6 we will list the periodic elements in Ext which survive to Em and correspond to nontrivial homotopy elements. First we have

3.4.15 Lemma. F o r p = 2 and n ? 2 , I I , ( h ~ " - ' h n + , ) = hr+'-'hn+2.F o r p > 2 and n 2 1, II,(a$-'h,) = ag""-'h,+, up to a nonzero scalar. [ I t is not true that IIo(ho)= &'h1 .]

Prooj We d o not know how to make this computation directly. However, 3.4.6 says the indicated operators act isomorphically on the indicated

108

3. The Classical A d a m Spectral Sequence

elements, and 3.4.21 below shows that the indicated image elements are nontrivial. Since the groups in question all have rank one the result follows. (3.4.6 does not apply to no acting on h, for p > 2.) W 3.4.16 Theorem.

( a ) For p > 2 the set of elements in the ASS Em-term on which all iterates of some periodicity operator n, are nontrivial is spanned by rI;(a;n-Jh,,)with n r O , O < j s n + l , and i f - 1 mod ( p ) . (For i = - l these elements vanish for n = 0 and are determined by 3.4.15 for n > 0.) The corresponding subgroup of 7r*(So) is the image of rhe J-homomorphism (1.1.12). (Compare 1.5.19.) ( 6 ) Forp = 2 the set is generated by all iterates of II, on h , , h:, h: = h&, hohz, h Z , c,, and h,co (where c o = ( h , ,ho, h : ) Ext3*”) ~ and by n ~ h , , h ~ - ’ - J with n 2 3, i odd, and 0 <j 2 n + 1. (For even i these elements are determined by 3.4.15.) The corresponding subgroup of .rr*(So) is .rr,(J) (1.5.22). In particular, im J corresponds to the subgroup of E, spanned by all of the above except n ; h , for i > 0 and n ; h : for i 2 0. This can be proved in several ways. The cited results in Section 1.5 are very similar and their proofs are sketched there; use is made of K-theory. The first proof of an essentially equivalent theorem is the one of Adams [ 13, which also uses K-theory. For p = 2 see also Mahowald [ 151 and Davis and Mahowald [l]. The computations of Section 5.3 can be adapted to give a BP-theoretic proof. The following result is included because it shows that most (all i f p > 2) of the elements listed above are not hit by differentials, and the proof makes no use of any extraordinary homology theory. We will sketch the construction for p = 2. It is a strengthened version of a result of Maunder [ I]. Recall (3.1.9) the spectrum bo (representing real connective K-theory) with H,(bo) = A, CIA(,)* Z/(2) = P(t:,i:,f 3 , * . ). For each i r Othere i s a m a p to Z4’H (where H is the integral Eilenberg-Mac Lane spectrum) under which 6:’ has a nontrivial image. Together these define a map f from bo to W=VlzoZ4’H. We denote its cofiber by There is a map of cofiber sequences 3.4.17

So-H-H

in which each row induces an SES in homology and therefore an LES of Ext groups. Recall (3.1.26) that the Ext group for bo has a tower in every fourth dimension, as does the Ext group for W One can show that the former map injectively to the latter. Then it is easy to work out the ASS

109

4. Some General Properties of Ext

&-term for W,namely 3.4.18

b

ExtS+l,'( H , ( b o ) ) if t - s f 0 mod (4) ExtsVf( H,( W))= Z/ ( 2 ) if f - s = 0 and Ext""( If,( b o ) )= 0 otherwise,

where Ext(M) is an abbreviation for ExtA*(Z/(2),M ) . See Fig. 3.4.20. Combining 3.4.17 and 3.4.8 gives us a map 3.4.19

Ext"*'(Z/(Z))+ Exts-'*'( H,( W))

t - s > 0.

for

Since this map is topologically induced it commutes with Adams diff erentials. Hence any element in Ext with a nontrivial image in 3.4.19 cannot be the target of a differential. 15

10

I 5

1

I

I

5

10

15

t--S

Figure 3.4.20.

-

I 20

Exts-'.' H*( W).

I 25

110

3. The Classical Adam Spectral Seqnence

One can show that each h, for n>O is mapped monomorphically in 3.4.19, so each h, supports a tower going all the way up to the vanishing line as is required in the proof of 3.4.15. Note that the vanishing here coincides with that for Ext given in 3.4.5. A similar construction at odd primes detects a tower going up to the vanishing line in every dimension - 1 mod ( 2 p - 2). To summarize 3.4.21 Theorem. ( a ) For p = 2 there is a spectrum W with ASS E,-term described in 3.4.18 and 3.4.20. The resulting map 3.4.19 commutes with Adams differentials and is nontrivial on h, for all n > O and all II, iterates of h , , h : , h i = hih,, h,hz, h Z , and h i h , . Hence none of these elements is hit by Adams diferentials. ( b ) A similar construction f o r p > 2 gives a map as above which is nontrivial on h, for all n 2 0 and on all the elements listed in 3.4.16(a). m

The argument above does not show that the elements in question are permanent cycles. For example, all but a few elements at the top of the towers built on h, for large n support nontrivial differentials, but map to permanent cycles in the ASS for ii! We do not know the image of the map in 3.4.19. For p = 2 it is clearly onto for t - s = 2" - 1. For t - s + 1= (2kf 1)2" with k > 0 the image is at least as big as it is for k = 0, because the appropriate periodicity operator acts on h,. However, the actual image appears to be about $ as large. For example, the towers in Ext in dimensions 23 and 39 have 6 elements instead of the 4 in dimension 7, while the one in dimension 47 has 12. We leave this as a research question for the interested reader.

5. Survey and Further Reading In this section we survey some other research having to do with the classical ASS, published and unpublished. We will describe in sequence results related to the previous four sections and then indicate some theorems not readily classified by this scheme. In Section 1 we made some easy Ext calculations and thereby computed the homotopy groups of such spectra as MU and bo. The latter involved the cohomology of A ( I ) , the subalgebra of the mod (2) Steenrod algebra generated by Sq' and Sq'. A pleasant partial classification of A ( 1)-modules is given in section 3 of Adams and Priddy [lo]. They compute the Ext groups of all of these modules and show that many of them can be realized

111

5. Survey and Further Reading

as bo-module spectra. For example, they use this result to analyze the homotopy type of bo A bo. The cohomology of the subalgebra A(2) was computed by Shimada and Iwai [2]. Recently, Davis and Mahowald [4] have shown that A//A(2) is not the cohomology of any connective spectrum. In Davis and Mahowald [5] they compute A(2)-Ext groups for the cohomology of stunted real projective spaces. More general results on subalgebras of A can be found in Adams and Margolis [ l l ] and Moore and Peterson [l]. The use of the ASS in computing cobordism rings is becoming more popular. The spectra MO, MSO, MSU, and MSpin were originally analyzed by other methods (see Stong [ l ] for references) but in theory could be analyzed with the ASS; see Pengelley [ 1,2] and Giambalvo and Pengelley [I]. The spectrum MO(8)(the Thom spectrum associated with the 7-connected cover of BO) has been investigated by ASS methods in Giambalvo [2], Bahri [l], Davis [3,6], and Davis and Mahowald [4]. In Johnson and Wilson [5] the ASS is used to compute the bordism ring of manifolds with free G-action for an elementary abelian p-group G. The most prodigious ASS calculation to date is that for the symplectic cobordism ring by Kochman [ 1-31. He uses a change-of-rings isomorphism to reduce the computation of the E2-term to finding Ext over the coalgebra

3.5.1

=P(&, 52,.

* *

)/(t3,

for which he uses the May SS. The &-term for MSp is a direct sum of many copies of this Ext and these summands are connected to each other by higher Adams differentials. He shows that MSp is indecomposable as a ring spectrum and that the ASS has nontrivial d,’s for arbitrarily large r. In Section 2 we described the May SS. The work of Nakamura [I] enables one to use algebraic Steenrod operations (A1.5) to compute May differentials. The May SS is obtained from an increasing filtration of the dual Steenrod algebra A,. We will describe some decreasing filtrations of A, for p = 2 and the SSs they lead to. The method of calculation these results suggest is conceptually more complicated than May’s but it may have some practical advantages. The E2-term (3.5.2) can be computed by another SS (3.5.4) whose E2-term is the A ( n ) cohomology (for some fixed n ) of a certain trigraded comodule 7: The structure of T is given by a third SS (3.5.10) whose input is essentially the cohomology of the Steenrod algebra through a range of dimensions equal to 2-“-’ times the range one wishes to compute. This method is in practice very similar to Mahowald’s unpublished work on “Koszul resolutions.”

112

3. The Classical Adam Spectral Sequence

3.5.2 Proposition.

For each n 2 0 , A , has a decreasing filtration (Al.3.5) {F'A,} where F" is the smallest possible subgroup satisfying E F2k-' for i +j = k + n + 1. In particular, Fo/ F' = A( n ) , , so A( n ) , c E,A, where - ,f n + z , . . . ). Hence there is a spectral A ( n ) , =A,/(E'+',f:",. . . , -2 &+, sequence (A1.3.9) converging to ExtA (Z/(2), M ) with EI.'," = Extk;,*(Z/(2), E o M ) and d,: E:'," + E',+'.'"+*' , where the third grading is that given by theJiltration, M is any A,-comodule, and EoM is the associated E,A,-comodule (A1.3.7).

(.,

NOWlet G ( n ) , = EoA, OA(")* Z/(2). It inherits a Hopf algebra structure from EoA, and A ( n ) * + &A*+ G ( n ) *

3.5.3

is an extension of Hopf algebras (A1.1.15). Hence we have a CartanEilenberg spectral sequence (Al.3.14), i.e., 3.5.4 Lemma. Associated with the extension 3.5.3 there is a spectral sequence

with s s

E;'

IU-

2"

- Ext2(,,)*(Z/(2),Ext&$,(Z/(2),

M))

with d,: E;lJ2,1,u ES~+r*S2-r+'.',U r converging to E ~ t ~ ~ ~ l ' " ( ZM/ ()2for ) , any EoA, comodule M. [ExtG(,,,*(Z/(2),M ) is the T referred to aboue.] ~

According to A1.3.11(a) the cochain complex W used to compute Ext over G(n ) , can be taken to be one of A( n),-comodules. The E2-term of the spectral sequence is the A ( n ) , Ext of the cohomology of W, and the &-term is the cohomology of the double complex obtained by applying CX(,,,,( ) (A1.2.11) to W. This W is the direct sum [as a complex of A( n),-comodules] of its components for various u (the filtration grading). T h e differentials are computed by analyzing this W.

3.5.5 Remark.

Next observe that EoA, and G ( n ) , contain a sub-Hopf algebra A(*"+') isomorphic up to regrading to A,; i.e., A(*n+l)c EoA, is the image of P(gn*')c A,. The isomorphism follows from the fact that the filtration coincides with the topological degree of Hence we degree 2' - 1 of have

c.

sf""

3.5.6

Ext>;( Z/ (2), Z/(2)) = Extl::~'?'(

Z/ ( 2 ) , Z/ (2))

and we can take these groups as known inductively. Let L( n ) , = G ( n)* Z/(2) and get an extension 3.5.7

A(*"+') + G (n)* + L( n ) , .

L ( n ) , is easily seen to be cocommutative with 3.5.8

Ext>;',",*(Z/(2),Z/ (2)) = P(h,,, :0 s j In, i 2 n + 2 -j ) ,

113

5. Survey and Further Reading

where h,, E Ext 1 , 2 J ( 2 ' - 1 ) , 2 ' + I - ' ' '- I corresponds as usual to 4;'. This Ext is a comodule algebra over A(*"+') (A1.3.14) with coaction given by 3.5.9

Hence by A3.1.14 we have 3.5.10 Lemma. with

The extension 3.5.7 leads to a spectral sequence as in 3.5.4

EY2*'." - Ext l(c+i(Z/(2), Ext?(i;*( Z/ (2), M ) s+s

I U

converging to Ext&,j*' (Z/(2), M ) for any G(n),-comodule M . For M = Z/(2), the Ext over L( n ) , and its comodule algebra structure are given by 3.5.8 and 3.5.9. Moreover, this spectral sequence collapses from E 2 ,

All is clear but the last statement, which we prove by showing that G ( n ) , possesses an extra grading which corresponds to s2 in the spectral sequence. It will follow that differentials must respect this grading so d , = 0 for r z 2 . Let g,,E G ( n ) , be the element corresponding to (f'. The extra grading is defined by ProoJ

Since the &,j for j 5 n are all exterior generators, the multiplication in G (n ) , respects this grading. The coproduct is given by A(E.8) =I ck,j@gs=k,k+j-

If j L n + 1 then all terms have degree 0, and if j 5 n, we have k + j L n + 2 so all terms have degree 1 , so A also respects the extra grading. We now describe how to use these results to compute Ext. If one wants to compute through a fixed range of dimensions, the isomorphism 3.5.6 reduces the calculation of the spectral sequence of 3.5.10 to a much smaller range, so we assume inductively that this has been done. The next step is to compute in the spectral sequence of 3.5.4. The input here is the trigraded A(n),-comodule Ext&'$)*(Z/(2), Z/(2)). We began this discussion by assuming we could compute Ext over A ( n ) , , but in practice we cannot do this directly if n > 1. However, for O s m < n we can reduce an A(n), calculation to an A( m ) , calculation by proceeding as above, starting with the mth filtration of A( n ) , instead of A,. We leave the precise formulation to the reader. Thus we can compute the A ( n ) , Ext of Ext.&'i",*(Z/(2), Z/(2)) separately for each u ; the slogan here is divide and conquer.

114

3. The Classical A d a m Spectral Sequence

This method can be used to compute the cohomology of the Hopf algebra B (3.5.1) relevant to MSp. Filtering with n = 1, the SS analogous to 3.5.4 has E2=

E x t ~ , , , * ( Z / ( 2 )P(h21, , h30, h 3 1 , h o , . . *

with I)( hi+,,o)= t10hi,,+ 1 0 and $(hi,,)= 10 h,,, for i 2 2. This Ext is easy to compute. Both this SS and the analog of the one in 3.5.2 collapse from E 2 . Hence we get a description of the cohomology of B which is more concise though less explicit than that of Kochman [l]. In Section 3 we described A and hinted at an unstable ASS. For more on this theory see Bousfield and Kan [3], Bousfield and Curtis [4], Bendersky, Curtis, and Miller [ 13, Curtis [ 11, and Singer [3-51. A particularly interesting point of view is developed by Singer [2]. In Mahowald [3] the double suspension homomorphism

A(2n - 1 ) + A ( 2 n + 1 ) is studied. He shows that the cohomology of its cokernel W (n ) is isomorphic to Extik(Z/(2), Z2"-'A(0),) for t - s < 5 s k for some constant k, i.e., above a line with slope 4. This leads to a similar isomorphism between H*(h(2n + l ) / A ( 1 ) ) and ExtA*(Z/(2),I?*( RP'")). In Mahowald [4] he proves a geometric analog, showing that a certain subquotient of T * ( S ~ ~ + ' ) is isomorphic to that of RP'"). The odd primary analog of the algebraic result has been demonstrated by Harper and Miller [ 11. The geometric result is very likely to be true but is still an open question. This point was also discussed in Section 1.5. Now we will describe some unpublished work of Mahowald concerning generalizations of A. In 3.3.3 we defined subcomplexes A( n ) c A by saying that an admissible monomial A,, . . . A,, is in A( n ) if i, < n. The SES

+

n:.

A(n - l ) + A ( n ) + Z n A ( 2 n - 1 ) led to the algebraic EHPSS of 3.3.7. Now we define quotient complexes A(n> by A(n>=A/A(A, ,...,. An-,), so A ( O ) = A and l & M A ( n ) = 2 / ( 2 ) . Then there are SESs 3.5.1 1

O + Z"A((n + 1)/2)+ A(n)+ A(n + l ) - + 0

+

where the fraction ( n 1)/2 is taken to be the integer part thereof. This leads to an SS similar to that of 3.3.7 and an inductive procedure for computing H * ( A ) . Next we define A,-comodules B, as follows. Define an increasing filtration on A, (different from those of 3.5.2) by 6, E F24 and let B. = F,. The B, is realized by the spectra of Brown and Gitler [3]. They figure critically

115

5. Survey and Further Reading

in the construction of the qj’s in Mahowald [ 6 ] and in the Brown-PetersonCohen program to prove that every closed smooth n-manifold immerses in ~ Z n - a ( n ) , where a ( n ) is the number of ones in the dyadic expansion of n. Brown and Gitler [3] show that ExtA*(Z/(2),B , ) = H * ( A ( n ) )and that the SES 3.5.1 1 is realized by a cofibration. It is remarkable that the Brown-Gitler spectra and the unstable spheres both lead in this way to A. Now let N = (n,, n2, . . . ) be a nonincreasing sequence of nonnegative integers. Let A( N ) = A*/([;"', . . . ). This is a Hopf algebra. Let M ( N )= A, Z/(2), S O M ( N ) = P(l:", S:"', . . . ). The filtration of A, defined above induces one on M ( N ) and we have

e:"',

where N kis the sequence (nk+l, nk+2,. . . ). For N and this is equivalent to 3.5.11.

= (0,

0, . . . ) A( N ) = A,

3.5.13 Proposition. The SES 0 + F,-,M( N ) + F,M( N ) + Fn/F,-l + 0

is split over A( N ) . This result can be used to construct an LES of A,-comodules 3.5.14

o+ z/ (2) -+ c: -+ c', + c: + . .

such that Ck is a direct sum of suspensions of M ( N k )indexed by sequences (il , i2,. . . , ik) satisfying 1 + 5 = 0 mod 2"1+*-~ and i, 5 2$-1. Equation 3.5.14 leads to an SS (A1.3.2) converging to Ext with 3.5.15

EF* = ExtL*(Z/(2), C",.

The splitting of C: and the change-of-rings isomorphism A1.3.13 show that EF* is a direct sum of suspensions E ~ t ~ ( ~ k ) ( Z /Z/(2)). (2), The El-term of this SS is a "generalized A" in that it consists of copies of A( N k )Ext groups indexed by certain monomials in A. The d , is closely related to the differential in A. We will describe the construction of 3.5.14 in more detail and then discuss some examples. Let M ( N ) be the quotient in 0-,2/(2)+ M ( N ) + M ( N ) + O .

In 3.5.14 we want C & = M ( N ) and C h = @ , , 0 2 1 2 " ' M ( N 1so ) , we need to embed a(N ) in this putative C', . The filtration on M ( N ) induces ones on a ( N ) and C L ; in the latter F, should be a direct sum of suspensions

116

3. The Classical A d a m Spectral Sequence

of M ( N ’ ) . Consider the commutative diagram

O+

F,-,fi(N)

0-

F,-,C’,

+

F I M ( N )-+ F , / F , - , f i ( N )

1 /I /’/

+

X,

+

l

-.O

F,/F,_,M(N) +O

with exact rows. The upper SES splits over A ( N ) (3.5.13) and hence over A( N ’ ) . Since Fl-,C’, splits as above, the change-of-rings isomorphism A1.3.13 implies that the map

, HomA,(F,G(N ) , F n - C’,)

-+

Hom,,(F,-,

fi(N ) , F,-, C’,)

is onto, so the diagonal map exists. It can be used to split the middle SES, so the lower SES can be taken to be split and C’, is as claimed. The rest of 3.5.14 can be similarly constructed. Now we consider some examples. If N = (0, 0, . . . ) the SS collapses and we have the A-algebra. If N = (1,1,. . . ) we have ExtA(N) = P(a,, a,, . . . as computed in 3.1.9, and the El-term is this ring tensored with the subalgebra of A generated by A, with i odd, which is isomorphic up to regrading with A itself. This is also the E,-term of an SS converging to the ANSS E,-term to be discussed in Section 4.4. The SS of 3.5.15 in this case can be identified with the one obtained by filtering A by the number of A , with i odd occurring in each monomial. For N = ( 2 , 2 , . . . ) we have A ( N )= B as in 3.5.1, so the El-term is ExtB tensored with a regraded A. Finally, consider the case N = (2, 1,0,0,. . . ). We have E Y ’ = Ext&,,, and E:,” X4 Ext>(0),. One can study the quotient SS obtained by setting EFs = 0 for k > 1. The resulting E2= Em is the target of a map from Ext, and this map is essentially the one given in 3.4.19. More generally, the first few columns of the SS of 3.5.15 can be used to detect elements in Ext. In Section 4 we gave some results concerning vanishing and periodicity. In particular we got a vanishing line of slope f (for p = 2) for any connective comodule free over A(O), . This result can be improved if the comodule is free over A( n), for some n > 0; e.g., one gets a vanishing line of slope for n = 1, p = 2. See Anderson and Davis [ I ] and Miller and Wilkerson [S]. The periodicity in Section 4 is based on multiplication by powers of h20 ( p = 2 ) or a , ( p > 2) and these operators act on classes annihilated by some power of h,,, or a,. As remarked above, this corresponds to u,-periodicity in the A N S S (see Section 1.4). Therefore one would expect to find other

=el,,,

5. Survey and Further Reading

117

operators based on multiplication by powers of h,+l,o or a, corresponding to u,-periodicity for n > 1. A u,-periodicity operator should be a Massey product defined on elements annihilated by some on-, -periodicity operator. For n = 2, p = 2 this phenomenon is investigated by Davis and Mahowald [I] and Mahowald [lo-121. More generally one can ask if there is an ASS version of the chromatic SS (1.4.8). For this one would need an analog of the chromatic resolution (1.4.6), which means inverting periodicity operators. This problem is addressed by Miller [4,7]. A u,-periodicity operator in the ASS for p = 2 moves an element along a line of slope 1/(2"+'-2). Thus u,-periodic families of stable homotopy elements would correspond to families of elements in the ASS lying near the line through the origin with this slope. We expect that elements in the E,-term cluster around such lines. Now we will survey some other research with the ASS not directly related to the previous four sections. For p = 2 and t - s 5 45, differentials and extensions are analyzed by Mahowald and Tangora [9], Barratt, Mahowald, and Tangora [l], and Tangora [5]. Some systematic phenomena in the E,-term are described in Davis [2], Mahowald and Tangora [14], and Margolis, Priddy, and Tangora [ 11. Some machinery useful for computing ASS differentials involving Massey products is developed by Kochman [4] and Section 12 of Kochman [2]. See also Milgram [2] and Kahn [l]. The ASS was used in the proof of the Segal conjecture for Z/(2) by Lin [l] and Lin et al. [2]. Computationally, the heart of the proof is the startling isomorphism Ext>:(Z/(2), M ) = Ext>T'(Z/(2), Z/(2)), where M is dual to the A-module A/(2)[x, x-'1 with dim x = 1 and Sqkx' = (this binomial coefficient makes sense for any integer i). This isomorphism was originally conjectured by Mahowald (see Adams [ 141). The analogous odd primary result was proved by Gunawardena [I]. The calculation is streamlined and generalized to elementary abelian p-groups by Adams, Gunawardena, and Miller [18]. This work makes essential use of ideas due to Singer [ 13 and Li and Singer [ 11. In Ravenel [4] we proved the Segal conjecture for cyclic groups by means of a modified form of the ASS (MASS) in which the filtration is altered. This method was used by Miller and Wilkerson [9] to prove the Segal conjecture for periodic groups. The general Segal conjecture has been proved by Carlsson [l]. Finally, we must mention the Whitehead conjecture. The n-fold symmetric product S p " ( X )of a space X is the quotient of the n-fold Cartesian product by the action of the symmetric group Z,. Dold and Thom [ l ] showed that

118

3. The Classical Adam Spectral Sequence

Sp"(X) = S p " ( X ) is a product of Eilenberg-Mac Lane spaces whose homotopy is the homology of X Symmetric products can be defined on spectra and we have Sp"(So) = H,the integer Eilenberg-Mac Lane spectrum. After localizing at the prime p one considers SO-, SpP( SO) + SpP2(SO) -9

'

and 3.5.16

H ~ S 0 ~ Z ~ ' S p P ( S 0 ) / S 0 ~ Z ~ * S p p 2 ( S 0 ) */ .S.p p ( S 0 ) ~

Whitehead conjectured that this diagram induces an LES of homotopy groups. In particular, the map Z-'Spp(So)/So+So should induce a surjection in homotopy in positive dimensions; this is the famous theorem of Kahn and Priddy [ 2 ] .The analogous statement about Ext groups was proved by Lin [3]. Miller [6] generalized this to show that 3.5.16 induces an LES of Ext groups. The LES of homotopy groups was established by Kuhn [ 11. The spectra in 3.5.16 were studied by Welcher [1,2]. He showed that H*(Spp""(So)/Sppn(So)) is free over A ( n ) , , so its Ext group has a vanishing line given by Anderson and Davis [ l ] and Miller and Wilkerson [8] and the LES of 3.5.16 is finite in each bigrading.

CHAPTER 4

BP-THEORY AND THE ADAMS-NOVIKOV SPECI'RAL SEQUENCE

In this chapter we turn to the main topic of this book, the ANSS. In Section 1 we develop the basic properties of MU and the Brown-Peterson spectrum BP, using the calculation of r , ( M U ) (3.1.5) and the algebraic theory of formal group laws as given in Appendix 2. The main result is 4.1.19, which describes B P , ( B P ) , the BP-theoretic analog of the dual Steenrod algebra. Section 2 is a survey of other aspects of BP-theory not directly related to this book. In Section 3 we study BP,( B P ) more closely and obtain some formulas, notably 4.3.13, 4.3.18,4.3.22,and 4.3.33, which will be useful in subsequent calculations. In Section 4 we set up the ANSS and use it to compute the stable homotopy groups of spheres through a middling range of dimensions, namely 5 2 4 for p = 2 and 1 2 p 3 - 2 p - l forp>2.

1. Quillen's Theorem and the Structure of BPJBP) In this section we will construct the Brown-Peterson spectrum BP and determine the structure of its Hopf algebroid of cooperations, B P , ( B P ) , i.e., the analog of the dual Steenrod algebra. This will enable us to begin computing with the Adams-Novikov spectral sequence (ANSS) in Section 1 I9

120

4. BP-Theory and the AdnmsNovikov Spectral Sequence

4. The main results are Quillen's theorem 4.1.6, which identifies r*(M U ) with the Lazard ring L(A2.1.8); the Landweber-Novikov theorem 4.1.11, which describes MU,( M U ) ; the Brown-Peterson theorem 4.1.12, which gives the spectrum BP; and the Quillen-Adams theorem 4.1.19, which

describes BP,( B P ) . We begin by informally defining the spectrum MU. For more details see Milnor and Stasheff [S]. Recall that for each n 2 0 the group of complex unitary n x n matrices U (n ) has a classifying space BU( n). I t has a complex n-plane bundle yn over it which is universal in the sense that any such bundle 5 over a paracompact space X is the pullback of -yn induced by a mapf: X + BU( n). Isomorphism classes of such bundles 6 are in one-to-one correspondence with homotopy classes of maps from X to B U f n ) . Any C"-bundle 5 has an associated disc bundle D( 6) and sphere bundle S(6). The Thom space T (6) is the quotient D ( l ) / S ( & )Alternatively, . for compact X , T ( 6 )is the one-point compactification of the total space of 5. M U ( n ) is T ( y , ) , the Thom space of the universal n-plane bundle y,, over B U ( n ) . The inclusion U ( n ) c sU ( n + 1) induces a map B U ( n ) + B U ( n + l ) . The pullback of under pullback of yn+lunder this map has Thom space Z 2 M U ( n ) .Thom spaces are functorial so we have a map C 2 M U ( n )+ M U ( n + 1). Together these maps give the spectrum MU. It follows from the celebrated theorem of Thom [l]that a , ( M U ) is isomorphic to the complex cobordism ring (see Milnor [4]) which is defined as follows. A stably complex manifold is one with a complex structure on its stable normal bundle. (This notion of a complex manifold is weaker than others, e.g., algebraic, analytic, and almost complex.) All such manifolds are oriented. Two closed stably complex manifolds M , and M 2 are cobordant if there is a stably complex manifold W whose boundary is the disjoint union of M , (with the opposite of the given orientation) and M z . Cobordism, i.e., being cobordant, is an equivalence relation and the set of equivalence classes forms a ring (the complex cobordism ring) under disjoint union and Cartesian product. Milnor and Novikov's calculation of r*(M U ) (3.1.5) implies that two such manifolds are cobordant if they have the same Chern numbers. For the definition of these and other details of the above we refer the reader to Milnor and Stasheff [S] or Stong [l]. This connection between M U and complex manifolds is, however, not relevant to most of the applications we will discuss, nor is the connection between M U and complex vector bundles. On the other hand, the connection with formal group laws (A2.1.1) discovered by Quillen [2] (see 4.1.6) is essential to all that follows. This leads one to suspect that there is some unknown formal group theoretic construction of M U or its associated infinite loop space. For example, many well-known infinite loop spaces have been constructed as classifying spaces of certain types of categories

1. Quillen's Theorem and the Structure of BPJBP)

121

(see Adams [9], section 2.6), but to our knowledge no such description exists for MU. This infinite loop space has been studied in Ravenel and Wilson [2]. In order to construct BP and compute BP,( BP) we need first to analyze MU. Our starting points are 3.1.4, which describes its homology, and the Milnor-Novikov theorem 3.1.5, which describes its homotopy and the behavior of the Hurewicz map. The relevant algebraic information is provided by A2.1, which describes universal formal group laws and related concepts and which should be read before this section. The results of this section are also derived in Adams [ 5 ] . Before we can state Quillen's theorem (4.1.6), which establishes the connection between formal group laws and complex cobordism, we need some preliminary discussion.

Let E be an associated commutative ring spectrum. A complex orientation for E is a class xE E i 2 ( C P " ) whose restriction to i 2 ( C P 1 )= E2(S2)= no(E)is 1, where C P " denotes n-dimensional complex projective space. 4.1.1 Definition.

This definition is more restrictive than that given in Adams [5] (2.1), but it is adequate for our purposes. Of course, not all ring spectra (e.g., bo) are orientable in this sense. Two relevant examples of oriented spectra are the following. 4.1.2 Example.

Let E = H, the integral Eilenberg-Mac Lane spectrum. Then the usual generator of H2(CP") is a complex orientation x H . Let E = MU. Recall that M U is built up out of Thom spaces M U ( n ) of complex vector bundles over B U ( n ) and that the map BU( n ) += M U ( n ) is an equivalence when n = 1. The composition

4.1.3 Example.

CP"= BU( 1) 5 MU(1) .+ MU gives an orientation xMUE MU2(CPm).Alternatively, xMU could be defined to be the first Comer-Floyd Chern class of the canonical complex line bundle over CP" (see Conner and Floyd [l]). 4.1.4 Lemma.

Let E be a complex oriented ring spectrum.

( a ) E*(CP")= E*(pt.)[[X,]]. ( b ) E*(CP" x CP") = E*(pt.)[[xE @ 1, 1@xE]]. be the H-space structure map, i.e., the ( c ) Let t : CP"xCP"+CP" map corresponding to the tensor product of complex line bundles, and let FE(X,y)EE*(pt.)[[X,y]] be dejined by t * ( x E ) = F E ( x E O 1 , l o x E ) . Then FE is a formal group law (A2.1.1)ozler E* (pt.).

122

4. BP-Theory and the AdamcNovikov Spectral Sequence

ProoJ: For (c), the relevant properties of FE follow from the fact that CP" is an associative, commutative H-space with unit. For (a) and (b) one has the Atiyah-Hirzebruch spectral sequence (AHSS) H * ( X ; E * ( p t . ) ) J E * ( X ) (see section 7 of A d a m [4]). For X = CP" the class xE represents a unit multiple of X , E H 2 ( C P m ) .Hence xH and all of its powers are permanent cycles so the spectral sequence collapses and (a) follows. The argument for (b) is similar. Hence a complex orientation xE leads to a formal group law FE over E*(pt.). Lazard's theorem A2.1.8 asserts that FE is induced by a homomorphism O E : L+E*(pt.), where L is a certain ring over which a universal formal group law is defined. Recall that L = Z[x,, xz, . . .], where xi has degree 2i. There is a power series over L@Q Iog(x)=

1

mixi+'

LZO

where mo= 1 such that L O Q = Q[m,, m 2 , . . .I and Iog(F(x, Y ) ) =log(x)+log(Y).

This formula determines the formal group law F(x, y ) . The following geometric description of OMMU, while interesting, is not relevant to our purposes, so we refer the reader to Adams [ 5 , Theorem 9.21 for a proof. 4.1.5 Theorem (Mischenko [ 13). The elemenr ( n + 1 ) OMU ( m , j E T*(M U ) is represented by the complex manifold CP". 4.1.6 Theorem (Quillen [2]).

OMU is an isomorphism.

H

We will prove this with the help of the diagram L

A

n,(MU)

M

h

H,(MU)

where M = Z[ m , , m z , . . .] is dqfined in A2.1.9(b) and contains L. The map 4 will be constructed below. Recall [A2.1.10(b)] that modulo decomposables in M, the image of j is generated by pmi

m,

if

i = p k - I for some prime p,

otherwise.

123

1. Quillen's Theorem and the Structure of B P J B P )

Recall also that H , ( M U ) = Z [ b , , b 2 , . . .] [3.1.4(a)] and that modulo decomposables in H,( M U ) , the image of h is generated by if

-pb, - b,

i = p k - 1 for some prime p,

otherwise.

Hence it suffices to construct 4 and show that it is an isomorphism. Before doing this we need two lemmas. First we must compute E , ( M U ) . It follows easily from 4.1.4(a) that E,(CP") is a free .rr,(E) module on elements j3: dual to X;. We have a stable map CP"+ X'MU and we denote by b f the image of j35,.

4.1.7 Lemma.

Zf E is a complex oriented ring spectrum then

E , ( M U ) = T*(E ) [ b F ,b f , . . .]. Prooj We use the AHSS H,( MU, T,(E ) ) J E , ( M U ) . The b f represent unit multiples of bi E H 2 , ( M U )[3.1.4(a)], so the b, are permanent cycles and the AHSS collapses. 1 If E is complex oriented so is E A MU. The orientations xE and xMU both map to orientations for E A MU which we denote by .?E and iMU, respectively. We also know by 4.1.7 that .rr,(E 4.1.8 Lemma.

A

~r,.E)[bf]

M U ) =E,(MU)=

Let E be an oriented spectrum. Then in ( E A MU)2(CP"). .?Mu=

1 bfi?Ef', I20

where b, = 1. This power series will be denoted by g E ( 2 E ) .

Prooj

We will show by induction on n that after restricting to CP" we get

1

.?Mu=

bE.?iE+l.

Oci 2 by more algebraic means, i.e., without using the Sullivan-Baas construction. This is done in Wurgler [ 11, where the structure of P ( n ) , ( P ( n ) ) is also obtained. k ( n ) , ( k ( n ) ) is described in Yagita [2]. We now turn to the important work of Peter Landweber on the internal algebraic structure of MU- and BP-theories. The starting point is the invariant prime ideal theorem 4.3.2, which first appeared in Landweber [4], although it was probably first proved by Morava. It states that the only prime ideals in T*(B P ) which are invariant (A1.1.21), or, equivalently, which are subcomodules over BP,(BP),,are the I, = (p, z),, v q , . . . , z),,-,) for 0 s n I00. In Conner and Smith [3] it is shown that for a finite complex X, B P , ( X ) is finitely presented as a module over T , ( B P ) . [The result there is stated in terms of MU,( X), but the two statements are equivalent.] From commutative algebra one knows that such a module over such a ring has a finite filtration in which each of the successive subquotients is isomorphic to the quotient of the ring by some prime ideal. Of course, as anyone who has contemplated the prospect of algebraic geometry knows, a ring such as .rr,(BP) has a very large number of prime ideals. However, Landweber [3] shows that the coaction of BP,(BP) implies that the filtration of B P , ( X ) [or of any BP,(BP)-comodule which is finitely presented as a module over r,(BP)] can be chosen so that each successive subquotient has the form T,( B P ) / I,,for some finite n. {The corresponding statement about MU,( X ) appeared earlier in Landweber [ 5 ] . } The submodules in the filtration can be taken to be subcomodules and n (the number of generators of the prime ideal) never exceeds the projective dimension of the module. This useful result is known as the Landweber filtration theorem. It leads to the Landweber exact functor theorem, which addresses the following question. For which r,(BP)-modules M is the functor

135

2. A Survey of BP-Theory

BP,(-) M a generalized homology theory? Such a functor must be exact in the sense that it converts cofiber sequences into long exact sequences of modules. This will be the case if M is flat, i.e., if Torp(BP)(M,N ) = 0 for all modules N. However, in view of the filtration theorem it suffices for this Tor group to vanish only for N = r * ( B P ) / I , , for all n. This weaker (than flatness) condition on M can be made more explicit .rr*(BP)/I, is as follows. For each n, multiplication by v, in M monic. Thus Landweber [3] shows that any M satisfying this condition gives a homology theory. For example, the spectrum E ( n ) mentioned above (in connection with Johnson-Wilson spectra) can be so obtained since

n * ( E ( n ) )= Z ( p ) [ v , ,~

. ., On, v i ' I

2 , .

satisfies Landweber's condition. [Multiplication by u, is monic in T* ( E (n ) ) i t s e l f f o r i s n , whilefori>n, .rr,(E(n))OT*(BP) ?r,(BP)/I,=Osothecondition is vacuous.] As remarked earlier, E ( 1 ) is a summand of complex K-theory localized at p . The exact functor theorem can be formulated globally in terms of MU-theory and .rr,(K) [which is a .rr,(MU)-module via the Todd genus t d : T , ( M U ) + Z] satisfies the hypotheses. Thereby one recovers the Conner-Floyd isomorphism

K * ( W = M U * ( X ) @ = * ( M U ) .rr*(K) and similarly for cohomology. In other words, complex K-theory is determined by complex cobordism. This result was first obtained by Conner and Floyd [l], whose proof relied on an explicit K-theoretic orientation of a complex vector bundle. Using similar methods they were able to show that real K-theory is determined by both symplectic and special unitary cobordism. Landweber's results have been generalized as follows. Let J = r * (B P ) be an invariant regular ideal (see Landweber [7]), and let BPJ be the spectrum obtained by killing J ; e.g., P( n ) above is BPI,,. Most of the algebra of BP-theory carries over to these spectra, which are studied systematically in a nice paper by Johnson and Yosimura [4]. The case J = I , was treated earlier by Yagita [3] and Yosimura [l]. The mod I,, version of the exact functor enables one to get K ( n ) from P(n). Johnson and Yosimura [4] also prove some important facts about .rr.+.(BP) modules M which are comodules over B P J B P ) . They show that if an element m E M is v,-torsion (i.e., it is annihilated by some power of v,) then it is v,_,-torsion. If all of the primitive elements in M [i.e., those with $( m )= 1 0 m] are v,-torsion, then so is every element, and, if none is, then M is v,-torsion free. If M is a v,-,-torsion module, then v;'M is still a

136

4. BP-Theory and the Adarns-Novikov Spectral Sequence

comodule over B P , ( B P ) . Finally, they show that u , ' B P , ( X ) = O if E ( n ) , ( X ) = 0. This last result may have been prompted by an erroneous claim by the author that the spectrum u,'BP splits as a wedge of suspensions of E ( n ) . It is clear from the methods of Wiirgler [2] that one must complete the spectra in some way before such a splitting can occur. Certain completions of MU are studied in Morava [ 2 ] . We now turn to the last topic of this section, the applications of BP-theory to unstable homotopy theory. This subject began with Steve Wilson's thesis (Wilson [ l , 21) in which he studied the spaces in the a-spectra for M U and BP. He obtained the splitting mentioned above (in connection with the Johnson-Wilson spectra) and showed that all of the spaces in question have torsion-free homology. Both the homology and cohomology of each space are either an exterior algebra on odd-dimensional generators or a polynomial algebra on even-dimensional generators. These spaces were studied more systematically in Ravenel and Wilson [ 2 ] . There we found it convenient to consider all of them simultaneously as a graded space. The mod ( p ) homology of such an object is a bigraded coalgebra. The fact that this graded space represents a multiplicative homology theory implies that its homology is a ring object in the category of bigraded coalgebras; we call such an object a Hopf ring. We show that the one in question has a simple set of generators and relations which are determined by the structure of MU*(CPm), i.e., by .rr,(MU) and the associated formal group law. We obtain similar results for the value on this graded space of any Adams oriented (4.1.1) generalized homology theory. As mentioned earlier, the complex bordism of the graded space associated with MU is the cobordism group of maps between stably complex manifolds. We show that it is a Hopf ring generated by maps from a manifold to a point and the linear embeddings of CP" in C P " + ' . The Hopf ring point of view is also essential in Ravenel and Wilson [ 3 ] , where we calculate K ( n ) , ( K ( G , m)). We show that the Hopf ring K ( n ) , ( K ( Z / ( p J )*)) , is a certain type of free object on K ( n ) , ( K ( Z / ( p ' ) , 1)). The ordinary homology of K ( Z / ( p ' ) , *) can be described in similar terms and the methods of our paper may lead to simpler proofs of the classical theorems about it (see Wilson [3], section 11.8). Knowing the BP homology of the spaces in the BP spectrum is analogous to knowing the mod ( p ) homology of the mod ( p ) Eilenberg-Mac Lane spaces. This information, along with some ingenious formal machinery, is needed to construct the unstable Adams spectral sequence, i.e., a spectral sequence for computing the homotopy groups of a space X rather than a spectrum. This was done in the BP case by Bendersky, Curtis, and Miller [l]. Their spectral sequence is especially convenient for X = S2"+'. In that

3. Some Calculations in BPJBP)

137

case they get an &-term which is a subcomplex of the usual E,-term for the sphere spectrum, i.e., of the cobar complex of A1.2.11. Their &-term is Ext in an appropriate category. For SZnflthey compute Ext', which is a subgroup of the stable Ext', and get some corresponding information about rr*( S2"+l).

In Bendersky [2] the spectral sequence is applied to the special unitary groups S U ( n ) . In Bendersky, Curtis, and Ravenel [3] the E2-terms for various spheres are related by an analog of the EHP sequence.

3. Some Calculations in BP,(BP) In this section we will prove the Morava-Landweber theorem (4.3.2), which classifies invariant prime ideals in r , ( B P ) . Then we will derive several formulas in BP,(BP) (4.1.18 and 4.1.19). These results are rather technical. Some of them are more detailed than any of the applications in this book require and they are included here only for possible future reference. The reader is advised to refer to this material only when necessary. Theorem 4.3.3 is a list of invariant regular ideals that will be needed in Chapter 5. Lemma 4.3.8 gives some generalizations of the Witt polynomials. They are used to give more explicit formulas for the formal group law (4.3.9), the coproduct (4.3.13), the conjugation (4.3.16), and the right unit (4.3.18). We define certain elements, b,,, (4.3.14) and c,,,. (4.3.19), which are used to give approximations (modulo certain prime ideals) of the coproduct (4.3.15) and right unit (4.3.20). Explicit examples of the right unit are given in 4.3.21. The coboundaries of b,,, and c,,, in the cobar complex are given in 4.3.22. In 4.3.23 we define a filtration of BP,(BP)/Z, which leads to a May spectral sequence which will be used in Section 6.3. The structure of the resulting bigraded Hopf algebroid is given in 4.3.32-34. From now on r , ( B P ) will be abbreviated by BP,. Recall (A2.2.3) that we have two sets of generators for the ring BP, given by Hazewinkel [2] (A2.2.1) and Araki [ l ] (A2.2.2). The behavior of the right unit vR:BP,+ B P , ( B P ) on the Araki generators is given by A2.2.5, i.e., 4.3.1

For the Hazewinkel generators this formula is true only mod ( p ) . This formula will enable us to define some invariant ideals in BP,. In each case it will be easy to show that the ideal in question is independent

138

4. BP-Theory and the AdamcNovikov Spectral Sequence

of the choice of generators used. The most important result of this sort is the following.

4.3.2 Theorem

(P,~

1 * ,* *

9

Un-1)

(Morava c BP*.

[3],

Landweber

[4]).

Let

I,,=

( a ) I, is inuarianr. ( b ) For n > 0 ,

EXtOsp,

B P 1 ( BP*

9

Bp*/ I n

= Z/ ( P ) [ u n

I

and

EXtOsP*cBP)W*,BP,)

=&p,.

( c ) o + p ' p " - ' ) BP,/ I,, --%BP,/ I, + BP,/ I,+1+ 0 is a short exact sequence of comodules. ( d ) The only invariant prime ideats in BP, are the I , for 0 n CO.

Prooj Part (a) follows by induction on n, using (c) for the inductive step. Part (c) is equivalent to the statement that vn

E

ExtOsp*(BP)(BP*,Bp*/fn)

and is therefore a consequence of (b). For (d) suppose J is an invariant prime ideal which properly contains some I,. Then the smallest dimensional element of J not in I,, must be invariant modulo I,, i.e., it must be in Ext;p*cBp,(BP,, B P * / l , ) , so by (b) it must be a power of u, (where uo = p ) . Since J is prime this element must be u, itself, so J 2 I,+l.If this containment is proper the argument can be repeated. Hence, if J is finitely generated, it is I,, for some n < 00. If J is infinitely generated we have J 3 I,, which is maximal, so (d) follows. Hence it remains only to prove (b). It is clear from 4.3.1 that vR(u,,)= u, mod I,, so it suffices to show that Ext~p*csp,(BP,,BP,/ I,,) is no bigger than indicated. From 4.3.1 we see that in BP,( B P ) / I,,, T R ( un+,)

v,+,

-tunf:" - u!'q

mod ( 4 ,

t2,

. ..,$ - I ) ,

so the set { un+,, q R (u,+,)Ij > 0}u { u,} is algebraically independent. It follows that if vR(u) = z, then u must be a polynomial in u,. Now we will construct some invariant regular ideals in BP,. Recall that an ideal (xo, x,, . . . ,x,,-,) c BP, is regular if x, is not a zero divisor in BP,/(x,, . . . ,x , - ~ )for 0 s i < n. This means that the sequence

. . . , x , - , ) L B P , / ( x o , . . . , x , - ~ ) +B P , / ( x o , . . . ,x,)+O is exact. The regular sequence (x,, x,, . . .) is invariant if the above is a short O+BP,/(xo,

139

3. Some Calculations in BPJBP)

exact sequence of comodules. Invariant regular ideals have been studied systematically by Landweber [7]. He shows that an invariant regular ideal with n generators is primary with radical I n , and that any invariant ideal with n generators and radical In is regular. Invariant ideals in general need not be regular, e.g., Zf: for k > 1. 4.3.3 Theorem. Let i, , i2, . . . be a sequence of positive integers such that for each n > 0, in+, is divisible by the smallest power of p not less than in, and let k 2 0. Then for each n > 0, the regular ideal ( pl+k,uillPk,u>p2k,. . . , v i , " p k n )is invariant.

In order to prove this we will need the following. 4.3.4 Lemma.

Let B, A,, A 2 , . . . be ideals in a commutative ring. Then if x=ymodpB+CAi, I

then x P " =-

y

P"

modp"+'B+

n

k=O

pkZAAPn-k. i

ProoJ The case n > 1 follows easily by induction on n from the case n = 1. For the latter suppose x = y + pbI; a, with b E B and a, E A i . Then xp=yp+

C

(~)yp-J(pb+Cai)'+(pb+~a,)P

O<jCp

and we have (y)yP-'(pb+C a , ) ' E p 2 B + p C A i and

( p b + C a i ) pE p*B+p

Ai +C A:

Proof of 4.3.3. We have u, = q R ( u n )mod I n , so we apply 4.3.4 to the ring BP,(BP) by setting B = ( l ) , A, = ( u i ) . Then we get

’-0 i = l

To prove the theorem we invariant modulo the ideal replace this ideal by the i,, . . . , in-l by the smallest the form

must show that the indicated power of u, is generated by the first n elements. It suffices to smaller one obtained by replacing each of power of p not less than it, i.e., by an ideal of

with j , =prk+kl where 0 s k , 5 k, * - 5 k n - , . Then the hypothesis on in is that it is divisible by p k n - l , so it suffices to assume that in = p k " - l . Hence we

140

4. BP-Theory and the AdamsNovikov Spectral Sequence

must show u$-1Pk=

vR(v,)'n-~Pk

mod 1.

We have u,, = v R ( v , , ) mod I,,, so we apply 4.3.4 to the ring BP,(BP) by setting B = ( l ) , A i= ( v , ) . Then we get j=o

I = ]

We are interested in the case m = kn + k,-, . Careful inspection shows that the indicated ideal in this case is contained in I. Theorem 4.3.3 leads to a list of invariant regular ideals which one might hope is complete. Unfortunately, it is not. For example, it gives { ( p k + l , v i ~ k) k 2 0, i > 0) as a list of I,-primary regular ideals, and this list 1 can be shown to be a complete for p > 2 , but at p = 2 the ideal (16, u:+ 8 ~ v2) is regular and invariant but not in the list. Similarly, for p > 2 the ideal

I

(p,v ; 2 + P - 1

9

v22P*-2uP21 u2P2-Pu;-2*f-1

u22 P 2 -

P+ I )

is invariant, regular, and not predicted by 4.3.3. This example and others like it were used by Miller and Wilson [ 3 ] to produce unexpected elements BP,, BP,. I,,) (see Section 5.2). in Extbp*(BP)( Now we will make the structure of BP,(BP) (4.1.19) more explicit. We start with the formal group law. Recall the lemma of Witt (see, e.g., Lang [ l ] , pp. 234-235) which states that there are symmetric integral polynomials w, = w,,(xI, x 2 , . . .) of degree p" in any number of variables such that wo = 2 x,

4.3.5

and

p'wf-'

xf" = 1

1

For example, 4.3.6

and for p

w1

=2

= (C ( X f ) - ( C X 1 f P ) l P

with two variables, w2

= -x:x2

2 2 -2X,X, - x,x:

.

Witt's lemma can be restated as follows. Let G be the formal group law with logarithm C l g ox p ' / p i . Then 4.3.1

G

C G x t = C w,,.

This formula is in some sense more explicit than the usual log(C" x,) = C log x,. We will derive a similar formula for the universal group law.

141

3. Some Calculations in BPJBP)

First we need some notation. Let I = ( i, , i z ,. . . , i,) be a finite (possibly empty) sequence of positive integers. Let (I1= m and 11 I ( (= X if. For positive integers n let n(n ) = p - P’~"’and define integers n(Z ) recursively by n(4) = 1 and n(Z)= n(llZ l l ) l I ( i , , . . . , Note that n(Z)= plrl mod plll+'. Given sequences I and J let ZJ denote the sequence ( i l , . . . ,i,, j , , .. . ,j , , } . Then we have lZJl= IZl+lJI and IIZJII = 11111 + 11J11. We will need the following analog of Witt's lemma (4.3.5), which we will prove at the end of this section. 4.3.8 Lemma.

( a ) For each sequence I as above there is a symmetric polynomial of degree pllll' in any number of variables with coeficients in Z, p , , w I= w I ( x l ,x 2 , . . .) with w, = Z, x, and

( b ) Let w ibe the polynomial dejined by 4.3.5. Then III II I l l

w f = wptl

~

mod(p).

Now let u, be Araki's generator and define vI by v, = 1 and vI = vi,(vl,)" where a =pil and 1'= ( i z ,i,, . . .). Hence dim vI =2(p111ii-1 ) . Then our analog of 4.3.7 is 4.3.9 Theorem.

With notation as above, c ' x , = ~ F v I W I ( X , , X *. .). ,. I

I

( A n analogous formula and proof in terms of Hazewinkel's generators cal;l,b$ obtained by replacing n(I ) by pl'l throughout. Zn this case W, becomes wbl precisely.)

ProoJ: Araki's formula (A2.2.1) is PA,=

c

Ai&,

Osisn

which can be written as n(n)A,=

2 O5tm

m

which completes the inductive step and the proof.

+F

’ 1

n>m

a,i

143

3. Some Calculations in BP,(BP)

Recall now the coproduct in B P , ( B P ) given by 4.1.18(e), i.e.,

C

log(A(ti))=

irO

1

log( tiOtg'),

i,jrO

which can be rewritten as

C'

4.3.12

A( ti) =

iZ0

C"

ti@ tf'.

i.jr0

To apply 4.3.1 1, let Mn= {ti0t zli1 0 Ii 5 n } ( M here stands for Milnor since these terms are essentially Milnor's coproduct 3.1.1) and

Then we get from 4.3.11 and 4.3.12 4.3.13 Theorem.

With notation as above,

A ( t , ) = w , ( A , ) E E P * ( B P ) O B P , BP,(BP).

For future reference we make 4.3.14 Definition. In B P , ( B P ) O B , B P , ( B P ) let b,,, = w , ( A i ) . When J = (j),bi,, will be written as b,,j.

For example,

This biSjcan be regarded as an element of degree 2 in the cobar complex (A1.2.11) C ( B P , ) . It will figure in subsequent calculations and we will give a formula for its coboundary (4.3.22) below. If we reduce modulo In, 4.3.13 simplifies as follows. 4.3.15 Corollary. A(tk)=

In BP,( B P ) OB, BP,( B P ) / I nfor k 5 2n

C tiOt[',+ Osisk

Vn+jbk-n-J,n+j-l. Osjsk-n-1

Now we will simplify the conjugation formula 4.1.18(f). Define tI as we defined v l , replacing vi by ti. Then we have 4.3.16 Lemma.

and

In BP,(BP),

144

4. BP-Theory and the Adams-Novikov Spectral Sequence

( I t can be shown that f o r p > 2, [-1](x) law. [ n ] ( x )is deJined in A.2.1.19.)

= -xfor

any p-typical formal group

Proof: In the first expression, for each I = ( i , , i2, . . . , i n ) with n > 0, the f"' expression tl appears twice: once as tlto and once as t l ~ ( t i n ) p "where 1'=(i,, . . . , in-,). These two terms have opposite formal sign and hence cancel, leaving 1 as the value of the first expression. The argument for the second expression is similar. The conjugation formula 4.1.18(f) can be written as

c log tic(

$)PI = log

1

or ZF tic( $)"' = 1, so the stated formula for ZF c( 2,) follows from the second expression above. Next we will simplify the right unit formula 4.3.1. Combining 4.3.1 with 4.3.16 gives 4.3.17

1 ’

[{-1)111](t l ( tiu;’)P"f")

i, j,l I 120

Then we get 4.3.18 Theorem. In EP,( E P ) , we have

7]R(

u,) = w+( R,,).

~ w J ( R , ) .For J 4.3.19 Definition. In BP,( E P ) , c , , = as Cq,.

=(

j ) this will be written

Again we can simplify further by reducing modulo I,,. 4.3.20 Corollary.

c

Osick

In

BP,. E P ) / In for 0 < k 5 2n,

un+,tkP'I-nR(U,~;,~,)f,

=

c

0515 k - n - I

( N o t e thar the right-hand side vanishes i f k n.) ~

un+jCk-J.n+~-l

.

145

3. Some Calculations in BP,(BP)

and

+

+ U 3 t l 3+ u*t;2 - u:t, - U f 2 t 2- U , P 3 t 3 - u; t :+p4 u; tl t 2p3 - U2P2f f4t2+ u;'t :+p + U2P3tl t i

V R (u s )= us

UJf4

-

+ U;3fp2f*+

&+P+P4_

+u2w1(u3,

U2P3t;+P+P2

u*tp2, - u , P t , ) P

for

n = 2, p > 2

(add u:t: for p

=2).

1

Now we will calculate the coboundaries of bij (4.3.14) and cij (4.3.19) in the cobar complex C ( B P , / Z , ) (A1.2.11).

Prooj (a) It suffices to assume i = n. Recall that in C ( B P , / I , ) , d ( t , )= t i @ l + l @ t i - A ( t i ) a ndd(u,+,)= n R ( u , + , ) - u n + i . A ( t 2 n ) - 1 @ t 2 n - t 2 n 0 1 , g i v e n by 4.3.13, is a coboundary and hence a cocycle. Calculating its coboundary term by term using 4.3.13 and 4.3.17 will give the desired formula for d ( b n , , - ] )and the result will follow. The details are straightforward and left to the reader.

146

4. BP-Theory and the Adnms-Novikov Spectral Sequence

For (b) we assume i = n if i + n is even and i = n - 1 if i + n is odd. Then we use the fact that d ( v 2 , , + , )is a cocycle to get the desired formula, as in the proof of (a). Now we will construct an increasing filtration on the Hopf algebroid BP,(BP)/Z,. We will use it in Section 6.3. To do this we first define integers d , , by 4 . t

=

{Omax( i, pd,,,-,,)

for i s 0 for i > 0.

We then set deg tp’=deg v,P:,=d,,,, for i, j s 0 . The subgroups F, c BP,. B P ) / I, are defined to be the smallest possible subgroups satisfying the above conditions. The associated graded algebra E,BP,( B P ) / I , , is defined by EbBP,( B P ) / Z,, = F,/ F , - , . Its structure is given by

4.3.23 Proposition. &BP*(BP)/In

= T(t,,,,U n + r , j : i > O , j z O ) ,

where tlPJand u ~ + ,are , ~ elements corresponding to t/” and va:, , respectively, T ( x )= R [ x ] / ( x P and ) R =Z/(p)[v,].

4.3.24 Theorem. With the above Jiltration, BP,( B P ) / I,, is a jiltered Hopf algebroid, and E,BP,( B P ) / I,, is a Hopf algebroid. ProoJ: For a set of elements X in BP*(BP)/Z,, or BP,(BP) Be, BP,( B P ) / I , , , let deg X be the smallest integer i such that X c F,. It suffices to show then that deg A, = deg R,+, = d,,,,.We do this by induction on i, the assertion being obvious for i = 1. First note that

4.3.25

dn.a+b

d , a + dn,b

and

4.3.26

dn,a+bn

b dn,a*

It follows from 4.3.25 that deg M, = deg Nn+, = d , ,. It remains then to show i that for 1)1(1< 4.3.21

and 4.3.28

Since 4.3.29

147

3. Some Calculations in BPJBP)

both 4.3.21 and 4.3.28 reduce to showing 4.3.30

Now if

dn,i 2

U, f

deg 0, +PIJldn,i-ll,II

0 mod I,, we can write

J

= ( n+ j i 7

rn + j k , . . . ,n + j i )

with j i 2 0 , so

IJI

llJll

= I,

I

=In+

I

j:,

and deg u, =

1 d,,j;. 1=1

1=1

If we set k = J)Jl/ - n / J \ , then 4.3.25 implies dn.k2 deg v,.

4.3.31

However, by 4.3.25 and 4.3.26

so 4.3.20 follows from 4.3.31.

ai,

We now turn to the Hopf algebroid structure of EoBP.JBP)/I,,. Let Rnfi denote the associated graded analogs of M i , h i , N,,+i, and R,,+i, respectively, with trivial elements deleted. (An element in one of the latter sets will correspond to a trivial element if its degree is less than &.) All we have to do is describe these subsets. Let ti, fir, and i 3 r ( x ) denote the associated graded elements corresponding to f,, V I , and w , ( x ) , respectively.

&, N,,+i7and

4.3.32 Lemma.

-1

M-=

U

05J5i

{ ~ , ~ 6 3 t ~ - for ~ ~ }i s m

{ ti,063 1,163 ti,J

for

i>rn

where rn = pn/ ( p - 1).

Proof: This follows from the fact that equality holds in 4.3.25 if a+bsrn.

148

4. BP-Theory and the Adams-Novikov Spectral Sequence

4.3.33 Lemma.

ii;= fi; M1 u { u,wy'-'( A4--n)}

Rnii

=I

for for

i<m i= m

for for

i=m i> m

.. . .

N,,;v { ~ , , w ~ " ~ ~ ( R ~ ~ , , ) } I-

O < IIJ 11 1 are primitive, so we could not get a Hopf algebroid with deg t,., < d,,, once we have set det t,,, = 1. Note finally that the structure of EoBP,(BP)/Z,, depends in a very essential way on the prime p. Theorem 4.3.34 implies that EoBP,( B P ) / I , , is cocommutative for n = 1 and p > 2. For any n and p we can use this filtration to construct a spectral

149

3. Some Calculations in BPJBP)

sequence as in A1.3.9. The cocommutativity in the case above permits a complete, explicit determination of the E2-term, and hence a very promising beginning for a computation of Ext Bp*( B p ) ( BP, , BP,/ I , ) . However, after investigating this method thoroughly we found the &-term to be inconveniently large and devised more efficient strategies for computing Ext, which will be described in Chapter 7. Conceivably the approach at hand could be more useful if one used a machine to do the bookkeeping. We leave the details to the interested reader. Proofof4.3.8. We will prove (a) and (b) simultaneously by induction on rn = IKI. If K ' = (1 + k,,k Z ,. . . , k,) then it follows from (b) that wK'= wi

mod (P).

Let K " = ( k , + kZ,k3,.. . , k,) and K"'= ( k 2 ,k3,. . . , k , ) . Then by the inductive hypothesis w K "and wK".exist with wK"-

where a , = p k l . Since 11 K

w",l,.

mod ( p ) ,

(1 = 11 K"IJ we have

Expanding both sides partly we get

Note that the same wf"’ occur on both sides, and one can use the definition of n(k ) to show that they have the same coefficients so the sums cancel. The remaining terms give

Since n(k,)= p mod ( p 2 ) and wK,, = w",l,, mod ( p ) , we get an integral expression when we solve for w K . This completes the proof of (a). For (b) we have

150

4. BP-Theory nod the Adams-Novikov Spectral Sequence

mod ( p1+IKI) and II( I)= pl'l mod ( pl+l'l), we get mod (p""I). By the inductive hypothesis

Since I I ( K ) I I ( K ) / I I ( I )= p

pIIJIl-IJI

wJ I[J)1 =

IlKII-lJl

so w5 - w:,~

mod ( p )

Wljl

mod (p'+I'l). Combining these two statements gives

Hence the defining equation for wK becomes

1x;""" = p l K ' w K+ C IJ

11

pllW-IJI

p wlJl

mod (p*+lKi).

=K

lll>O

Let n = IIKJJ -1KI. Substituting xf" for x, in 4.3.5 gives

Since w j ( x f ' ) = wp" mod (p),

wplKl-'(xf")= w PI ' + ' K ' - '

mod ( p l + l K l - j ) ,

so we get

Comparing this with the defining equation above gives I1 K il-I

WK

3

WIKl(Xf')

=W L I

KI

mod ( p )

as claimed.

4. Beginning Calculations with the Adams-Novikov Spectral Sequence In this section we introduce the main object of interest in this book, the Adams-Novikov spectral sequence (ANSS), i.e., the BP,-Adams spectral sequence (2.2.4). There is a different BP,-theory and hence a different ANSS for each prime p. One could consider the MU,-Adams SS (as Novikov [ l ] did originally) and capture all primes at once, but there is no apparent advantage in doing so. Stable homotopy theory is a very local (in the arithmetic sense) subject. Even though the structure formulas for BP,(BP) are more complicated than those of M U , ( M U ) (both are given in Section 1) the former are easier to work with once one gets used to them.

4. Beginning Calculations with the Adams-Novikov Spectral Sequence

151

(Admittedly this adjustment has been difficult. We hope this book, in particular the results of Section 3, will make it easier.) The ANSS was first constructed by Novikov [ l ] and the first systematic calculations at the primes 2 and 3 were done by Zahler [l]. In this section we will calculate the E2-term for t - s 5 25 at p = 2 and for t - s 5 ( p 2 + p ) q for p > 2, where q = 2 p - 2. In each case we will compute all the differentials and extensions and thereby find ..(So) through the indicated range. At p = 2 this will be done by purely algebraic methods based on a comparison of the ANSS and ASS E,-terms. At odd primes we will see that the ASS E2-term sheds no light on the ANSS and one must compute differentials by other means. Fortunately, there is only one differential in this range and it is given by Toda [2,3]. The more extensive calculations of later chapters will show that in a much larger range all nontrivial differentials follow formally from the first one. In Section 2.2 we developed the machinery necessary to set up the ANSS and we have 4.4.1 ANSS Theorem (Novikov [ 11). For any spectrum X there is a natural SS E $ * ( X ) with d , : E:‘+ Es+r*‘+r-lsuch that

( a ) E2 = EXtgp*(BP)(BP* B P * ( X ) ) and ( 6 ) ifX is connective andp-local then E$* is the bigraded group associated with the followingjltration of n*(X): a map f : S” + X hasjltration 2 s if it can be factored with s maps each of which becomes trivial after smashing the target with BP. 3

The fact that B P , ( B P ) , unlike the Steenrod algebra, is concentrated in dimensions divisible by q = 2 p - 2 has the following consequence.

4.4.2 Proposition: Sparseness. Suppose BP*( X ) is concentrated in dimensions divisible by q = 2p - 2 (e.g., X = So). Then in the ANSS for X, E:‘=O for all r and s except when t is divisible by q. Consequently d , is = E$z+q+, for all m 2 0. nontrivial only if r = 1 mod (4)and E:+*

For p = 2 this leads to the “checkerboard phenomenon”: E:‘ = 0 if t - s and s do not have the same parity. To compare the ASS and ANSS we will construct two trigraded spectral sequences converging to the ASS and ANSS E,-terms. The former is a Cartan-Eilenberg spectral sequence (CESS) (A1.3.15) for a certain Hopf algebra extension involving the Steenrod algebra, while the latter arises from a filtration of B P , ( B P ) (A1.3.9). The point is that up to reindexing these two SSs have the same E2-term. Moreover, at odd primes (but not at p = 2 ) the former SS collapses, which means that the ASS E2-term when suitably reindexed is a trigraded E2-term of an SS converging to the ANSS

152

4. BP-Theory and the AdamsNovikov Spectral Sequence

E,-term. It is reasonable to expect there to be a close relation between differentials in the trigraded filtration SS, which Miller [2] calls the "algebraic Novikov SS," and the differentials in the ASS. Miller [4] has shown that many Adams d,'s can be accounted for in this way. At any rate this indicates that at odd primes the ASS E,-term has less information than the ANSS ,!?,-term. To be more specific, recall (3.1.1) that the dual Steenrod algebra A, as an algebra is I,

E(7,,,

'.

,,...)

71,

with d i m & = 2 ' - 1 for p = 2 with dim T,= 2p' - 1 and d i m t , = 2 p 1 - 2 for p > 2 .

. .)O P ( [ , , &, . . .)

e:,

f : , . . .) for p = 2 and P( &, &, . . .) for p > 2, and let Let P* c A, be P ( E, = A, O,Z/( p), i.e. E, = E ( t , , &, . . .) for p = 2 and E , = E ( T " , 7 1 , . . .) for p > 2. Then we have

4.4.3 Theorem.

With notation as above

( a ) Ext,*(Z/(p), Z / ( p ) ) = P(ao, a , , . . .) with a, E Ext',2pt- 1 represented in the cobar complex (A1.2.11 ) by [[,I for p = 2 and [ T , ] for p > 2, ( b ) P, + A, + E, is an extension of Hopf algebras ( A I .1.15) and there is a CESS (A1.3.15) conuerging to Ext,*(Z/(p), Z / ( p ) ) with

ES2' 3s .I - Ext;*(Z/(P), Ext;;(Z/(p),

Z(P)))

and

d,: E ; , + . I E;l+r.~2-r+1,~ ~

9

(c)

the P,-coaction on Ext,*(Z/(p), Z ( p ) ) is given by

( d ) for p > 2 the CESS coilapses from E , with no nontrivial extensions. Proof: Everything is straightforward but (d). We can give A, a second grading based on the number of 7,'s which are preserved by both the product and the coproduct (they d o not preserve it at p = 2). This translates to a grading of Ext by the number of ai's which must be respected by the differentials, so the SS collapses.

For the algebraic Novikov SS, let I = ( p , ul, u 2 , . . .) c BP,. We filter BP,(BP) by powers of I and study the resulting SS (A1.3.9).

153

4. Beginning Calculations with the Adams-Novikov Spectral Sequence

4.4.4 Algebraic Novikov SS Theorem (Novikov [l], Miller [2]).

There is an SS converging to ExtBp*(Bp)(BP*,BP,) with Es,"'"= Ext>:(Z/(p), I"'/,"'+') and d , : E:"','+, Es+l*r+m*' . The ET"" of this SS coincides with the ET** of 4.4.3.

Prooj A1.3.9 gives an SS with El = ExtGBP*(BP)(EoBP,,EoBP,). Now we have BP,( B P ) / I = EoBP,/( B P ) O,Bp* Z/( p ) = P*. We apply the changeof-rings isomorphism A1.2.12 to the Hopf algebroid map (EoBP,, EoBP,(BP))+ ( Z / ( P ) , P,) and get

= Ext,BP*(

BPI( EOBP*

9

= EXt&BP,(BP)(EOBP*, P*

z/ ( p

( B P ) @&BP*

( EOBP*

) 0P*

EOBP*)

UP*EOBP*)

= EX~&BP,(BP)(E&'*, EoBP*).

The second statement ExtP*(Z/(P), Z/(P)).

=

follows

from

the

fact

that

EoBP, =

In order to use this SS we need to know its E,-term. For p > 2, 4.4.3(d) implies that it is the cohomology of the Steenrod algebra, i.e., the classical Adams E,-term suitably reindexed. This has been calculated in various ranges by Oka [2], May [l], and Liulevicius [2], but we will compute it here from scratch. Theorem 4.4.3(d) fails for p = 2 so we need another method, outlined in Miller [2] and used extensively by Aubry [l]. We start with Ext,*(Z/(p),Z/(p)). For p = 2 we have Ext:;(Z/(2), Z/(2)) = Ext;;'(Z/(2), Z/(2)), so the latter is known if we know the former through half the range of dimensions being considered. For p > 2 we will make the necessary calculation below. Then we compute Extp*(Z/(p), EoBP,/In) by downward induction on n. To start the induction, observe that through any given finite range of dimensions BP,/I,, = Z/( p ) for large enough n. For the inductive step we use the SES 0 +, p

i m

un

EOBP,/L

+

E,BP*/In

-+

&BPJIn+,+,O,

which leads to Bockstein SS (BSS) of the form 4.4.5 P(an 10Extp*(Z/ ( P ), &BP*/ 1, + 1 1

*EXtp*(Z/ ( p 1, EoBP*/

I n ),

The method we will use in this section differs only slightly from the above. We will compute the groups ExtBp*(B p ) ( BP,, BP,/ I , ) by downward induction on n ; these will be abbreviated by Ext(BP,/I,,). To start the induction we note that Ext""(BP,/ I,,) = Ext>l(Z/( p ) , Z/( p ) ) for t < 2( p" - 1). For

154

4. BP-Theory and the Adams-Novikov Spectral Sequence

the inductive step we analyze the LES of Ext groups induced by the SES 4.4.6

O+ZdimV~BP,/l,+ BP,/In -+ BP,/I,,+l-+O,

either directly or via a BSS similar to 4.4.5. The LES and BSS are related as follows. The connecting homomorphism in the former has the form 6,: Ext"(BP,/l,+,)

+ ExtS+1(Z2p"-2 BP*/L).

The target is a module over Exto(BP,/ I,) which is Z / (p ) [ u n ] for n > 0 and Z ( p ) for n = O by 4.3.2. Assume for simplicity that n > 0 . For each x E Ext(BP,/ I,,+l)there is a maximal k such that 6,(x) = u:y, i.e., such that Y E Ext(BP,/I,) is not divisible by u,. (This y is not unique but is only ~ determined modulo elements annihilated by u: .) Let j j Ext(BP,/l,+,) denote the image of y under the reduction map BP,/ I , + BP,/ I,+, . Then in the BSS there is a differential dl+k(x)= a r k j . Now we will start the process by computing E x t> l( Z /( p ) , Z /( p ) ) for p > 2 and t < ( p 2 + p + 1)q. In this range we have P* = P ( [ , , t2). We will apply the CESS (A1.3.15) to the Hopf algebra extension 4.4.1

P(51)+P(51,52)+ P ( 5 2 ) .

The E2-term is Extp(S,)(Z/(p),Ext,,&,(Z/(p), Z /( p ) ) . The extension is cocentral (A1.1.15) so we have

E, = EXtP(&d Z / ( P ) , z/ ( P))0EXtP(&/

( P 1,

z/ ( P ) ).

By a routine calculation this is in our range of dimensions E(hI0, h l , ,

h 1 2 , h20,

h21)0P(bIO, b l , , b 2 0 )

with

= The differentials are ( u p to sign) d 2 ( h 2 , j=) h l , j h l , j + l and d3(bzo) hlzbIo- h l , b l l [compare 4.3.22(a)]. The result is

4.4.8 Theorem. F o r p > 2 and t < ( p 2 + p + l ) q , Ext>l(Z/( p ) , Z / ( p ) ) is a free module ouer P(b,,) on the following 10 generators: 1 , h l o , h l l , g o =

h,, ha,

~ 1 0 ) , ~ 0 = ( ~ 1 1 h ,l , , ~ l o ~ l ~ l o ~ o = ~ ~ l l ~ o , ~ , 2 , ~ 1 0 ~ 1 2 1 ~ l l ,

There is a multiplicative relation h l l b l l = hI2blo and ( f o r p = 3 ) hllk,,= *hObll. H

The extra relation for p = 3 follows easily from Al.4.6. For p > 3 there is a corresponding Massey product relation (k,,,h , , ,. . . , h l l ) = hlobIlup to a nonzero scalar, where there are p - 2 factors h , , . The alert reader may observe that the restriction t < ( p 2 + p + l ) q is too severe to give us Ext",' for t - s < ( p 2 + p ) q because there are elements in

4. Beginning Calculations with the Adams-Novikov Spectral Sequence

155

this range with s > q, e.g., by,. However, one sees easily that in a larger range all elements with s > q are divisible by b,, and this division gets us back into the range t < ( p 2 + p + 1)q. One could make this more precise, derive some vanishing lines, and prove the following result. 4.4.9 Theorem. Let p > 2.

( a ) Ext>t(Z/(p),Z/(p))=Ofor t - s < f ( s ) where

{

(P2-P-1)S f ( s )= 2 p - 3 + ( p 2 - p - l ) ( s - 1 )

(b) Let R,= P*/(&, where

c2). Then Ext:k(Z/(p),

{2( Pp43--P3-+ ( p 4 - p - l ) ( s - 1 ) 1)s

d s )= (c)

The map P ( t , ,

t2)+ P*

for s even for s odd. Z(p))=O for t - s < g ( s ) , for s even for s odd.

induces an epimorphism in ExtS3'for

( t - s) < h(s) and an isomorphism for ( t - s) < h ( s - 1) - 1, where

h ( s )=2p3-3+f(s-l)

={

2p3 -3 + ( p 2 - p - 1)(s - 1) 2p3+ 2p - 6 + ( p 2 - p - 1)( s - 2)

for s o d d for seven.

This result is far more than we need, and we leave the details to the interested reader. Now we start feeding in the generators v, inductively. In our range 4.4.8 gives us Ext( BP,/ 13).Each of the specified generators is easily seen to come from a cocycle in the cobar complex C(BP,/12) so we have Ext( BP,/ 12) = Ext( Bp*/ 1 3 ) 0 P(v2 ), i.e., the BSS collapses in our range. The passage to Ext(BP,/I,) is far more complicated. The following formulas in C(BP,/ I , ) are relevant. 4.4.10

(a)

d(v2)= vltf'-u7tl

and

(b) d(t,)=-t,

1 t7-uu,b,,.

These follow immediately from 4.3.20 and 4.3.15. From 4.4.10(a) we get 4.4.11

(a)

a,( u:) = iu;-'h,

(b)

a,( ug) = vy-lh,, mod ( u y ) .

mod ( u , )

and

156

4. BP-Theory and the AdamsNovikov Spectral Sequence

Next we look at elements in Ext'(BP,/12). Clearly, h,,, h , , , and h , , are in ker 6, as are u:h,, for i < p - 1 by the above calculation. This leaves uSh,, for 1 5 i s p - 1 and u$-'hll. For the former 4.4.10 gives

d ( uir, + iu,ui-'(

- t2)) = iu:u;-'b,,

The expression in the second term is a multiple of ko, so we have 4.4.12

To deal with u$-'h,, we use 4.4.10(a) to show

so 81(u$-'hll)= fuf-'b,, .

4.4.13

This is a special case of 4.3.22(b). Now we move on to the elements in Ext2(BP,/12). They are hloh12,b,,, uib,,, u:go, and u;ko for suitable i. The first two are clearly in ker 8,. Equation 4.4.12 eliminates the need to consider u;blofor i < p - 1, so that leaves V $ - ' b , , , uhg,, and uhk,. Routine calculation with 4.4.10 gives (a)

8,(&go)= *(u ~ h l o b l o *iul?-'hlob) mod ( u : )

and

( b ) 6 1 ( u ~ k o ) ~ * u ~ h l l b , o( m u ;o ) .d We have to handle V : - ' b , o more indirectly. 4.4.14 Lemma.

8 1 ( u ~ - 1 b l oI * p~- 2u k,) 2

=c

~ f - ~ h , ~ b for ,,

some

nonzero

cEZ/(P).

ProoJ: By 4.4.13, u f ~ ' b , , = Oin Ext(BP,.I,), so u,P-lhlobll=O and ufhlobll= 8,(x) for some i < p - I and some x E Ext2(BP*/12). The only remaining x is the indicated one. From 4.4.14 we get 8 1 ( u ~ ~ 1 h 1 0 b 1 0 * ~ u ~= ~ 02 ~mod l o k(up-'). o) All other elements in Exts(BP,.12) for s 2 3 are divisible by h , , or b,, and they can all be accounted for in such a way that the above element, which we denote by 6, must be in ker 8,. Hence 6, is completely determined in our range. Equivalently, we have computed all of the differentials in the BSS. However, there are some multiplicative extensions which still need to be worked out.

157

4. Beginning Calculations with the Adams-Novikov Spectral Sequence

4.4.15 Theorem.

For p > 2 Ext(BP,/Z,) = P ( u , ) O E ( h , , ) O M , where M is a free module over P ( b,,) on the following generators:

Pz = 6,(4),

hlOPI,

6 = u;'6l(~;hl,)(e.g.9

h,&

for

P1

=*ho),

and

1sisp-1;

p p / l= u:-'S,(uq) and hlOPpl, for 1 5 i ' p ; /3p,l= u;-’S,(u$-’h,,) for 2 1 i s p ; hlOBp,l

6

for 3 5 i 5 p ;

and

P1Pplp.

(Here 6, is the connecting homomorphism for the YBP*/ II-%BP*/ II+ BP*/ z, + 0.) Moreouer, h1oPz = U I P ; , U l P I = 0, uPPp/p= 0, v ~ - l & ~=p0,

and

SES

O+

v P - 2 h l o ~ p=l p0.

( This description of the multiplicative structure is not complete.)

Roo$ The additive structure of this Ext follows from the above calculations. The relations follow from the way the elements are defined. H

Figure 4.4.16 illustrates this result for p = 5 . Horizontal lines indicate multiplication by u l , and an arrow pointing to the right indicates that the element is free over P ( v l ) . A diagonal line which increases s and t / q by one indicates multiplication by h,, and one which increases t / q by 4 indicates the Massey product operation (-, hlo,h,,, h,,, h,,). Thus two successive diagonal lines indicate multiplication by b,, = *(hlo,hlo, hlo, hlo,hlo).The broken line on the right indicates the limit of our calculation. Now we have to consider the LES or BSS associated with

O + B P , A BP,+ BP*/I,+O. First we compute 6 , ( u ; ) . Since d ( v l ) = p t l in C(BP,) we have d ( u i ) = ipvlp't, mod (ip2),so 4.4.17

So(u i ) = iulp1h,,

mod (ip2).

Moving on to Ext'(BP,/I,) we need to compute 6, on P, and Ppll.The former can be handled most easily as follows. So(/?,) = 0 because there is no element in the appropriate grading in Ext3. 6, is a derivation mod ( p ) so G o ( u , ~ ,=) h,,B,. Since u,B, = hloP, we have h,,P, = 6,(h,,P,) = hl06,(P,) so 4.4.18

SO(P,) =

PI.

t

4

m I I

4

rj

1

I

4

r(

I

I

4

0 1

I

m 1 1

W 1 1

.r

-w

I

I

m

< 1

1

I

I

(?

I

I

-4 1

1

I

4

I

C

159

4. Beginning Calculations with the Adams-Novikov Spectral Sequence

Now P p l p= h 1 2 - v f Z - p h , ,and v f 2 - P h l lis cohomologous to v f 2 - 1 h 1 0 , which by 4.4.16 is in ker So. Hence 4.4.19

8 0 ( P p / p= ) S o ( h 2 ) = b,, =

*Pp/p.

It follows that SO(Pp/p-n)

= So(U;P,/,)

= iV;-lhlOPp/pf . ; P p / p .

This accounts for all elements in sight but So(h l O P p I l )which , vanishes mod ( p ) . We will show that it is a unit multiple of p @ below in 5.1.24. Putting all this together gives 4.4.20 Theorem. For p > 2 and t - s 5 ( p 2 + p ) q , Ext( BP,) is as follows. Ext' = 2,p ) concentrated in dimension zero. Ext'-4' = Z(p J ( p i ) generated by ti,= i-'SO(vi),where a , = h , ? . For s 2 2 Ext" is generated by all biox, where

xisoneofthefollowing: P t = 6 0 ( ~ , ) ( w h e r e ~ , = * b o ) a n d a , P , lf os ri s p - 1 ; ~ p ~ p ~ , = S o ( ~for p ~ Op s~ i ls )p - 1 ; a l p p l p - ,for O ~ i s p - 3 ;and + = p - ' S o ( h , o ~ p which ~ , ) has order p 2 . is a unit multiple of ( P P l 2 ,a , , a,) and p(p is a unit multiple of a l p p I ,Here . PI/, denotes the image under So of the corresponding element in Ext( BP,/ I , ).

For p = 5 this is illustrated in Fig. 4.4.21, with notation similar to that of and Fig. 4.4.16. It also shows differentials (long lines originating at P I P s I 5 ) , which we discuss now. By sparseness (4.4.2) E 2 = E2p-1 and d2,_, : E $ - , + E $ ~ ~ - 1 ~ t - .2 It p cis2 clear that in our range of dimensions E2p= Embecause any higher (than differential would have a target whose filtration (the s-coordinate) would be too high. Naively, the first possible differential is d 2 p - I ( a p 2 -=l )cPP. However, d,,-, respects multiplication by a , and a1aP2-,= 0 so caIPP = 0 and c = 0. Alternatively one can show (see 5.3.7) that each element in Ext' is a permanent cycle. 4.4.22 Theorem (Toda [ 2 , 3 ] ) . d 2 p - l ( / 3 p / p=) aa#f

aEZl(p).

for some nonzero

rn

Toda shows that any x E r*(S)of order p must satisfy a l x P= 0. For x = PI this shows alp: = 0 in homotopy. Since it is nonzero in E2 it must be killed by a differential and our calculation shows that Pplp is the only possible source for it. We do not know how to compute the coefficient a, but its value seems to be of little consequence. Theorem 4.4.22 implies that d Z p - l ( P I P p l=p )alp:". Inspection of 4.4.20 or 4.4.21 shows that there are no other nontrivial differentials. Notice that the element a l p p I survives p to E , even though P p I pdoes not. Hence the corresponding homotopy element, usually denoted by E ' , is indecomposable. It follows easily from the definition of Massey products (A1.4.1) that ( a , , a , , Pf) is defined in E2p, has trivial indeterminacy, and

m I I

i

-

J

d I

1

~

a

l

h

9 I

~ I 1

~ I

m I

c 1

r

1

I I

1

I

1

1

I

I

I

I

1

I

1

I

1

I

I

I

I I

I

r

1

1

t

d

I

I

1

1

d

I

I 1

I

d

I

N

F

E

u ,

i

c

J

Ph 1 1

0

Figure 4.4.23. (a) Ext(BP,/I,) (c) Ext(BP,/I,) for 1 - s c 2 7 .

t - 5 -

30

20

for p = 2 and t - s i 2 9 . (b) Ext(BP,/I,)

161

PO I1

for r - s s 2 8 .

162

4. BP-Theory and the Adams-Novikov Spectral Sequence

contains a unit multiple of a,p,,,. It follows from 7.4.9 that corresponding Toda bracket. Using Al.4.6 we have

E'

is the

(a,,...) a , , d ) = ( a l)...)a,)p;=p;+'

with p - 2 at's on the left and p at's on the right. Looking ahead we can see this phenomenon generalize as follows. For l s i s p - 1 we have d 2 p - I ( ~ ~ , p ) = i a aFor , ~ ~i s~ p' .- 2 this leads to (a1,. . . , a,,pip) [with ( i + l ) a , ' s ] being a unit multiple of and ( a , , a , , . . . , a l , E " ) ) with ( p - i- 1) at's is a unit multiple of / 3 : + I P . In particular, ~ , E ' P - ~is) a unit multiple of p:+(p-2)p . Since a l p : = 0 (4.4.22), --P+l= o since it is a unit multiple of a L p 7 X ( p - 2 . However, ) in the E2-term all powers of PI are nonzero (Section 6.4), so /37 2- + I must be killed by a differential, more precisely by d ~ p - l ) q + l ( a l P ~ j ~ ) . Now we will make an analogous calculation for p = 2 . The first three steps are shown Fig. 4.4.23. In (a) we have Extp*(Z/(2),Z(2)), which is Ext(BP,/ I,) for t - s 5 29. Since differentials in the BSSs and the ANSS all lower t - s by 1, we lose a dimension with each SS. In (a) we give elements the same names they have in ExtA*(Z/(2),Z/(2)). Hence we have co= ( h , , , hlo) and Px = (x, hfo, hI2). Diagonal lines indicate multiplication by hlo, h , , , and h12.The arrow pointing up and to the right indicates that all powers of h , , are nontrivial, The BSS for Ext(BP,/f,) collapses and the result is shown in Fig. 4.4.23(b). The next BSS has some differentials. Recall that 62 is the connecting homomorphism for the SES € ( I ) =

0 + E6BP,/ I2

"2

BP,/ l2+ BP,. I3 + 0.

Since q R ( u j ) = u,+ u2t:+u:f1 mod I2 by 4.3.1 we have 4.4.24

( a ) S 2 ( u 3 h ~ , ) = ( h , 2 + u 2 h , o ) h ~ o for

(b)

62(~3h;0)=~2hif;l

(c)

82(u3h12)=

(d)

is2,

for i 2 3 ,

for

i = 1,2,

62(v:)=v2hl3+U:hll-

This accounts for all the nontrivial values of S2. In Ext(BP,/12) we denote 6,(u;) by y, and u ; ' S 2 ( u ~ ) by y 2 / 2 . The elements u 3 h , , , u3hI0hl2E Ext( BP,/I,) are in ker 62 and hence lift back to Ext( BP,/I2), where we denote them by J2 and x~~~respectively. They are represented in C(BP,/I2) by 4.4.25 ( a ) ~ 2 = u , t ~ + u 2 ( t ~ + t ~ ) + u and ~f2

( b ) xZ2= u3tlIt':+u2(t21t:+ t,lt:t:+ t21f:+ t,lt:+ t;ltf)

+ u:(f21f2+

t l l f : f 2 +t l t 2 ( f : ) .

4. Beginning Calculations with the Adams-Novikov Spectral Sequence

163

Now we pass to Ext(BP,/Z,). To compute S, on Exto(BP,/Z,) we have ulr;+ u:t, mod I , , so

vR(u2)= uz+ 4.4.26

SI(U2)'

h l

mod

(Vl),

= v l h l z = U , ( Y , + u2hlO)mod (ui), a,( u:) = v:hll mod ( u , ) , S1(&

S l ( v ; ) = u:h13= ~ ~ ( ~ ~ , ~ + v ~ h ~ ~ ) m o d ( u ~ ) ,

a,( u:) = u:h,, mod ( ul). This means that in Ext1(BP,/Z2) it suffices to compute 61 on u2h,,, 12, u:hlo, r2,and u&. We find S,(u2h1,)=0 and the element pulls back to

+

x7 = uZrl u,( t,+ t i ) .

4.4.27

For l2we compute in C ( B P , / I , ) and get d ( 5 2 + u 1 t : t : ) = u , ( t ~ ( t ~ + u ~ t mod l ( t l )( u : )

so 4.4.28

S*(l2)

= r: mod ( U I )

For ulh,, we compute d ( u : t l + u , u : ( t 2 + t : ) + u : u 3 t l ) = v:u:tl(tl mod ( u : ) so

4.4.29

S,(u:h,,) = u,u:h:, mod ( u : ) .

Similar calculations give 4.4.30

S,(Y,) = h l l Y 2 , 2 mod

(ui)

and sl(u252)'h,152+u:h:omod In Ext2BP,/12 it suffices to compute 4.4.31

(u:)

a1 on x2,. We will show

Sl(X22) = co

using Massey products. Since x2* projects to u3hl0hl2 we have ~ 2 (uz, 2 Y:, ~ hi,), SO ~ I ( x ~ , ) E ( ~d ~ ,(hu i dz ) by , A1.4.11. This is ( h , , , $, h,,), which is easily seen to be c,. This completes our calculation of 8, . The resulting value of Ext( BP,/Z,) is shown in Fig. 4.4.32. The elements 1 and x7 are free over P ( ul , hlo). As usual we denote u : - ' ~ , ( u ; )by pilj. x7 is defined by 4.4.27. 77, and v2 (not to be confused with the vj of Mahowald [ 6 ] ) denote 6 , ( 1 2 )and 61(u212).

164

4. BP-Theory and the Adams-Novikov Spectral Sequence

Pi?',

0

20

i0 t

Figure 4.4.32.

-

s-

Ext( BP,/ I , ) for p = 2 for f - s 5 26.

4 30

165

4. Beginning Calculations with the AdamsNovikov Spectral Sequence

Now we pass to Ext(BP,) by computing Exto(BP,/ I t ) = P ( u l ) . By direct calculation we have

a0,

beginning with

4.4.34

To handle larger even powers of u l , consider the formal expression u = u: - 4u;'u2. Using the formula (in terms o f Hazewinkel's generators A2.2.1) qJdu2) = tlz- 5u1t:- 3 v : t 1 +2t2-4r:,

we find that d ( u ) = ~

u F in ~ xC ~(U ; ' B P , / ( ~ ~ )It) .follows that

d ( ~ ~ - 2 ~ u ; *=u24(x7+p2,2) ;) mod ( Z 5 )

and for i > 2 d ( u ' ) = 8 i ~ : ' - ~mod x ~ (16i)

so So(u:) = 23(x7+ P2,J mod (24)

4.4.35

and

a,( uf') = 4 i ~ : ' - ~ mod x, (8i)

for i 2 3.

Combining this with 4.4.36

2i+l

ao(u1

j

2i

j+l

ho)= u1 ho

accounts for all elements of the form u';h',xS for i, j we have 4.4.37 Theorem.

20

and

E

= 0,

1 and

For p = 2 Ext'( BP,) is generated by cUj for i 2 1 where

(1/2i)80(ui)

for i odd for i = 2 for even i r 4

In particular 6 ,= hlo. Moreover LY{LYi # 0 for all j > 0 and i # 2.

166

4. BP-Theory and the Adams-Novikov Spectral Sequence

Moving on to Ext'(BP,/Z,) we still need to compute So on h12, u l h l 2 , i 3. An easy calculation gives

p3, and u { h , , for 0 ' j 4.4.38

SO(hl2)

=

mod (21, mod f2),

6o(fJlhl2)= h l O h 2

So(h,,) = h:2

mod ( 2 ) ,

= Ulh:2+ hob,,

mod ( 2 ) ,

= 2 ( h ,+ ~ l ~ , O ) h , ,

mod (4),

60(~lhI,)

So(U:hl,)

and 80(u:h13)

For

mod ( 2 ) .

v:h10h13

P3 we have

4.4.39

So(P3) = P $ 2 +

771.

The proof is deferred until the next chapter (5.1.25). In Ext2(BP,. I , ) all the calculations are straightforward except q 2 and x7P4/,.The former gives 4.4.40

So( 772) = co

9

which we defer to 5.1.25. For the latter we have s0(x7P4/4)

= x7h:2 mod ( 2 ) .

Computing in C (BP,/12) we get d(t:lt,+

so

r2)r:+

r:lt:+

t:rzlr:) = t,lt:lr:+

u2tI(rl[r, 2 2 2

= P3P: mod ( u : ) and

~7h:2

4.4.41

S0(x7P4/4)

P:P3+ Ch:OP4/2

mod ( 2 )

for c = O or 1. Note that SO(hIOP4)

= h:OP4/2.

We also get from 4.4.37 4.4.42

Mx,P4/3) = ch:oP4+

60(x,p4/2) must be a multiple of h,&P4/, 4.4.43

s0(x7p4/2)

mod ( 2 ) .

hi0~7P414

but the latter is not in ker So so = 0.

Of the remaining calculations of all are easy but p:p4/4and h&P4= /3:/34,4. It is clear that sO(P:P4,4) and 6O(p:P4/4) are multiples of elements which reduce to h:Op4/3 and PP, , respectively. Since P:P:,, = 0 and

4. Beginning Calculations with the Adams-Novikov Spectral Sequence

167

P1Ph:OP4/3 = 0 we have S0(&34/4) = 0 mod (2) and So(P:p4/4) = mod (4). Thus the simplest possible result is 4.4.44

i80(P;P4/4) 3 0 ( P k / 4 )

h:OP4/3

mod ( 2 ) ,

= PP, mod ( 2 ) .

We will see below that larger values of the corresponding Ext groups would lead to a contradiction. The resulting value of Ext(BP,) is shown in Fig. 4.4.45. Here squares denote elements of order greater than 2. The order of the elements in Ext' have order 4 while is given in 4.4.37. The generators of Ext2,20and has order 8. that of

S

20

10 t

-

30

s-

Figure 4.4.45. Ext( BP,) for p

= 2, t

- s I25.

We compute differentials and group extensions in the ANSS for p = 2 by comparing it with the ASS. The €?,-term of the latter as computed by Tangora [ l ] is shown in Fig. 4.4.46. This procedure will determine all differentials and extensions in the ASS in this range as well. The Adams element h, corresponds to the Novikov E l . Since h: = 0, a: must be killed by a differential, and it must be d3((Y3).It can be shown that the periodicity operator P in the ASS (see 3.4.6) corresponds to multiplication by v:, so Pihl corresponds to so d3(cU4i+3) = cY:cY4i+l. The relation h;h,= hi gives a group extension in the ANSS, 2 c ~ ~ ~ + ~ ~ c U : c Uin4 ~ + 1 homotopy. The element P'h, for i > 0 corresponds to 2Eqi+2. This element

168

4. BP-Tbcory a d the Adams-Novikov Spectral Sequence

S

0

10 t

Figure 4.4.46.

I

I

20

30

- s-

ExtA*(Z/2, 2/2) for I - s 5 25.

is not divisible by 2 in the ASS so we deduce d3(&4,+2)= (Y:CT,~ Summing up we have

for i > 0.

4.4.41 Theorem. The elements in Ext( BP,) for p = 2 listed in 4.4.37 behave in the ANSS as fo2lows. d3(c?,~+J = C?:CT4,+, for i 2 0 and d3((Yqi+*) = ci:Gqi

for i 2 1. Moreover the homotopy element corresponding to a 4 , + 2 = rci4i+zdoes not have order 2; twice it is C?;E,, for i 2 1 and 6 ; for i = 0. As it happens, there are no other ANSS differentials in this range, although there are some nontrivial extensions. These elements in the ANSS E,-term correspond to Adams elements near the vanishing line. The towers in dimensions congruent to 7 mod (8) correspond to the groups generated by ~5,~. Thus the order of a,, determines how many elements in the tower survive to the Adams Em-term.For example, the tower in dimension 15 generated by h, has 8 elements. C8 has order 2’ so only the top elements can survive. From this we deduce d3(hbh,) = h6do for i = 1, 2 and either d3(h,) = do or d,( h,) = h,h:. To determine which of these two occurs we consult the ANSS and see that fi3 and fi4/, must be permanent so 7r7,=2(2)02/(2). If d3(h4)= do the ASS would give 7rf4= Z/(4), so we must have dz(h4)= hob:. One can also show that Pico corresponds to crldi4i+4 for i 2 1 and this leads to a nontrivial multiplicative extension in the ASS. For example, the homotopy element corresponding to Pco is a1times the one corresponding to hih,.

169

4. Beginning Calculations with the AdamsNovikov Spectral Sequence

The correspondence between ANSS and ASS permanent cycles is shown in the following table. 4.4.48 Table Correspondence between ANSS and ASS Permanent Cycles for p=2, 14St-sS24 A N S S element

ASS element

A N S S element

ASS element

4.4.49 Corollary. The ANSS has nontrivial group extensions in dimensions 18 and 20 and the homotopy product p462 is detected in filtration 4. I

4.4.50 Corollary. ASS for p = 2. d2(h4)=

For 14 It - s 5 24 the following diflerentials occur in the hOh:,

d2(f,) = hie,,

d2(eO)=

d3(hOh4)=

d2(i ) = h,Pd,,

and

h:dO,

d2(Pe,) = h:Pd.

There are nontrivial multiplicative extensions as follows: h, . hih4 = Pc,,

h, * g = Pd,,

and

h, . hie, = h, Pd, = h, . hid,.

I

CHAPTER 5

THE CHROMATIC SPECTRAL SEQUENCE

The SS of the title is a mechanism for organizing the ANSS E2-term and ultimately m*(S0) itself. The basic idea is this. If an element x in the &-term, which we abbreviate by Ext(BP,) (see 5.1.1), is annihilated by a power of p, say pi,then it is the image of some X ' E Ext(BP,/p') under a suitable connecting homomorphism. In this latter group one has multiplication by a suitable power of v , (depending on i ) , say v? . x' may or may not be annihilated by some power of v?, say v;". If not, we say x is v,-periodic; otherwise x' is the image of some X"E Ext(BP,/(p', v?')) and we say it is u,-torsion. In this new Ext group one has multiplication by v; for some n. If x is v,-torsion, it is either v2-periodic or v,-torsion depending on whether X" is killed by some power of v ; . Iterating this procedure one obtains a complete filtration of the original Ext group in which the nth subgroup in the vn-torsion and the nth subquotient is v,-periodic. This is the chromatic filtration and it is associated with the chromatic spectral sequence (CSS) of 5.1.8. The CSS is like a spectrum in the astronomical sense in that it resolves stable homotopy into periodic components of various types. Recently we have shown that this algebraic construction has a geometric origin, i.e., that there is a corresponding filtration of r * ( S 0 ) . The CSS is based on certain inductively defined SESs of comodules 5.1.5. In Ravenel [9] we show that each of these can be realized by a cofibration N n

+

Mn

+

170

Nn+,

Introduction

171

with No= So so we get an inverse system

) the one we want. The filtration of r , ( S 0 ) by the images of T ~ * ( E Y " N " is Applying the Novikov Ext functor to this diagram yields the CSS, and applying homotopy yields a geometric form of it. For more discussion of this and related problems see Ravenel [8]. The CSS is useful computationally as well as conceptually. In 5.1.10 we introduce the chromatic cobar complex CC(BP,). Even though it is larger than the already ponderous cobar complex C(BP,), it is easier to work with because many cohomology classes (e.g., the Greek letter elements) have far simpler cocycle representatives in CC than in C. In Section 1 the basic properties of the CSS are given, most notably the change-of-rings theorem 5.1.14, which equates certain Ext groups with the cohomology of certain Hopf algebras E(n), the nth Morava stabilizer algebra. This isomorphism enables one to compute these groups and was the original motivation for the CSS. These computations will be the subject of the next chapter. Section 1 also contains various computations (5.1.20-22 and 5.1.24) which illustrate the use of the chromatic cobar complex. In Section 2 we compute various Ext' groups (5.2.6, 5.2.14, and 5.2.18) and recover as a corollary the Hopf invariant one theorem (5.2.8), which says almost all elements in the ASS Ei3* are not permanent cycles. Our method of proof is to show they are not in the image of the ANSS E$* after computing the latter. In Section 3 we compute the v,-periodic part of the ANSS and its relation to the J-homomorphism and the p-family of Adams [l]. The main result is 5.3.7, and the resulting pattern in the ANSS for p = 2 is illustrated in 5.3.8. In Section 4 we describe Ext2 for all primes (5.4.5), referring to the original papers for the proofs, which we cannot improve upon. Corollaries are the nontriviality of y, (5.4.4) and a list of elements in the ASS E?" which cannot be permanent cycles (5.4.7). This latter result is an analog of the Hopf invariant one theorem. The ASS elements not so excluded include the Arf invariant and vj families. These are discussed in 5.4.8-10. In Section 5 we compile all known results about which elements in Ext2 are permanent cycles, i.e., about the P-family and its generalizations. We survey the relevant work of Smith and Oka for p 2 5 , Oka and Toda for p = 3, and Davis and Mahowald for p = 2. In Section 6 we give some fragmentary results on Ext" for s 2 3 . We describe some products of a's and P's and their divisibility properties. We close the chapter by describing a possible obstruction to the existence of the &family.

172

5. The Chromatic Spectral Sequence

1. The Algebraic Construction In this section we set up the chromatic spectral sequence (CSS) converging to the ANSS E,-term, and use it to make some simple calculations involving Greek letter elements (1.3.17 and 1.3.19). The CSS was originally formulated by Miller, Ravenel, and Wilson [ 13. First we make the following abbreviation in notation, which will be in force throughout this chapter: given a BP,( B P ) comodule M (A1.1.2), we define 5.1.1

Ext(M) = ExtBp*(BP)(BP*,MI-

To motivate our construction recall the SES of comodules given by 4.3.2(c) O+x2(P'-l)BP / I 5.1.2 * n BP*lIn + BP*/In+I + O ",I

+

and let 6, : Ext"( BP,/ I,,,)

-+ ExtS+'(B P J I , )

denote the corresponding connecting homomorphism. 5.1.3 Definition.

For t, n > 0 let , = 606, * * * 6 , - I ( u : ) ~ Ext"(BP,).

Here a ( " )stands for the nth letter of the Greek alphabet. The status of these elements in T : is described in 1.3.11, 1.3.15, and 1.3.18. The invariant prime ideals in I, in j 1 . 2 can be replaced by invariant regular ideals, e.g., those provided by 4.3.3. In particular we have C Y , ~ ~, E ~ ,Ext'sqsP'( + BP,) (where q = 2 p - 2 ) is the image of usp' under the connecting homomorphism for the SES

5.1.4 Definition.

O + BP,

CB P , + B P , / ( ~ ' + ~ ) + o .

We will see below that for p > 2 these elements generate Ext'( BP,) (5.2.6) and that they are nontrivial permanent cycles in im J. We want to capture all of these elements from a single SES; those of 5.1.4 are related by the commutative diagram

Taking the direct limit we get

173

1. The Algebraic Construction

we denote these three modules by N o , M o , and N ' , respectively. Similarly, the direct limit of the sequences O+ BP,/(p'+')

Llf'

,x - w ' + JBP,/(

+ x-qp'+' BP,/(

PI+')

PI+',

uy'+') + 0

gives us 0 + BP,/( p") + u;'BP,/( p") + BP,/( pa, u y ) + 0

and we denote these three modules by N ' , M ' , and N 2 ,respectively. More generally we construct SESs O + N" + M " + N " + ~ + o

5.1.5

inductively by M " = u i ' B P , OBP*N". Hence N " and M " are generated as Z(,,-modules by fractions x /y where x E BP, for N" and vi'BP, for M " and y is a monomial in the ideal ( p u , . . . of the subring Z,,,[ ul , . . . , u,-~] of BP,. The BP,-module structure is such that u x / y = 0 for v E BP, if this fraction when reduced to lowest terms does not have its 1 denominator in the above ideal. For example, the element Y E N 2 is

P

annihilated by the ideal

(PI,

u:

u:).

5.1.6 Lemma. 5.1.5 is an SES of BP,(BP)-comodules.

ProoJ: Assume inductively that N" is a comodule and let N ' c N" be a finitely generated subcomodule. Then N' is annihilated by some invariant regular ideal with n generators given by 4.3.3. It follows from 4.3.3 that multiplication by some power of u,, say u t , is a comodule map, so u;l

NI

-

= lim x - d l m

u,"'

N'

k

0.

is a comodule. Alternatively, N ' is annihilated by some power of I,, so multiplication by a suitable power of u, is a comodule map by Proposition 3.6 of Landweber [7] and v , ' N ' is again a comodule. Taking the direct limit over all such N' gives us a unique comodule structure on M " and hence on the quotient N"+'. W 5.1.7 Definition.

The chromatic resolution is the LES of comodules O+BP,+ MO-%

obtained by splicing the SESs of 5.1.5.

d

MIA..

.

W

By standard homological algebra (A1.3.2) we get There is a chromatic spectral sequence ( C S S ) converging to Ext( BP,) with E ?' = Ext"( M ") and d, : E + E:+r7S+'-rwhere d , is the map induced by d , in 5.1.7.

5.1.8 Proposition.

:"

174

5. The Chromatic Spectral Sequence

There is a CSS converging to Ext(F) where F is any comodule which is flat as a BP,-module, obtained by tensoring the resolution of 5.1.7 with F.

5.1.9. Remark.

5.1.10 Definition.

The chromatic cobar complex CC(BP,) is giuen by CC"(BP,) Cs((M"),where C ( ) is the cobar complex of A1.2.12, with d ( x ) = d : ( x ) + ( - l ) " d , ( x ) f o rX E C"(M,) where d : is the map induced by d, in 5.1.7 (the external component of d ) and d, (the internal component) is the differential in the cobar complex C(A4,).

=eS+,=,

It follows from 5.1.8 and A1.3.4 that H(CC(BP,)) = H ( C (BP,)) = Ext( BP,). The embedding BP, + M' induces an embedding of the cobar complex C (BP,) into the chromatic cobar complex CC( BP,). Although CC(BP,) is larger than C ( BP,), we will see below that it is more convenient for certain calculations such as identifying the Greek letter elements of 5.1.3. This entire construction can be generalized to BP,/I, as follows. Let NO, = BP,/I,

5.1.11 Definition.

and dejine BP,-modules N : and M :

inductively by SESs O - , N : + M : + N:+'+O where M :

= u;:,BP,

Os, N : .

Lemma 5.1.6 can be generalized to show that these are comodules. Splicing them gives an LES 0 -+ BP,/ I ,

+ ML

--%A4; A .

,

and a CSS as in 5.1.8. Moreover BP,/I, can be replaced by any comodule L having an increasing filtration { F i L } such that each subquotient F,/F,-, is a suspension of BP,. I,, e.g., L = BP,/ I", We leave the details to the interested reader. Our main motivation here, besides the Greek letter construction, is the computability of Ext( M : ) ; it is essentially the cohomology of the automorphism group of a formal group law of height n (1.4.3 and A2.2.17). This theory will be the subject of Chapter 6. We will state the first major result now. We have M ; = u;'BP,/ I,, which is a comodule algebra (A1.1.2), so Ext(M:) is a ring (A1.2.14). In particular it is a module over Exto(M;). The following is an easy consequence of the Morava-Landweber theorem, 4.3.2. 5.1.12 Proposition. For n > 0, Exto(M : ) = Z/( p ) [u,, u , ' ] . We denote this ring by K ( n),. [The case n = 0 is covered by 5.2.1, so it is consistent to denote

Q by K(O)*.I

175

1. The Algebraic Construction

5.1.13 Definition. Make K ( n), a BP,-module by dejning multiplication by vi to be trivial for i # n. Then let C(n)= rn K ( n ) * O B P * BP*(BP) O B P * K ( n ) * . Z( n), the nth Morava stabilizer algebra, is a Hopf algebroid which will be closely examined in the next chapter. It has previously been called K(n),K(n), e.g., in Miller, Ravenel, and Wilson [l], Miller and Ravenel [5], and Ravenel [ 5 , 6 ] . K ( n ) * is also the coefficient ring of the nth Morava K-theory; see Section 4.2 for references. We have changed our notation to avoid confusion with K(n),(K(n)), which is C(n) tensored with a certain exterior algebra. The starting point of Chapter 6 is

5.1.14 Change-of-Rings Theorem (Miller and Ravenel [ 5 ] ) . Ext(MO,)= Extx.,(K(n)*, K(n>*). We will also show (6.2.10)

5.1.15 Morava Vanishing Theorem. r f ( p - 1) Y n then Exts( M z ) = 0 f o r s > n2. Moreover this Ext satisfies a kind of PoincarC duality, e.g., Exts(M:) = Ext"'-"(M:), and it is essentially the cohomology of a certain n stage nilpotent Lie algebra of rank n2. If we replace C(n) with a quotient by a sufficiently large finitely generated subalgebra, then this Lie algebra becomes abelian and the Ext [even if ( p - 1 ) divides n] becomes an exterior algebra over K ( n ) , on n 2 generstors of degree one. To connect these groups with the CSS we have

5.1.16 Lemma.

-

There are SESs ofcomodules O+MkY',

J

y d i m urn

Mk-

urn

Mk+O

and Bockstein spectral sequences (BSSs) converging to Ext( M k ) with Ef"

= Ext"(Mk;l,)OP(a,)

where multiplication by a, in the BSScorresponds to division by v, in Ext( M",. d, is not a derivation but ifd,(akx) = y # O then d,(ak+'x)= uLy.

Roo$ The SS is that associated with the increasing filtration of M i defined by F,Mk= ker v; (see A1.3.9). Then E o M k = Mk;',OP(a,). Using 5.1.16 n times we can in principle get from Ext(M:) to Ext( M ; f )= Ext(M") and hence compute the chromatic El-term (5.1.8). In practice these computations can be difficult.

176

5. The Chromatic Spectral Sequence

We will not actually use the BSS of 5.1.13 but will work directly with the LES

5.1.17 Remark. -+

-

Ext'(M",-,) A Ext"( M r ) %Ext'(X-2pm'2M~) 6

EXfS*'(Mi;',) + . . .

by induction on s. Given an element x E Ext(MiY1,)which we know not to be in im 6, we try to divide J ( x ) by urn as many times as possible. When we find an X ' E Ext( M:) with u k x ' = j ( x ) and S(x') = y # 0 then we will know that j ( x ) cannot be divided any further by urn.Hence 6 serves as reduction mod This state of affairs corresponds to dr(cy;x) = y in the BSS of 5.1.16. We will give a sample calculation with 6 below (5.1.20). We will now make some simple calculations with the CSS starting with the Greek letter elements of 5.1.3. The SES of 5.1.2 maps to that of 5.1.5, i.e., we have a commutative diagram 0-t BP,/l,, -% Z-dim BP,/l"

O+

I

li -

M"

N"

+

x-dimvn

BP,I 1,+ 1 -+ 0

I

N"+I

-3

+O

with i(Ui+l) =

u:+,

p u , . . . u,

Hence a!") can be defined as the image of i(u:) under the composite of the connecting homomorphisms of 5.1.5, which we denote by a : Exto(N " ) Ext"(BP,). On the other hand, the CSS has a bottom edge homomorphism Exto(M " ) = E:"

T

T

Exto(N " ) ker d , -+ EZo-+ Ext"( BP,) -+

which we denote by K : E ~ ~ O ( N " Ext"(BP,). )-+ K

and

cy

differ by sign, i.e.,

5.1.18 Proposition. exceeding x.

K

= ( -1)["+"21a,where

[ x ] is the largest integral nor

Proof. The image yo of i ( u k ) in M " is an element in the chromatic complex (5.1.10) cohomologous to some class in the cobar complex C (BP,). Induc) , y v E C r ( M n - Ssuch ) that d,(x,) = y, tively we can find x, E C S ( M " - S p 'and and d,(x,) = y,+, . Moreover y, E C " ( M o )is the image ofsome x, E C " (BP,).

177

1. The Algebraic Construction

It follows from the definition of the connecting homomorphism that x, is a cocycle representing a ( i ( u i ) > = a j " ) . On the other hand, y, is cohomologous to (-l)"-sys+lin CC(BP,) by 5.1.10 and IIzZA (-1)"-" = ( - I ) [ ~ + " * ~so x, represents ( - l ) r n " " l ~ ( i ( u ~ ) ) . 5.1.19 Definition. I f x E Exto(M " ) is in the image of Exto(N " ) (and hence vi

giues a permanent cycle in the C S S ) and has the form

plov>

f

..

U L l

"-1

modulo I , (i.e., x is the indicated fraction plus terms with larger annihilator ideals) then we denote a ( x ) by a!y!n-l, . iffor some m < n, ik = 1 f o r k s m then we abbreviate a ( x ) by ,,,.+,. We will compute the image of p, E Ex?( BP,) in Ext2(BP,/12) for p > 2 in two ways.

5.1.20 Examplesand remarks.

(a) We regard p, as an element in Exto(M2)and compute its image under connecting homomorphisms So to Ext'f M f ) and then SI to Ext2(M:), which is E Y in the CSS for Ext(BP,/I,). To compute So, we pick an element in X E M 2 such that p x = p r , and compute its coboundary in the cobar complex C ( M 2 ) .The result is necessarily a cocycle of order p , so it can be pulled back to Ext'( M i ) . To compute 8, on this element we take a representative in C ' ( M i ) , divide it by u l , and compute its coboundary. Specifically p, is

4 M Z ,so

--E PUl

we need to compute the coboundary of

4

up-'

1

02

x =7 . It is convenient to write x as zp, then the denominator is the

P-ul P 01 product of elements generating an invariant regular ideal, which means that we need to compute q R on the numerator only. We have p-1

-

p-1

r l R ( u , ) = u1

-pvfp2t, mod ( p 2 )

and q R ( u ; ) = u: + t&'( u1f y + p t J mod ( p 2 , puI,u i ) .

These give d(

uy-'u:) P2Uf

-

-u:t,

*

PUI

tui-'

+-(t*-t:+p). PO1

This is an element of order p in C ' ( M 2 ) ,so it is in the image of C ' ( M : ) . In this group the p in the denominator is superfluous, since everything has order p , so we omit it. To compute 61 we divide b y u1 and compute the coboundary; i.e., we need to find

178

5. The Chromatic Spectral Sequence

Recall (4.3.15)

where

as in 4.3.14.From this we get

V;-2 + t ( t - 1) t:l(t2-

t:+P)

01

We will see below that Ex?( M:) has generators k, represented by 2tf I t2- 2t: I tl+P- t f p I t , and b,,. Hence the mod l2reduction of -PI is

(b) In the chromatic complex CC(BP,) (5.1.10), P I € M 2 is cohomologous to elements in C ' ( M ' ) and C 2 ( M o ) These . three elements pull back to N 2 , C ' ( N ' ) ,and C 2 ( N o ) respectively. , In theory we could compute the element in C2(N o )= C2(BP,) and reduce mod I , , but this would be very laborious. Most of the terms of the element in C"(BP,) are trivial mod I,, so we want to avoid computing them in the first place. The passage from Co(N 2 ) to C 2 ( B P , ) is based on the four-term exact sequence O + BP,+ M o + MI-+ N 2 + 0 . 0:

Since --E

N 2 is in the image of Z-qBP,.Z,, we can replace this sequence

PV1

with O + BP,S BP,'.

~~qBP,ll,+~~qBP,ll,+O.

179

1. The Algebraic Construction

We are going to map the first BP, to EP,/Z2; we can extend this to a map of sequences to 0 + BP,. I2f: B P J ( p 2 ,pu, , uf) -% Z-4BP,/( p, u:) + 2-9BP,.Z2 + 0,

which is the identity on the last comodule. [The reader may be tempted to replace the middle map by B P , / ( P 2 , u , ) -% x - 9 B P , / ( P ,

d,

but EP,/ ( p 2 , u,) is not a comodule.] This sequence tells us which terms we can ignore when computing in the chromatic complex, as we will see below. Specifically we find (ignoring signs) that ty

P

4 M 2 is cohomologous to

--E

Pul

+ higher terms.

Note that the first two terms are divisible by u1 and uf respectively in the image of C'(Z-qBP,/( p ) ) in C ' ( M ' ) . The higher terms are divisible by u: and can therefore be ignored. In the next step we will need to work mod Z: in the image of C'(BP,) in C 2 ( M o )via multiplication by p. From the first term above we get t ( t - 1)U:-2t21

t;+

tu:-'b,,,

while the second term gives

and their sum represents the same element obtained in (a). Our next result is 5.1.21 Proposition. For n

2 3,

a ( " )=

1 & (p"-- 1I )

.

rn

For n = 3 this gives y , = -a1p p - , . In the controversy over the nontriviality of y1 (cf. the paragraph following 1.3.18) the relevant stem was known to be generated by a l p p - , ,so what follows is an easy way (given all of our machinery) to show y1 # 0. Proofof5.1.21. a 1is easily seen to be represented by t , in C(BP,), while and ab"_;" are represented by

180

5. T h e Chromatic Spectral Sequence

respectively. Hence (-1)"ala:-;')

= -afd- 2 we represented x , by

with n - 3 t,’s. To compute d f l - , ( x , )let &=xX,+

“-2(uf+l-u~u;:1ui+l)fl~. 2Ul * . . ui-,v: ,=1

c

* *

It‘

E

CC( BP,),

where the ith term has ( n - 2 - i ) t,’s. Then one computes

so

unless this element is killed by an earlier differential, in which case x , would represent a nontrivial element in Extn-1*2n(BP,), which is trivial by 5.1.23 below. 5.1.23 Edge Theorem.

( a ) For all primes p ExtS”(BP,) = 0 for t < 2s, ( b ) for p = 2 ExtS.2s(BP,) = Z/(2) for s 2 1, and ( c ) for p = 2 ExtS32s+2( BP,) = 0 for s 2 2.

Proof: We use the cobar complex C ( B P , ) of A1.2.11. Part (a) follows from the fact that C”*‘=Ofor t < 2 s . CS, is spanned by t , I . . . )t , while (y,2s+2 is spanned by u , t , 1. . (t , and ej = t , 1. . tlI t:l t , . . . tl with t: in the j t h position, l s j s s . Since d(t~)=-3tlIt:-3t:t,, the ej’s differ by a coboundary up to sign. Part (b) follows from

-

I

d ( e , )= 2 t ,

1.

*

I zI = - d ( u l t l ( *. I t l )

182

5. The Chromatic Spectral %ueoce

and (c) follows from d ( t , ) t , I - .. ) t l ) = - v l t l l . * l r l - e l . We conclude this section by tying up some loose ends in Section 4.4.For p > 2 we need For odd primes, a Ip, is divisible by p but not by p2. (This gives the first element of order p 2 in Ext'( BP,) for s 2 2.)

5.1.24 Lemma.

V P tl

v;t1

is represented by -. Now 7 is not a cocycle, Pvl P u1 but if we can get a cocycle by adding a term of order p then we will have the desired divisibility. It is more convenient to write this element as -. v; tl , then the factors of the denominator form an invariant sequence

Proof: Up to sign

a 1p,

P24 [i.e., vR(v f ) = vf mod ( p ' ) ] , so to compute the coboundary it suffices to ~ v( p~2 ,)v f ) . We find compute ~ ~ ( u f -mod d

(P - l GP f l ) P v1

=----=2 -v;t,It,

1 d ( zv;t: )

Po:

so the desired cocycle is vf-'v;t,

1u;t:

p2vf

2 pv:.

This divisibility will be generalized in (5.6.2). To show that crl p, is not divisible by p 2 we compute the mod ( p ) reduction of our cocycle. More precisely we compute its image under the connecting homomorphism associated with O+ M i + Mi$ Mi+O

(see 5.1.16). To do this we divide by p and compute the coboundary. Our divided (by p ) cocycle is

and its coboundary is

+

u~(t:ltl+tllr:) v ~ - ~ f 2 ~ v ; f- ' lt ~- l t~: - vp-' 2 t,

Pv:

Pvl

2

PVI

1 vnt:

I tl

PVI

We can eliminate the first term by adding - 7(even if p 3 Pvi

= 3).

For p > 3

183

1. The Algebraic Construction

the resulting element in C 2 ( M i )is

Reducing this mod iz in a similar fashion gives a unit multiple of

4 in

- v2t: 4.1.14. For p = 3 we add -to the divided cocycle and get

3 v:

which still gives a nonzero element in Extz(M:). For p = 2 we need to prove 4.4.38 and 4.4.40, i.e., 5.1.25 Lemma.

In the notation of 4.4.32 for p

=2

S O ( /33) 71 mod (2), ( b ) 6,(r12) = co mod (2).

(0)

Roo$

For (a) we have v,v:

u:ff

uzvj

v3f:+ vzf:+ vffz+

=2v:+

d ( q + s )

2VI

02tf 9

which gives the result. For (b) we use Massey products. We have qz = ( v l , u , , PI) so by A1.4.11 we have So(T ~=) ( q l , hlo, PI) mod (2). Hence we have to equate this product with c,, which by 4.4.31 is represented by %, where xz2is defined by 4.4.25. 01

To expedite this calculation we will use a generalization of Massey products not given in A1.4 but fully described by May [3]. We regard q l as an element in Ext'(M;), and h , , and PI as elements in Ext'(BP,/Z,) and use the pairing M : 0BP,/I, 3 M i to define the product. Hence the cocycles representing ql, hlo, and P I are

respectively. The cochains whose coboundaries are the two successive products are v3(t 2 + t:,

+ v2(t 3 + t,r:+

t:t2+ t:)

+ V Z ( t:+

flf2)

and

f2.

211

If we alter the resulting cochain representative of the Massey product by

184

5. The Chromatic Spectral Sequence

the coboundary of u ~ t : t , + u , ( t : + t , t ~ + t ~ , + u : ( t ~ + t ~ )U : ( t 2 + t : )

+

u1

u:

u:t,

+T

we get the desired result.

2. Ext'(BP,/I,) and Hopf Invariant One In this section we compute Ext'( B P * / I , ) for all n. For n > 0 our main results are 5.2.14 and 5.2.18. For n = O this group is E:,* in the ANSS and is given in 5.2.6. In 5.2.8 we will compute its image in the classical ASS, thereby obtaining proofs of the essential content of the Hopf invariant one theorems 1.2.12 and 1.2.14. More precisely, we will prove that the specified hi's are not permanent cycles, but we will not compute d,( hi).The computation of Ext'(BP,/I,) is originally due to Novikov [ l ] for n = 0 and to Miller and Wilson [3] for n 2 1 (except for n = 1 and p = 2). To compute Extl(BP,) with the CSS we need to know Ext'(Mo) and Exto(M'). For the former we have 5.2.1 Theorem. ( a ) ExtS2'(M o )=

( b ) Exto.'(BP,) =

0 0

if s = t = O otherwise

otherwise.

ProoJ: (a) Since M o = QO BP,, we have Ext( Mo)= Extr(A, A) where A = M o and = QO BP,( B P ) . Since 1, is a rational multiple of q R (u,) - u, modulo decomposables, r is generated by the image of qR and q L and is therefore a unicursal Hopf algebroid (Al.l.11). Let V,= vR(u,),so r = A[T,,2]2,. . .I. The coproduct in r is given by A(u,) = u,Ol and A(V,) = 1 OK.The map vR:A + r = A OAr makes A a right r-comodule. Let R be the complex r O E ( y , , y,, . . .) where E(y,,y,,,. . .) is an exterior algebra on generators y i of degree 1 and dimension 2( p ' - 1 ) . Let the coboundary d be a derivation with d ( y , ) = a'(%) = 0 and d ( u,) = y n . Then R is easily seen to be acyclic with H"(R)=A. Hence R is a suitable resolution for computing Ext,(A, A ) (A1.2.4). We have Homr(A, R ) = A O E ( y , ,. . .) and this complex is easily seen to be acyclic and gives the indicated Ext groups for Mo.

185

2. Ext'(BP,/I,) and Hopf Invariant One

For (b) ExtOBP, = ker d, = Exto(M:) and d , ( x ) # 0 if x is a unit multiple of a negative power of p . H To get at Ext(M') we start with 5.2.2 Theorem.

( a ) For p > 2 , Ext(My) = K(l),@E(h,) where hoe Ext',¶ is represented by t , in C ' ( M y ) (see 5.1.12) and q = 2p -2 as usual. ( b ) For p = 2 , Ext(M~)=K(l),OP[ho]OE(p,),where ho is as above and p1 E Ext'.' is represented by uF3( f 2 - t : ) + ~ ; ~ u ~ t , . This will be proved below as 6.3.21. Now we use the method of 5.1.17 to find Exto(M'); in the next section we will compute all of Ext(M') in this way. From 4.3.3 we have v R ( ~ f l ) = uspa

u:' mod

5.2.3

(PI+'),

so +E

Exto(M I ) . For p odd we have

P

vR(us''') = us"' + s p i u f P ' - l t l

mod (pi+’)

so in 5.1.17 we have

for p $ s, and we can read off the structure of Exto(My) below. For p =2, 5.2.3 fails for i > O , e.g., q R ( u : ) = u:+4ult, +4t:

mod (8).

The element t i + u,t, E C ' ( M ? ) is the coboundary of u ;'u 2, so

i.e., we can divide by at least one more power of p than in the odd primary case. In order to show that further division by 2 is not possible we need to show that cr213 has a nontrivial image under 6 (5.1.17). This in turn requires mod (4). From 4.3.1 we get a formula for vR(u2) 5.2.4

T R (02)

= L)2

+ 13u1 f

- 3 0:

f ,- 14t2 - 4f:.

[This formula, as well as vR(u , ) = u1 - 2t,, are in terms of the u, defined by Araki's formula A2.2.2. Using Hazewinkel's generators defined by A2.2.1 gives v R ( u , ) =u,+2t1 and t 7 R ( u 2 ) = u 2 - ~ u r t : - 3 u ~ t l + 2 t 2 - 4 t : . ] Let x , , ~= u:+4u;'u2. 5.2.5

so S (

Then 5.2.4 gives

+

qR(kl,,)= xI,’ 8( u;'t,+ = u:p, # 0 E

Ext'( M y ) .

u;'t:+

u;2u2tl)mod (16)

5. The Chromatic Spectral Sequence

186

5.2.6 Theorem.

r

( a ) F o r p odd

if if if

Exto7'(M')= Q/Z,,, Z/(p'+')

where

q X t

9=2p-2

t =O

t=spl

and p Y s

These groups are generated by

u I"’ i+l E M ' . P ( b ) F o r p odd Ext'"( BP,)

=

if

t r O

(c) Forp=2

Exto.'( M I )

is odd

Yo

if

t

[Z/(2"3)

if if

t = 2 mod 4 t = 2'+2s for odd s

=

xz's

4

These groups are generated by - and 2r+3 M' where xl,' is as in 5.2.5 2 ( d ) Forp=2

Ext'.'(BP,)

=

r

if if if

Exto.'(M')

z/ (4)

and ExtIs4(BP,) is generated by

*-.v:4

L Y ~= , ~

t%O t>O

and

t24

t=4

1

We will see in the next section (5.3.7) that in the ANSS for p > 2 , each element of Ext'(BP,) is a permanent cycle detecting an element in the image of the J-homomorphism (1.1.18). For p = 2 the generators of Ext'." are permanent cycles for t = O and 1 mod (4) while for t = 2 and 3 the generators support nontrivial d,'s (except when t = 2) and the elements of order 4 in EX^',^'+^ are permanent cycles. The generators of E i 4 ' = Ek4' detect elements in im J for all t > 0. Proof of 5.2.6. Part (a) was sketched above. We get Q / Z ( , , in dimension zero because l/p' is a cocycle for all i > O . For (b) the CSS gives an SES 0 + Ek0 + Ext'( BP,)

+ E z ' +0

187

2. Ext'(BP,/I,) and Hopf Invariant One

and E Z = O by 5.2.1. Ek'= E:,'=ker dJim d,. An element in E:,'= Ext'(M') has a nontrivial image under d, iff it has terms involving negative powers of u l , so ker d, c E is the subgroup of elements in nonnegative dimensions. The zero-dimensional summand Q/Z(,,, is the image of d,, so E Y = Ext'(BP,) is as stated. For (c) the computation of Exto(My)is more complicated for p = 2 since 5.2.3 no longer holds. From 5.2.5 we get

:.'

5.2.7

T~(x:;)=

xi:; +2i+3x::;-'(v ; ' t , +

v;'t:+

mod (2i+4)

u;2u2tl)

X:;;

for odd s, from which we deduce that 2l+3 is a cocycle whose image under 6 (see 5.1.17) is ~ : ' + ' ~ pEquation ,. 5.2.3 does hold for p

=2

when i = 0, so

4 Ext0*2s(M:) is generated by - for odd

s. This completes the proof of (c). 2 For (d) we proceed as in (b) and the situation in nonpositive dimensions x2's

is the same. We need to compute de($).

L

Since xl,’= u : + 4 u ; I v 2 , we have

L

For 2's = 1 (but for no 2's > 1) this expression has a negative power of u, and we get d, - = - € M 2 .

,;;

This gives a chromatic d, (compare 5.1.21) and accounts for the discrepancy BP,). between EX^',^( M I ) and ExtLt4( Now we turn to the Hopf invariant one problem. Theorems 1.2.12 and 1.2.14 say which elements of filtration 1 in the classical ASS are permanent cycles. We can derive these results from our computation of Ext'( BP,) as follows. The map B P + H/( p ) induces a map @ from the ANSS to the ASS. Since both SSs converge to the same thing there is essentially a one-to-one correspondence between their Em-terms. A nontrivial permanent cycle in the ASS of filtration s corresponds to one in the ANSS of filtration 5 s.

To see this consider BP, and mod ( p ) Adams resolutions (2.2.1 and 2.1.3) S"=X,+X,+. .. I

,

188

5. The Chromatic Spectral Sequence

where the vertical maps are the ones inducing CP. An element x E n*(So) has Adams filtration s if it is in im r,( Y,) but not in im n,( Y S + ' )Hence . it is not in im r , ( X S + ' )and its Novikov filtration is at most s. We are concerned with permanent cycles with Adams filtration 1 and hence of Novikov filtration 0 or 1. Since Exto(BP,) is trivial in positive dimensions [5.2.l(b)] it suffices to prove

5.2.8 Theorem. The image of Q: Ext'(BP*)+

Extk*(Z/(p), Z l ( p ) )

is generated by h , , h Z , and h, for p = 2 and by h o E Ext'*qfor p > 2. (These elements are permanent cycles; CJ 1.2.11 and 1.2.13.)

Pro05 Recall that A, = Z / ( p ) [ t , , t2.. . . ] O E ( e o ,e l , . . .) with A ( t , ) = Co515nt, 0r:', and A( e n )= 10en+ l , 5 1 e,5 0t!', n where to = 1 . Here t, and en are the conjugates of Milnor's & and 7, (3.1.1). The map BP,(BP)+ A, sends t, E BP,(BP) to t, E A,. Now recall the I-adic filtration of 4.4.4. We can extend it to the comodules u'

M " and N" by saying that a monomial fraction 7is in F k iff the sum of U

the exponents in the numerator exceeds that for the denominator by at least k. (This k may be negative and there is no k such that FkM"= M " or F k N " = N " . However, there is such a k for any finitely generated subcomodule of M " or N " . ) For each k E Z the sequence o + F ~ N " +F ~ M " +F ~ N " + ' + o

is exact. It follows that a : Ext'( N " )+ Ext'+"( BP,) (5.1.18) preserves the I-adic filtration and that if x E F' Exto(N ' ) then @ a ( x ) = 0. Easy inspection of 5.2.6 shows that the only elements in Exto(M') not in F' are a I and, for p = 2, and a4,4,and the result follows. Now we turn to the computation of Ext'( BP,/I,) for n > 0; it is a module over Exto(BP,/In) which is Z / ( p ) [ u , ] by 4.3.2. We denote this ring by k ( n ) , . It is a PID and Ext'(BP,/I,,) has finite type so the latter is a direct sum of cyclic modules, i.e., of free modules and modules of the form k( n ) J ( u h ) for various i > 0. We call these the u,-torsion free and u,-torsion summands, respectively. The rank of the former is obtained by inverting u,,, i.e., by computing Ext'(M:). The submodule of the u,-torsion which is annihilated by u, is precisely the image of Ext"( BP,/l,,+,) = k( n + l), under the connecting homomorphism for the SES 5.2.9

O+Xd'"L~BP,/I,

c,,

BP,/I,,+ BP*/I,,+l -0.

We could take these elements in Ext'( BP,/I,,) and see how far they can

189

2. Ext'(BP,/I,) and Hopf Invariant One

be divided by v, by analyzing the LES for 5.2.9, assuming we know enough about Ext'( BP,/ I,,,) to recognize nontrivial images of elements of Ext'(BP,/I,) when we see them. This approach was taken by Miller and Wilson [3]. The CSS approach is superficially different but one ends up having to make the same calculation either way. From the CSS for Ext(BP,/I,) (5.1.11) we get an SES O + ELo + Ext'( B P , / I , )

5.2.10

+

E:'

+ 0,

where E L o = E Y is a subquotient of Exto(M!,,,) and is the u,-torsion summand, while E z ' = Ext'( M f ) is the v,,-torsion free quotient. To get at Exto(M!,+,) we study the LES for the SES

-

O + Mf,, & CdlrnUnMkv,, Mi+()

as in 5.1.17; this requires knowledge of Exto(Mf,,) and Ext'( M:,,). To determine the subgroup E z ' of Ex?( M f ) we need the explicit representatives of generators of the latter constructed by Moreira [ l , 31. The following result (to be proved later as 6.3.12) then is relevant to both EZ' and ELo in 5.2.10. 5.2.11 Theorem. Ext'(M:) for n > O is the K(n),-vector space generated by h, E Ext13P'4for 0 i i In - 1 represented by t f , 5, E Ext'.' ( f o r n 2 2) represented for n = 2 by u ; ' t Z + u;"( tP - t f 2 + P) - u;'-Pv3t;, and (if p = 2

and n 2 1) p,

E

5.2.12 Remark.

Ext'.'. (5, and p, will be dejned in 6.3.11).

H

For ir n, h, does not appear in this list because the

equation T ~ ( U , + ~ v,+,+u,t~'-uU::t1 )= mod I,,

leads to a cohomology between h,,, and ~ ( , p - " ~ ' h I.

Now we will describe Exto(M:) and ELo. The groups are v,-torsion modules. The submodule of the former annihilated by v, is generated by

{%:

t E Z } . Only those elements with t > O will appear in ELo; if t = O

the element is in im d , , and ker d , is generated by those elements with t 2 0. We need to see how many times we can divide by u, and (still have a dl+1

.

cocycle). An easy calculation shows that if t = sp' with p $ s, then -is

4

a cocycle whose image in Ext'(M:+,) is S U ~ ~ ~ ) ~ but ’ ~ by ~ +5.2.12 , , these are not linearly independent, so this is not the best possible divisibility

190

5. The Chromatic Spectral Sequence

result. For example, for n = 1 we find that

is a cocycle. The general result is this.

5.2.13 Theorem. As a k(n),-module, Exto(M:) is the direct sum of

(i)

the cyclic submodules generated by

xS,+I + for i z 0,p k s ; and U,"+’.’ i

1

(ii) K ( n ) , / k ( n ) , , generated b y - f o r j 2 1 . v',

The

are deJined as follows.

x1.0 = 0 1 9

if p > 2

q l =uf

and

if p = 2 ,

v:+4v,'u2

for

X1.i = X lP. i - ~

i22,

where

and i = 1 mod ( n - 1 ) where b .=

- l ) ( p "- 1 ) ( f - 1

- 1)

for

i=lmod(u-1).

191

2. Ext'(BP,/I,) and Hopf Invariant One

The a,,, are dejined by a1.n = 1,

a,,, = i + 2

for p = 2

and

i21,

u,,,= i + 1

for p > 2

and

irl,

a2,,= p ' + p ' - ' - l

for

p>2

and

i r l

a2,,= 3 ' 2l-I

for p = 2

and

ir2,

a2,n = 1,

an.0

=1

or p = 2

for

n>2,

for

i>l

and

i+lmod(n-I),

for

i>l

and

i=Imod(n-I).

and

i=l,

a , , = P7 4 . 1

= Pan,r- I

and a,,,=pa,,,-,+p-l

This is Theorem 5.10 of Miller, Ravenel, and Wilson [l], to which we refer the reader for the proof. Now we need to compute the subquotient Ei3' of Exto(M!,). It is clear that the summand of (ii) above is in the image of d , and that kerd, is XA+l,I

generated by elements of the form -for s 2 0. Certain of these elements v:

for s > O are not in ker d , ; e.g., we saw in 5.2.6 that d,

(

" n ~ ~ " ’ > # 0 iff s = 1 and p' < j generally we find d , -

(see Miller and

Ia,,,,,

Wilson [3]), so we have 5.2.14 Corollary.

The u,,-torsion summand of Ext'( BP,/I,,) is generated by

the elements listed in 5.2.13(i) for s > 0 with (when s = 1)

&+I,*

replaced by

n

Now we consider the k(n),-free summand E 2 ' c Ext'(M:). We assume n > 1 ( n = 1 is the subject of 5.2.6); 5.2.11 tells us that 152’ has rank n + 1 for p > 2 and n + 2 forp = 2. We need to determine the image of Ext'( BP,/I,) in Ext'(M;). To show that an element in the former is not divisible by u, we must show that it has a nontrivial image in Ext'(BP,/I,,+,). The elements hi E Ext'( M : ) clearly are in the image of Ext'( BP,/ I,,) and have nontrivial images in Ext'(BP,/I,+,). The elements J,, and p, are more complicated.

192

5. The Chromatic Spectral Sequence

The formula given in 5.2.11 for l2 shows that u:+"f; pulls back to Ext1(BP,/12) and projects to u,h, E Extl(BP*/Z3). This element figures in the proof of 5.2.13 and in the computation of Ext2(BP*) to be described in Section 4. The formula of Moreira [ l ] for a representative of I , is k-s

C

T,, =

5.2.15

u~,-kt~"-'c(tk-,)P"-'+'

1 s ISIS k s n

where the u,+, E M: are defined by 5.2.16

u, = u;'

and

C

W , + , U ~ ~ ~ - , = O for

k>0.

Osrck 1

One sees from 5.2.16 that U , , + ~ - ~ U ~ ' ~ E~ )BP,/I,, / ( ~ ~ ' ) so T, = U ~ ~ " - ~ ) ' (E~BP,(BP)/I,,. -~)T,, In 5.2.15 the largest power of 0,’ occurs in the term with i =j = k = 1; in T, this term is u (,p" - 1 )/( p - 1) ~ ~ ~ - ~ tand y " its - ~ image in Ext'(BP,/Z,,+,) is ( - l ) n + l ~ ~ ~ " l ~ ' - l ) ' (nP- I-' ' ) h The formula of Moreira [3] for a representative U, of p,, is very complicated and we will not reproduce it. From it one sees that uY"-l+*"-'u n E B P , ( B P ) / I , reduces to u?:1’-’t:"-l E BP,(BP)/I,,+, . Combining these results gives 7 3 e k( n),-free quotien! Ezl of Extl( BP,/ I,,)for n 2 1 is generated by h, E Ext'7P'qfor 0 5 i 5 n -?, 5, = u ? ~ - ' ) ' ( ~ - 'l )n , and ( f o r p = 2) 22"+2"-1-1 P n = ’Jn p,. The images of 1 and there is only one basis element of Ext'( M y ) not in im 6, namely -I

v, t, Since Ext2(My)= 0, there P

u;'ho. Its image u n d e r j is represented by -.

is no obstruction to dividing j ( u ; ' h , ) by any power of p , so we have

5.3.3 for any odd prime p . We can construct a representative of an element of order p k in Ext'*'(M') as follows. From 4.3.1 we have vR(vl)= v, = p u t , where u = 1 - p p - l . Then a simple calculation shows that 5.3.4

is the desired cocycle. (This sum is finite although the ith term for some i > k could be nonzero if p I i.) The group Ext"@(M I ) + E : * I v ocannot survive in the CSS because it would give a nontrivial Ext2"( BP,) contradicting the edge theorem, 5.1.23. It can be shown (lemma 8.10 of Miller, Ravenel, and Wilson [l]) that this group in fact supports a d , with trivial kernel. Hence we have 5.3.5 Theorem.

( a ) For p > 2 the group Ext'.'( M I ) is 1

Q/z,,,,generated by -i;for (s, t ) = (0, 0), P

Z/( p ' + ' ) generated by

V

P

I"'

f o r p 4' r and (s, t ) = (0, rp'q),

Q/Z,,, generated by yk (5.3.4)for ( s , t ) = (1, o), and 0 otherwise.

( b ) In the CSS, where Ext'"(M') = E~*s,fE:.07@c im d , and ker d , =&, E:,@*', so Ek* = Ext'(BP,) and ker d , E;,',' so W EL* = Ext'(BP,) is generated by the groups Ext0*'(M1)for t > 0.

=err,

7

194

5. The Chromatic Spectral Sequence

We wit1 see below that each generator of Ext'(BP,) for p > 2 is a permanent cycle in the ANSS detecting an element in J (1.1.18). The situation for p = 2 is more complicated because Ext(My) has a polynomial factor not present for odd primes. We use 5.3.2 and 5.2.2 to compute Ext"(M') for s 2 1. The elements of order 2 in Ext'( M I ) are the images under j (5.3.1) of uih, for t odd and u ; p l for t odd and t = 0. We claim j ( p l ) is divisible by any power of 2, so Ext'*'( M’)contains a summand isomorphic to Q/Z,,, as in the odd primary case. To see this use 5.2.4 to compute

showing that y 2 (5.3.4) representsj(p,); the same calculation shows that y , = u;1t,+u;2t:

2

is a coboundary. Hence the yk for k 1 2 give us the cocycles we

need. Next we have to deal with j ( u : h o ) and j ( u i p l ) for odd t. These are not divisible by 2 since an easy calculation gives S , ( v { x ) = u;-’h,x for t odd and x = h:"' or hhp, for any i 2 0. Indeed this takes care of all the remaining elements in the SES for 5.3.1 and we get. 5.3.6 Theorem. ( a ) For p = 2 Ext'r'(M') is

1

Q/Z(,, generated by - f o r (s, t ) = (O,O), 2' v; Z/(2) generated by - f o r 2

( s , t ) = (0,2r) and r odd,

xr2'

Z/(2i'3) generated by

21+3

for ( s , t ) = (0, r2'+*) and r odd, u-'t

Q/Z,,,OZ/(2) generated b y y k ( k 2 2 ) a n d 2

2

for (s, t ) = ( l , O ) ,

Z/(2) generated by j ( u;hi) for s > 0, t = 2( r + s ) , and r odd, Z/(2) generated b y j ( u i p , h i - ' ) f o r s > 0, t + 2 ( r + s - I ) , and rodd, and 0 otherwise.

195

3. Ext(M’) and the J-Homomorphism

( b ) In the CSS for p

=2

E;?

is

Ext”‘(h4’) for t = 2 s + 2 r a n d r r l , r f 2 , v: Z/(4) generated by - for (s, t ) = (0,4), and 4

0 otherwise.

(See 5.1.22 for a description of diferentials originating in E :7s*2s+4.) In other words the subquotient o j Ext(BP,) corresponding to EL* is generated by Extl( BP,) (5.2.6) andproducts of its generators (excluding a 2 / 2E EX^'.^) with all positive powers of (Y, E Ext’,’.

ProoJ: Part (a) was proved above. For (b) the elements said to survive, for s > 0 with odd r 2 5 and j ( vlh;) for i.e., those in E and j ( v ; p , s>O with odd r r 1, are readily seen to be permanent cycles. The other elements in E for s > 0 have to support nontrivial differentials by the edge theorem, 5.1.23. B

:.” :,’

Now we describe the behavior of the elements of 5.3.5(b) and 5.3.6(b) in the ANSS. The result is

5.3.7 Theorem. ( a ) For p > 2 , each element in Ext’(BP,) is a permanent cycle in the ANSS represented by an element of im J (1.1.18) having the same order. ( b ) For p = 2 the behavior of Ext’,2‘(BP,) in the ANSS depends on the residue o f t mod (4) as follows. Zft = 1 mod 4 the generator a, is a permanent cycle represented by the element p2r-1 E rrs”-’ of order 2 constructed by Adams [l]. In particular a 1 is represented by 7) (1.1.13). a,al is represented b,vpZl= v p 2 t - Ianda:a,isrepresentedbyanelementoforder2in im J C (the order of this group is an odd multiple of 8 ) . atc3al= d,(a~a,+,)for all sro.

I f t = 0 mod (4) then the generutor of Ext’’2’(BP,) is a permanent cycle represented by an element of im J having the same order, as are alF and a:%. = d3(~ I ~ ~ y 1 + 2 for / 3 ) s 2 0. In particular y4 is represented by cr E rr; (1.1.13). (twice the generator except when t = 2 ) is a permanent If t = 2 mod (4), cycle represented by an element in im J of order 8. ( a , / 2has order 4 and 4 times the generator of im J represents a:alP2 as remarked above). In particular aZI2is represented by V E rrs (1.1.13).

This result says that the following pattern occurs for p Em-term as a direct summand for all k > 0:

=2

in the ANSS

196

5.3.8.

5. The Chromatic Spectral Sequence

I 01 a41,

Q 4 k + 212

O4k + 1

8k - 1

8k

8k + 2

8k + 1 t-s

8k

+3

--L

Where all elements have order 2 except (Y4k+2/2, which has order 4, and (Y4k, whose order is the largest power of 2 dividing 16k; the broken vertical line indicates a nontrivial group extension. The image of J represents all elements shown except ( ~ 4 k + land ( Y l a d k + l . Our proof of 5.3.7 will be incomplete in that we will not prove that im J actually has the indicated order. This is done up to a factor of 2 by Adams [l], where it is shown that the ambiguity can be removed by proving the Adams conjecture, which was settled by Quillen [ 11 and Sullivan [ 13. We will actually use the complex J-homomorphism J : T*( U )+ T : , where U is the unitary group. Its image is known to coincide up to a factor of 2 with that of the real J-homomorphism. We will comment more precisely on the difference between them in due course. An element x E ~ r ~ ,U- )~corresponds ( to a stable complex vector bundle 6 over S". Its Thorn spectrum T ( 5 ) is a 2-cell CW-spectrum Soue2' with attaching map J ( x ) and there is a canonical map T([)+MU. We compose it with the standard map M U + BP and get a commutative diagram 5.3.9

BPABP

197

3. Ext(M') and the J-Homomorphism

where the two rows are cofibre sequences. The map S2' + BP is not unique but we do get a unique element e(x) E .rr2,(BPA m ) / i m r 2 , ( B P ) Now . EiS2' of the ANSS is by definition a certain subgroup of this quotient containing e(x), so we regard the latter as an element in Ext'92'(BP,). Alternatively, the top row in 5.3.9 gives an SES of comodules which is the extension corresponding to e(x). We need to show that if x generates rz,-,( U ) then e(x) generates Ext'(BP,) up to a factor of 2. For a generator x, of rz,-,( U ) we obtain a lower bound on the order of e(x) as follows. If j e ( x l )= 0 for some integer j then for the bundle given by x = j x , ~n 2 , - IU( ) the map S2'+m in 5.3.9 lifts to BP, so we get an element in .rr,,(BP). Now consider the following diagram 5.3.10

H,(BU)

= H,(MU)

-

Q

where the two left-hand vertical maps are the Hurewicz homomorphisms and 8 is some ring homomorphism; it extends as indicated since .rr,(MU)O Q = H , ( M U ) O Q by 3.15. Let 4 be the composite map (not a ring homomorphism) from r,(BCJ) to Q. If 4(x,) has denominator j,, then j , divides the order of e(x,). According to Bott [2] the image of x, in H 2 , ( B U )is ( t - l ) ! s , where s, is a primitive generator of H 2 , ( B U ) .By Newton's formula z db(z) b(z) dz where s(z) = X,rO s,z' and b( z) = X l z o b,z', the b, being the multiplicative generators of H,( B U ) = H , ( M U ) (3.1.4). Now by Quillen's theorem, 4.1.6, 8 defines a formal group law (FGL) over Z (see Appendix 2), and by 4.1.11 s(z)=-----,

Z

so z d exp(z) - 1, O(s(z)) =exp(z) dz where exp(z) is the exponential series for the FGL defined by 8, i.e., the functional inverse of the logarithm (A2.1.5). The 0 we want is the one defining the multiplicative FGL (A2.1.4) x + y + xy. An easy calculation shows exp( z) = ez - 1 so ze e'-1

8(s(z)) =--

1.

198

5. The Chromatic Spestral Sequence

This power series is essentially the one used to define Bernoulli numbers (see appendix B of Milnor and Stasheff [ 5 ] ) , i.e., we have e(s(2))

2 =-+ C 2

k z l

BkZZk (-l)k+'(2k)!

where Bk is the kth Bernoulli number. Combining this with the above formula of Bott we get 5.3.11 Theorem. The image of a generator x, of T,,_~( U ) = 7r2,(B U ) under the map 4:.rr,BU+Q of 5.3.10 is 4 i f t = l , 0 f o r odd t > l , and * B k / 2 k for t = 2k. Hence the order of x, in Ext'( BP,) is divisible by 2 for t = 1, 1 for odd t > 1, and the denominator j,, of B k / 2 k for t = 2 k This denominator j2k is computable by a theorem of von Staudt proved in 1845; references are given in Milnor and Stasheff [5]. The result is that p Ij,, iff ( p - 1) I2k and that if p' is the highest power of such a prime which divides 2 k then p'+' is the highest power of p dividing j 2 , . Comparison with 5.2.6 shows that EX^',^^( BP,) also has order p'+I except when p = 2 and k > 1, in which case it has order 2'+,. This gives 5.3.12 Corollary. The subgroup of ExtIS2'(BP,) generated by e ( x , ) (5.3.9), i.e., by the image of the complex J-homomorphism, has index 1 for t = 1 and 2, and 1 or 2 for t 2 3. Moreover each element in this subgroup is a permanent cycle in the ANSS. This completes our discussion of im J for odd primes. We will see that the above index is actually 2 for all t 2 3, although the method of proof depends on the congruence class of tmod(4). We use the fact that the complex J-homomorphism factors through the real one. Hence for i = 3 mod (4),e ( x , )= 0 because 7r2,-1(S0) = 0. For t = O the map a 2 , - l ( U ) + ~ 2 , - , ( S Ohas ) degree 2 in Bott [ l ] (and for t = 2 it has degree 1 ) so e ( x , ) is divisible by 2 and the generator y of Ext'(BP,) is as claimed in 5.3.7.This also shows that rlyl and v 2 y , detect elements in im J. Furthermore T~ kills the generator of 7r2,-,(SO)by 3.1.26, so a : y , must die in the ANSS. It is nonzero at E,, so it must be killed by a higher differential and the only possibility is ~ f ~ ( a , +=*a, :~y ), [here we still have t = 0 mod (4)]. For t = 1 the generator of T ~ , - ~ ( S=OZ) / ( 2 ) is detected by q 2 y , _ , as observed above, so e ( x , )= 0. For t = 2 we just saw that the generator a,,3 of ExtlT2'supports a nontrivial d , for t > 2, so we must have e ( x , )= a,,> . To complete the proof of 5.3.7 we still need to show three things: for t = 1 mod (4) a, is a permanent cycle, for t = 3, d 3 ( a , )= a : ( ~ , - and ~ , for t = 2, a, is represented by an element of order 4 whose double is detected

199

3. Ext(M') and the J-Homomorphism

by To do this we must study the ANSS for the mod (2) Moore spectrum M(2). The result we need is 5.3.13 Theorem.

Let M(2) be the mod (2) Moore spectrum. Since BP,(M(2)) = BP,/(2) is a comodule algebra, the ANSS E,-term for M ( 2 ) , Ext(BP,/(2)), is a ring (A1.2.14).However, sinceM(2) is not a ring spectrum, the ANSS diferentials need not respect this ring structure. ( a ) Ext(BPJ(2)) contains Z/(2)[uI, h o ] O { l ,u } as a direct summand where ul E Ext0,2, ho€ Ext',', and u E Ext',' are represented by u l , t , and t:+ u, t:+ u:t:+ u,t,+ u2tl, respectiuely. This summand maps isomorphically to EZ* in the CSS for Ext(BPJ(2)) (5.1.11). ( b ) In the ANSS uihlu' is a permanent cycle for s 2 0, e = 0, 1 , and t = 0 or 1 mod (4). If t = 2 or 3 then d,(vth&') = ~ i - ~ h i + ~For u ' . t = 1, U:U' is M(2)) whose double is detected represented by an element of order 4 in T*,+,~( by hi u i-'u '. ( c ) Under the reduction map BP, + BP,/(2) induced by So+=M(2), if t is odd then the generator a,of Ext'*''(BP,) maps to ui-'ho. r f t is even and at least 4 then the generator y , of Ext',2L(BP,) maps to u : - ~ u . ( d ) Under the connecting homomorphism 6 : Ext"( BP,/(2)) += Ext"+'(BP,) induced by M ( 2 ) += S' (2.3.4), u: maps to a , E Ext'.''(BP,) for all t > 0; uui maps to a,y,+, if r is odd and to 0 if t is even. In other words, the ANSS .&-term for M(2) has the following pattern as a summand in low dimensions: 5.3.14

I 2

t

1

S

0

1

V,

~~

0

1

3

2 t-5

4

--t

where the broken vertical line represents a nontrivial group extension. [Compare this with 3.1.29(a) and 5.3.8.1 The summand of (a) also contains the products of these elements with u f r u efor t 2 0 and e = 0, 1. The only

200

5. The Chromatic Spectral Sequence

other generators of ExtZ-'(BP,/(2)) for t - s 5 13 are p, E EX^',^, p: E Ext2.', hip,,,^ Ext'+s*8+2s for s = 0, 1 , 2 (where hip,,, = p;), and h ; p , ~Ext1+S*10+2s for s=O, 1. Before proving this we show how it implies the remaining assertions of 5.3.7 listed above. For t = 1 mod (4), a , = 6( u l ) by (d) and is therefore a permanent cycle by (b). For t =3, a,= S ( u l ) and S commutes with differentials by 2.3.4, so dj((Yf)= Sd,( u : ) = S(h:U:y) =ff;a*-2

For the nontrivial group extension note that for t = 1 a f a , maps to an element killed by a differential so it is represented in 7r,(So) by an element is not the image under S of a permanent divisible by 2. Alternatively, cycle so it is not represented by an element of order 2. hmfof5..?.l.?. Recall that in the CSS converging to Ext( BP,/(2)), Ext?* = Ext( M y ) , which is described in 5.2.2. Once we have determined the subgroup E k * c Et'* then (c) and ( d ) are routine calculations, which we will leave to the reader. Our strategy for proving (b) is to make low-dimensional computations by brute force (more precisely by comparison with the ASS) and then transport this information to higher dimensions by means of a map a :ZsM(2) + M ( 2 ) which induces multiplication by u: in BP-homology. [For an odd prime p there is a map a :Z q M (p ) + M ( p ) inducing multiplication by u, . u: is the smallest power of u, for which such a map exists at p = 2.1 To prove (a), recall (5.2.2) that Ext( u,'BP,/(2)) = K(l),[ho, p,]/(p?) with ho€ Ext1,2and p I E Ext'.'. We will determine the image of Ext( BP,/(2)) in this group. The element u maps to u:p,. [Our representative of u differs from that of u:pl given in 5.2.2 by an element in the kernel of this map. We choose this u because it is the mod (2) reduction of y, E Ext',*(BP,).] It is clear that the image contains the summand described in (a). If the image contains u;'h; or u:-’hgpl for any t > 0, then it also contains that element times any positive power of h,. One can show then that such a family of elements in Ext( BP,/(2)) would contradict the edge theorem, 5.1.23. To prove (b) we need some simple facts about 7r*( So) in dimensions I 8 which can be read off the ASS (3.2.11). First we have q 3 = 4 v in 7rj(So). This means h i x must be killed by a differential in the ANSS for M ( 2 ) for any permanent cycle x. Hence we get d3(uf) = h: and d3(u : ) = q h ; . Next, if we did not have 7r2(M ( 2 ) ) = Z/(4) then u, E 7r2(M(2)) would extend to a map Z 2 M ( 2 ) + M ( 2 ) and by iterating it we could show that all powers of u1 are permanent cycles, contradicting the above.

20 1

4. Ext2 and the Thorn Reduction

Now suppose we can show that u: and u are permanent cycles representing elements of order 2 in r * ( M ( 2 ) ) i.e., , maps S" + M ( 2 ) which extend to self-maps Z"M(2)+ M(2). Then we can iterate the resulting a:Z8M(2)-+ M(2) and compare with the map extending u to generalize the lowdimensional results above to all of (b). A simple calculation with the ASS shows that 7r,(M(2)) and 7r8(M(2)) both have exponent 2 and contain elements representing u and u?, respectively, so we have both the desired self-maps.

4. Ext2 and the Thorn Reduction In this section we will describe Ext2(BP,) and what is known about its behavior in the ANSS. We will not give all the details of the calculation; they can be found in Miller, Ravenel, and Wilson [l] for odd primes and in Shimomura [ ll'for p = 2. The main problem is to compute Exto(M 2 )and the map d z from it to Exto(M'). From this will follow (5.4.4) that the Y,E Ext3(BP*) are nontrivial for all t > O if p is odd. (We are using the notation of 5.1.19.) They are known to be permanent cycles f o r p 2 7 (1.3.18). We will also study the map Q from Ext2 to E:.* of the ASS as in 5.2.8 to show that most of the elements in the latter group, since they are not im Q, cannot be permanent cycles (5.4.7). The result is that im Q is generated by { @ ( P p - / p " - l L@( P p " / , n ) : n 2 11 and a certain finite number of other generators. It is now known that for p # 3 the Q( p p n / p " - - l ) are permanent cycles [3.4.4(d)]; for p = 2 they are the 77n+2 E ! J $ + 2 constructed by Mahowald [ 6 ] and for p 2 5 analogous elements have recently been constructed by Cohen and Goerss [ 11. Their construction involves the use of Brown-Gitler spectra. For odd primes it follows that some element closely resembling P p " l p ~ for r ~ i 1 5 i 5 p" - 1 is a nontrivial permanent cycle (5.4.9) and there is a similar more complicated result for p = 2 (5.4.10). For p = 2, Ca( P 2 n I 2 " ) = b:,, is known to be a permanent cycle iff there is a framed (2"+*-2)-manifold with Kervaire invariant one (Browder [ l]), and such are known to exist for 0 5 n I4 (Barratt et al. [2]). The resulting element in T 2 J + ' - 2 is known as 0, and its existence is perhaps the greatest outstanding problem in homotopy theory. It is known to have certain ramifications in the EHP sequence (1.5.29). For odd primes the situation with a( is quite different. We showed in Ravenel [7] that this element is not a permanent cycle for p z 5 and &n,pn)

202

5. The Chromatic Spectral Sequence

n 2 1 , and that / l p n , p n itself is not a permanent cycle in the ANSS for p 2 3 and n z l ; see 6.4.1. To compute Ext2 with the CSS we need to know E z 2 , 152, and Eke. The first vanishes by 5.2.1; the second is given by 5.3.5 for p > 2 and 5.3.6 for p = 2. For odd primes Ext'( M I ) = El.' vanishes in positive dimensions; for p = 2 it gives elements in Ext2(BP,) which are products of a 1with generators in Ext'(BP,). The main problem then is to compute EYS2= Exto(M 2 ) . We use the SES

O+ M i + M 2 2 M 2 + 0 and our knowledge of Exto(MI) (5.2.13). The method of 5.1.17 requires us to recognize nontrivial elements in Extl(M I).This group is not completely known but we have enough information about it to compute Exto(M 2 ) .We know Ext'(M;) by 5.2.11, and in proving 5.2.13 one determines the image of Exto(Mi) in it. Hence we know all the elements in Ext'(Mf ) which are annihilated by u1, so any other element whose product with some u ; is one of these must be nontrivial. To describe Exto(M2) we need some notation from 5.2.13. We treat the odd primary case first. There we have x2.0 = u2 9 x2,1=

u; - upu;Lu,,

xz.2 = X4,I -

x2,1=

1-1

pZ-p+1-

02

up2+p-1 1

-2v;zIu:p-l'p'-'+l

p2-2p u2 u3,

for

and

i23,

where b 2 , , = ( p + l ) ( p ' - ' - 1 ) . Also u,,,=l and ~ ~ , , = p ~ + p ' for - ~ i- rl l . Then 5.4.1 Theorem (Miller, Ravenel, and Wilson [l]). For odd primes p, Exto(M 2 ) is the direct sum of cyclic p-groups generated by

_, (i) xg,,/pk + l ulI w i t h p & s , j r l , k z O , such f h a t p k ( j a n d j ~ a 2 , ,and either p k f l $ j or az-k-,<j ; and 1 (ii) - f o r P 01

k z 0 , pklj, a n d j 2 1 .

Note that s may be negative. For p = 2 we define x2.#as above for 0 5 i 5 2, x ~=,x:,,~ I for i 2 3, a2,0= 1, = 2, and a2,,= 3 * 2I-l for i 2 2. We also need xI,o= u I , x ~=,u ~ ~+ 4 u ; ' u 2 , and xl,' = for i 2 2. In the following theorem we will describe elements in Exto(M 2 ) as fractions with denominators involving xl.,, i.e., with denominators which are not monomials. These expressions are to be read

203

4. Ext' and the Thorn Reduction

as shorthand for sums of fractions with monomial denominators. For 1

example, in -we multiply numerator and denominator by x,,, to get 8XIJ XI,,

-.

gx:,,

Now xi,l = u': mod (8) so we have 1 uf+4u;'u2 - _ 1 u2 ~8XI.l 8 u: su: 2 u ,

+?.

5.4.2 Theorem (Shimomura [ 11). cyclic 2-groups generated by

(i)

(ii)

(iii)

For p

= 2,

Exto(M 2 ) is the direct sum of

U; x"l __ xs*2 and x;

2 f o r s odd, j = 1 f o r 2 and k 8x1, 221,' 2 ~ : '2v:' 6 ( k = 2 is excluded because /34s/zis divisible by 2 ) ;

x;,,

-f o r s odd, i 2 3, j 2 u:

2

k+ x4l ' j j 2 k

3, 4, 5 , or

either j is odd or u ~ , ~ 2 and 1 Ij Ip' f o r p = 2; and ( f o r p = 2 only) y 1 and y 2 . I n particular 0 # y f E Ext3(BP,) f o r all t > 0

ifp>2andforallt>2 ifp=2. 5.4.5 Corollary.

( a ) For p odd, Ext2(BP,) is the direct s u m of cyclic p-groups generated bypsp~,J,l+d(h,) f o r s 2 1 , p Y s , j ? l , i20,a n d + ( i , j ) r O where+(i,j) is the largest integer k such that p h Ij and j=[;:I: This generator has order p'"""'

if s > l if s = l . and internal dimension 2( p2- 1)sp' -

2 ( p - 1)j. It is the image under a (5.1.18) o f t h e element

x''l

of 5.4.1.

Pl+*("J)V:

( b ) For p = 2, Ext2(BP,) is the direct s u m of cyclic 2-groups generated by a , Z , w h e r e q generates Ext1*2f(BP,)f o r t z 1 and t f 2 ( s e e 5.2.6),and by&2~IJ,l+d(,.,, f o r s r l , s o d d , j 2 1 , i r O , a n d + ( i , j ) r O where 0 0 2 k 22 k 21 -1

if

21j and ai-l < j s a,, i f j is odd and j 5 a,, if j = 2 and i = 2 , if j = 2k-' mod (2k),j Ia,-k, and i 2 3, if 2k ( j ,al-k-l <j Ia,-k, and i 2 3, otherwise,

unless= 1, in which case a,,, is replaced by 2" and + ( 2 , 2 ) = 1 , and p , is omitted. The order, internal dimension, and definition of this generator are as in ( a ) . H

4.

205

Ex9 and the Thom Reduction

Next we study the Thom reduction map @ from Ext2(BP,) to E:,” in the classical ASS. This map on Ext‘ was discussed in 5.2.8. The group E:** was given in 3.4.1 and 3.4.2. The result is The generators of Ext2(BP,) listed in 5.4.5 map to zero under the Thom reduction map @: Ext(BP,) + ExtA*(Z/(p),Z / ( p ) ) with the following exceptions. 5.4.6 Theorem.

(a)

(S.A. Mitchell). For p = 2 @(a:)= h : , @(P2’/2’))

@(

P2’,2’-1)

@(

Pq2.2)

= h?+I

for j z 1,

= hh,+2

for j 2 2 ,

= hzh,

hlh3,

@(ala4/4)=

and

@( P S j 6 . 2 ) = h 2 h 5 .

= -6, for ( b ) (Miller, Ravenel, and Wilson [l]). For p > 2 @( PPl/,,~) j ” 0 ; @ ( f l p ~ / p ~ ~ l ) = h fo oh r, +j r~l , and @ ( P 2 ) = f k o .

Pro05 We use the method of 5.2.8. For (a) we have to consider elements of Ext’( N ’ ) as well as Exto(N ’ ) . Recall (5.3.6) that the former is spanned by

VSPl u;t, 7 for odd s 2 5 and __ for odd s 2 1. We are looking for elements 2

with I-adic filtration 20, and the filtrations of t , and p1 are 0 and -4, u:Pl

Ultl

respectively. Hence we need to consider only -and --, which give the 2 2 first two cases of (a). The remaining cases come from Exto(N2).The filtration of x2,, is p ’ so has filtration i -j - k, and this number is positive in all cases except those indicated above. We will compute @( p2/2)and @( P 4 / 2 , 2 ) , leaving the other cases of (a) and (b) to the reader. [The computation of @(PI)and @( p 2 ) for p > 2 were essentially done in 5.1.20.1 Using the method of u 2 t:

5.1.20(a), we find that /32/2 reduces to -mod

(2), which in turn reduces

01

12, which maps under @ to h l . Similarly,

to t:(t:mod u:t: v:( t: +

@ to

+ u, u:

Ul

tl)

P4/2.2

reduces to

mod ( 2 ) and to u:t:l r:+ t; 1 t i mod I,, which maps under

h,h4.

This result limits the number of elements in Ext:*(Z/(p), Z / ( p ) ) which can be permanent cycles. As remarked above (5.2.8), any such element must

206

5. The Chromatic Spectral Sequence

correspond to one having Novikov filtration 5 2 . Theorem 5.4.6 tells us which elements in Ext(BP,)2 map nontrivially to the ASS. Now we need to see which elements in Ext'(BP,) correspond to elements of Adams filtration 2. This amounts to looking for elements in Exto(N ' ) with I-adic filtration 1. From 5.2.8 we see that and a4/4for p = 2 have I-adic filtration 0, so a2 and have filtration 1 and correspond to hoh, and hoh3, respectively. More generally, a,for all primes has filtration t - 1 and therefore corresponds to an element with Adams filtration 2 t. Hence we get 5.4.7 Corollary. Of the generators of Exti*(Z/(p), Z ( p ) ) listed in 3.4.1 and 3.4.2, the only ones which can be permanent cycles in rhe ASS are

( a ) f o r p = 2 , hg, hoh2, hoh3, h: f o r j r l , h,h, f o r j 2 3 , h2h4, and h 2 h 5 ; and (b) forp>2,a~,6,forj~O,a,,aohlforp=3,h,h,forj~2,andko. Part (a) was essentially proved by Mahowald and Tangora [8], although their list included h3h6. In Barratt, Mahowald, and Tangora [ l ] it was shown that h2h5is not a permanent cycle. It can be shown that d3(&6,2) # 0, while &2,2 is a permanent cycle. The elements h i , hoh2, hoh3 for p = 2 and a& a , , aoh, ( p = 3) for odd primes are easily seen to be permanent cycles detecting elements in im J. This leaves two infinite families to be considered: the 6, (or hf,, for p = 2) for j r 0 and the h& (or hlh,,, for p = 2 ) for j z 1. These are dealt with in 3.4.4 and 4.4.22. In Section 6.4 we will generalize the latter to 5.4.8 Theorem.

)= # O modulo a ( a ) In the ANSS for p 2 3 , & p - , ( p p ~ , p ~a1/3Pp~-lIp~-l certain indeterminacy for j 2 1. ( 6 ) In the ASS for p 2 5 , b, is not a permanent cycle for j 2 1. The restriction on p in 5.4.8(b) is essential; we will see (6.4.11) that b, isoa permanent cycle for p = 3. The proofs of 3.4.4(b) and (d) do not reveal which element in ExtZ(BP,) detects the constructed homotopy element. For example, EX^^,(^+^^)^ (BP,) is spanned by Pp3,&l and & - p 2 + p l p - l so the homotopy element of 5.4.1 l ( b ) could be detected by the sum of the former with any multiple of the latter. More generally, 5.4.5 implies that Ext2*(1+pJ)4 is a Z / ( p ) vector space of rank [ j / 2 ] ; i.e., it is spanned by elements of the form 6,(x) for X E Ext'(BP,/(p)). (This group is described in 5.2.14 and 5.2.18.) The x that = & ( u ; ~ ) (6, . and 6, aredefined in 5.1.2.) The we want must satisfy fact that the homotopy class has order p , along with 2.3.4, means that x itself [as well as 6 , ( x ) ] is a permanent cycle, i.e., that the map f : S" + So

207

4. Ext2 and the TLom Reduction

for m = q ( l + p J ) - 2 given by 5.4.11(b) fits into the diagram S"

L

so

where M ( p ) denotes the mod ( p ) Moore spectrum and the vertical maps are inclusion of the bottom cell and projection onto the top cell. Now f" can be composed with any iterate of the map a : Z4M(p ) + M ( p ) inducing multiplication by v1 in BP-homology, and the result is a map S m f q fSo + detected by 6 , ( v : x ) . Similar remarks apply to the maps given by 3.4.4(e), although the details are more difficult in this case. These give 5.4.9 Theorem. Let qJE .rrL a n d f ; - , E 7rL-l be theelementsgiven by3.4.4(d) and ( e ) , respectively, where m = ( l + p J ) q - 2 .

( a ) (R. Cohen and Goerss [ l ] ) . vJ is detected by an element x J € Ext2.2+m(Bp,) congruent to p , ~i / p J - i - l modulo elements of the form for r, t < p J - ' - l . Moreover for O < i < p ' - ' - l , q J , , E ( a , , pq, J ) ~ . r r : + qobtained by the composition indicated above, is detected by an element in E X ~ ~ ~ ~ + " + ~ 'congruent ( B P * ) to I ~ ~ J - ~ - ~ - , . (R. Cohen [ 2 ] ) . lJPl I S detected by an element yJ-l E Ex~~*~+"'(BP*) (b) congruent to al p p ~ - l / p ~ - lmodulo elements of higher I-adic jiltration (i.e., modulo kercP). Moreover for j r 3 and O < i < ~ ' - ' - p ~ - ~ - &pI , ,~ ~E ~ obtained as above, is nontrivial and detected by an p , a , )c element in Ext'*2+m+q' (BP,) congruent to a , ppJ-l/pJ-l-, . W The range of i in 5.4.12(b) is smaller than in (a) because a 1

~ , J , , J - I + ~ J

for j

2 2.

2

=0

v1

To see this compute the coboundary of

P

tp+,)p,-2. v1

The analogous results for p = 2 are more complicated. vJE is nor known to have order 2, so we cannot extend it to a map X2'M(2)+ So and compose with elements in 7r,(M(2)) as we did in the odd primary case above. In fact, there is reason to believe the order of qJ is 4 rather than 2. To illustrate the results one might expect, suppose p 2 ' / 2 J is a permanent cycle represented by an element of order 2. (This would imply that the Kervaire invariant element O,+, exists; see 1.5.29.) Then we get a map f :Z2J+2-2M(2)+ So which we can compose with the elements of T*(M ( 2 ) ) given by 5.3.13. In particular, fv:k would represent &J/2'-4k, which is nontrivial for k < 2J-2,fv, would represent p 2 J / 2 ' - 1 (i.e., would be closely related to vJt2),and 2fv, would represent ( Y : p 2 ' / 2 J , leading us to expect v,+~to have order 4. Since the elements o f 5.3.13 have filtration 5 3 , the &J

208

5. The Chromatic Spectral Sequence

composites with f would have filtration 5 5 . Hence their nontriviality in Ext(BP,) is not obvious. Now 5.3.13 describes 12 families of elements in Ext( B P J ( 2 ) ) (each family has the form { u : ~ x :k 2 0 ) ) which are nontrivial permanent cycles: the six shown in 5.3.14 and their products with u. Since we do not know O,+, exists we cannot show that these are permanent cycles directly. However, five of them ( u l a l , q a : , uuI, u u l a l , and U U , ~ : )can be obtained by composing u1 with mod (2) reductions of permanent cycles in Ext(BP,), and hence correspond to compositions of vJ+,with elements in n”, Four of these five families have been shown to be nontrivial by Mahowald [lo] without use of the ANSS. 5.4.10 Theorem (Mahowald [lo]). Let & k + l E be rhe generator constructed by Adams [ l ] and detected bv q k + l E Ext’*8k+2( BP,), and let Pk E

&-, be a generator of

im J detected by a generator y.,k of Ext‘.*k(BP*). Thenfor O < k 0, U;E Exto(BP,/12) is a permanent cycle in the ANSS for M ( p , q).Then 2.3.4 tells us that p, = 6,6,( u ; ) detects the composite u‘

S2“ p 2 - 1 )

2M ( p , u , ) + Z q + 1 M ( p ) + S q + 2 ,

where q = 2p - 2 as usual. One way to do this is to show that the third SES of 5.1.2 can be realized, i.e., that there is a map P : Z ’ ‘ p 2 - ” M ( p ,u l ) + M ( p , u , ) realizing multiplication by u 2 .This self-map can be iterated t times and composed with inclusion of the bottom cell to realize u:. To construct /3 one must first show that u2 is a permanent cycle in the ANSS for M ( p , u, 1. One could then show that the resulting map S21 pz-1) + M ( p , u , ) extends cell by cell to all of X Z(p2-1) M ( p , u , ) by obstruction theory. However, this would be unnecessary if one knew that M ( p , q ) were a ring spectrum, which it is for p 2 5 but not for p = 3. Then one could smash u2 with the identity on M ( p , zll) and compose with the multiplication, giving X2( p2-1)

M(P,~l)+M(P,~l)~M u ,()P+ ,M ( p , v , ) ,

which is the desired map p. Showing that M ( p , u l ) is a ring spectrum, i.e., constructing the multiplication map, also involves obstruction theory, but in lower dimensions than above. We will now describe this calculation in detail and say what goes wrong for p = 3. We need to know Ext””(BP,/I,) for t - s 5 2( p 2 - 1). This deviates from Ext( B P , / I ) = Extp(Z/( p ) , Z/( p ) ) only by the class D ~ Exto”‘p2-’’. E It follows from 4.4.8 that there are five generators in lower dimensions, ) ~ , and namely 1 E Extoso,ho€ Extl.q, boc Ext2*pq,hobo€E ~ t ~ ( ~ h+, El Extl*pq, Ext”‘ = 0 for t - s = 2( p 2 - 1) - 1 so u2 is a permanent cycle for any odd prime. To show M ( p , u , ) is a ring spectrum we need to extend the inclusion So+M ( p , u , ) to a suitable map from X = M ( p , u , ) A M ( p , u , ) . We now assume p = 5 for simplicity. Then X has cells in dimensions 0, 1, 2, 9, 10, 11, 18, 19, and 20, so obstructions occur in Exts7‘for t - s one Iess than any of these numbers. The only one of these groups which is nontrivial is

210

5. The Chromatic Spectral Sequence

ExtoT0=Z/(p).In this case the obstruction is p times the generator (since the 1-cells in X are attached by maps of degree p ) , which is clearly zero. Hence for p 2 5 M(p , u , ) is a ring spectrum and we have the desired self-map p needed to construct the &'s. However, for p = 3 obstructions occur in dimensions 10 and 11, where the Ext groups are nonzero. There is no direct method known for calculating an obstruction of this type when it lies in a nontrivial group. In Toda [ l ] it is shown that the nontriviality of one of these obstructions follows from the nonassociativity of the multiplication of M ( 3). We will sketch another proof now. If M ( 3 , ul) is a ring spectrum then each pt is a permanent cycle, but we will show that p4is not. In EX^^**^( BP,) one has p:p, and PI/3&3.These elements are actually linearly independent, but we do not need this fact now. It follows from 4.4.22 that d,( PIp $ 3 ) = * a l p:p,,, # 0. The nontriviality of this element can be shown by computing the cohomology of P, in this range. Now p: E EX^^*^^( BP,) is a permanent cycle since p2 is. If we can show 5.5.1

p: = * p 1 pi13 * P: p4

then p:p4 and hence p4will have to support a nontrivial d , . We can prove 5.5.1 by reducing to Ext(BP,/fJ. By 5.1.20 the images of P I , &, and p4 in this group are *blo, *u2bl0*k,, and *u:blo, respectively, and the image of fi3,3 is easily seen to be *b,, . Hence the images of p:p4, p1p;,3 and p: are *u:b:,, *blob:, , and *u:b:,* k i , respectively. Thus 5.5.1 will follow if we can show k i = *blob:, . (At any larger prime p we would have kg = 0.) ko is the Massey product * ( h o , h l , h,). Using A1.4.6 we have up to sign

G = (ho, h , , h,)(ho,

hl

hl)

hl),

hl, h l )

= ((ho, ha, h l P 1 ,

h l , hl)

= (hO(h0, h l ,

= (ha, ho, hl)(h*, hl = gob11

and

9

9

hl)

211

5. Periodic Families in Exp

5.5.2 Theorem (Smith [ 11). Let p 2 5. (BP,) is a nontrivial permanent cycle in the ANSS ( a ) pt EXf2.q((p+1)t-1) for all t > 0.

p, u , ) + M ( p , u , ) inducing multiplication ( b ) There is a mapp: X2'p2p1)M( by u, in BP-homology. pf detects the composite S2'( p 2 -

p, u , ) p ’,M ( p , u , ) + s 2 p .

1) + x2r'!2-I)M(

( c ) M ( p, u , ) is a ring spectrum.

H

To realize more general elements in Ext2(BP,) one must replace Z, in the above construction by an invariant regular ideal. For example a self-map p of M ( p 2 , v y ) inducing multiplication by vf (such a map does not exist) would show that pfp2/p,2 is a permanent cycle for each t > 0 . Moreover we could compose p' on the left with maps other than the inclusion of the bottom cell to get more permanent cycles. Exto(BP,.( p 2 , u y ) ) contains p u ; for 0 5 i < p , and each of these is a permanent cycle and using it we could show that P r p 2 1 p p , is a permanent cycle. u;"') for s > 0 and p odd. Showing that it It is easy to construct M ( is a ring spectrum and constructing the appropriate self-map is much harder. The following result is a useful step. P I + ' ,

5.5.3 Theorem. ( a ) (Mahowald [ 111). M(4, u l ' ) is a ring spectrum for t > 0. ( b ) (Oka [4]). M(21t2, U ~ ' ~ + ~ " ~ U ~ ' ' is - ~aUring ~ ) spectrum for i 5 2 and t 2 2. (c) (Oka [4]). For p > 2, M ( p i f l , uf") is a ring spectrum for i 2 0 and t 2 2. [Recall M ( p , u , ) is a ring spectrum for p 2 5 by 5.5.2(c).] Note that M ( p ' , u:) is not unique; the theorem means that there is a finite ring spectrum with the indicated BP-homology. Hence we have a large number of four-cell ring spectra available, but it is still hard to show that the relevant power of u2 is a permanent cycle in Exto. 5.5.4 Theorem.

( a ) (Davis and Mahowald [l], theorem 1.3). For p = 2 , there is a map X4'MM(2, u:) + M(2, v:) inducing multiplication by v i , so P s t / 4 and /3'1/3 are permanent cycles for all t > 0. ( b ) For p 2 5 the following spectra exist: M ( p , uf-', u;) (Oka [2, 11, Smith [2], Zahler [2]); M ( p, u y , v;')for t 2 2 (Oka [2,II]); M ( p , u:'-~, v;') (Oka [ 2 , I]); M ( p, v?, u F 2 ) for t 2 2 (Oka [3]); M ( p 2 ,u y , u:") for t 2 2 (Oka [3]); and consequently thefollowing elements in Ex?( BP,) are nontrivial p t 2 2; p l p 2 / , for t > 0, permanent cycles: ptpl,for t > 0, 1 5 i 5 p - 1; p f p I for 15 i I2 p - 2; p t p 2 , 2 p and pfp2/2p-, for t 2 2; and p t p 2 / p ,for 2 t 2 2.

212

5. The Chromatic Spectral Sequence

u;") for t 2 2 and n 2 3, ( c ) (Oka [7]). Forp 2 S the spectra M ( p , and M ( p, u;") for n 2 3 exist. Consequently the following elements are nontrivial permanent cycles: P p n r l s for t 2 2, n 2 3, and 1 5 s 5 2"-'p; and /3p",/rfor t 2 1, n 2 3, and 1 5 s 5 2"-'p. In particular the p-rank of T: can be arbitrarily large.

Note that in (a) M(2, u:) is not a ring spectrum since M ( 2 ) is not, so the proof involves more than just showing that U!E Exto(BP,/(2, u ; ) ) is a permanent cycle. When a spectrum M ( p ' , u : , u,") for an invariant ideal ( p ' , u : , u,")c BP, does not exist one can look for the following sort of substitute for it. Take a finite spectrum X with torsion-free homology and look for a finite spectrum X M ( p l , u : , u,") whose BP homology is B P , ( X ) OBp*B P J ( p ' , u : , u,").Then the methods above show that the image &],, of induced by the inclusion So+ X [assuming X is (-1)-connected with a single 0-cell] is a permanent cycle. The resulting homotopy class must "appear" on some cell of X, giving us an element in 7 r z which is related to Pk.,,,. The first example of such a result is

5.5.5 Theorem (Oka and Toda [S]). mapping cone of P I .

Let p

=3

and X

= Sou p ,e l ' ,

the

( a ) The spectrum X M ( 3 , u , , u2) exists so &EEx~ '(BP, ( X)) is a permanent cycle for each t > 0. ( b ) The spectrum X M ( 3 , u:, u:) exists so & , / 2 ~ Ext2(BP,(X)) is a permanent cycle for each t > 0. Let p = 5 and X = So u p ,e39. ( c ) The spectrum XM(5, u I ru2, u3) exists so 7, E Ext3(BP*(X)) is a permanent cycle for all t > 0. Hence fir detects an element in %-16,-6(x) which we also denote by The cofibration defining X gives an LES . - * + 7 r n ( S 07rn(X)& )~

6,.

7r,~,,(S0)-+ A T,-I(so)+. * .

where the last map is multiplication by ,f3, E ale( S o ) . If @,C im i then j ( P I ) f 0, so for each t > we ~ get an element in either 7&-6 or 7 r f 6 , - 1 7 . For example, in the ANSS for the sphere one has d,( p4)= a1 so p4aim i and j ( p4)E 7rt7is detected by a l p 1&3, i.e., j ( p4)= P I & '(see 5.1.1). We can regardj( p,) as a substitute for p, when the latter is not a permanent cycle. In the above example we had BP,(X) = BP,OE'BP, as a comodule, so Ext( BP,) is a summand of Ext( BP,( X ) ) . In the examples below this is not the case, so it is not obvious that &,,, # 0.

5.5.6 Theorem (Davis and Mahowald [ 13 and Mahowald [ 121). Let p = 2, X = So u, e2, W = So u e4, and Y = X A W Part ( a ) below is essentially

5. Periodic Families in Exe

213

theorem 1.4 of Davis and Mahowald [l], while the numbers in succeeding statements refer to theorems in Mahowald [12]. Their Y a n d A , are XM( 2) and XM(2, u , ) in our notation. ( a ) XM(2, u , , u:) exists and & E Ext2(BP,(X)) is a nontrivial permanent cycle, ( b ) (1.4) In the ANSS for So, P8, is not a permanent cycle and ps,E T ~ ~ , - ~ projects (X) under the pinching map X + S 2 to an element detected by if this element is nontrivial. ~ are non( c ) (1.5) $+'E Exto(BP,(X)/Z2) and P 8 , + ,Ext2(BP,(X)) trivial permanent cycles. &,+,E Ext2(BP,) is not a permanent cycle and &+, E T48'+2(X)projects to an element detected by ff](U4/4P8,/3 E Ext4(BP,) if this element is nontriuial. Proof.

(a) Davis and Mahowald [l] showed that XM(2, u , ) admits a self-map realizing u:. This gives the spectrum and the permanent cycles. To show & f 0 it suffices to observe that PsrE Ext2(BP,) is not divisible by a , . (b) Mahowald [12] shows that PSIE n481-4(x) projects nontrivially to TS ~ ~ ,By - ~duality . there is a map f :X48t-4(X)+ So which is nontrivial on the bottom cell. From 5.3.13 one can construct a map X48'-4X+ X48r-10M ( 2 ) which is u,v2on the bottom cell and such that the top cell is detected by u: E Exto(BPJ(2)). Now compose this with the extension of Ps,/4X48'-'oM(2)+ So given by 5.5.3(a). The resulting map g : E48r-4X+ So is on the bottom cell and the top cell is detected by P8r. Hence this # 0E map agrees with f modulo higher Novikov filtration. If Ext4(BP,) it follows that the bottom cell on f is detected by that element. [It is likely that a:Pst13 = 0 (this is true for t = l ) , so the differential on Psr is not a d,.] in T : ~ is , (c) As in (b) Mahowald [12] shows the projection of & + I nontrivial. To show that f f I f f 4 / 4 & , / 3 detects our element if it is nontrivial we need to make a low-dimensional computation in the ANSS for M(2, u:) where we find that u : u 2 E ~ X ~ " ~ ~ ' ( B P ,u?)) / ( ~ ,supports a differential hitting u , ( Y ~ , ~ L YExt3,14. :E It follows that u , q E n I l ( M ( 2 , u : ) ) extends to a map XloX+ M(2, u:) with the top cell detected by u2v:. Suspending 48t - 10 times and composing with the extension of P8r,4to E48r-10M(2,u:) gives the result. Now we consider products of elements in Ext'. 5.5.7 Theorem. (a )

Let 5, be a generator of Extlvq'(BP,) (see 5.2.6).

(Miller, Ravenel, and Wilson [ 11). For p > 2, 5@,= 0 for all s, t > 0.

2 14

5. The Chromatic Spectral Sequence

(b) Forp=2 - ~ (i) If s or t is odd and neither is 2 then G,CU, = C Y , ~ ~ ' , +f~ 0. (ii) G: = p 2 / 2 . (iii) a: = p4/4. (Presumably, all orher products of this sort uanish.)

Pro05 Part (a) is given in Miller, Ravenel, and Wilson [ l ] as theorem 8.18. The method used is similar to the proof of (b) below.

4

For (b) ( i ) assume first that s and t are both odd. Then a, =- and ihe 2 uf+l-l

mod ( 2 ) reduction of a, is u : - l t , . Hence gSCr,= -t , = a , + I - , G , . 2 For s odd and t = 2 we have

For

t

even and t > 2, recall that x1/2

6,=-

where

4t

x = u: -4u;'u2

and d ( x ) = 8p where p

= u ; ~ u , -~ u, ; ' ( t 2 + t : ) + 2 ( u lt , +u;'r;+ v;2rlt2+u ; ~ u , ~ mod : ) (4).

Hence for even t > 2 the mod ( 2 ) reduction of 5,is u:-2p and for odd s ( Y s Gu,f = + r2- 2p =

U,X(S+r-I)/2

P-

Since

so

CU,&,

= ( Y I t i s + , - , as claimed. 2

2

u:

V;'u:

For (ii) we have 6:= u ' ( t ' + u l t l ) . The coboundary of y+shows 4 2 2 this is cohomologous to p2/2. Ext2,I6which is (Z/(2))3 generated by ala7, P 3 , For (iii) we have and p4,4.a I a 7is not a permanent cycle (5.3.7) so a:/4 must be a linear

6. Elements in Ext3 and Beyond

215

combination of p4/4and p 3 . Their reductions mod I,, t:l t: and u2f:I t , , are linearly independent so it suffices to compute a:l4mod 1,. The mod Z, reduction of a4/4is f:, so the result follows.

6. Elements in Ext3 and Beyond We begin by considering products of elements in Ext' with those in Ext' and Ext2. If x and y are two such elements known to be permanent cycles, then the nontriviality of xy in Ext implies that the corresponding product in homotopy is nontrivial, but if xy = O then the homotopy product could still be nontrivial and represent an element in a higher Ext group. The same is true of relations among and divisibility of products of permanent cycles; they suggest but do not imply (without further argument) similar results in homotop y. Ideally one should have a description of the subalgebra of Ext(BP,) generated by Ext' and Ext2 for all primes p . Our results are limited to odd primes and fall into three types (see also 5.5.7). First we describe the subgroup of Ext3 generated by products of elements in Ext' with elements of order p in Ext' (5.6.1). Second we note that certain of these products are divisible by nontrivial powers of p (5.6.2). These two results are due to Miller, Ravenel, and Wilson [l], to which we refer for most of the proofs. Our third result is due to Oka and Shimomura [6] and concerns products of certain elements in Ext' (5.4.4-7). They show further that in certain cases when a product of permanent cycles is trivial in Ext4,then the corresponding product in homotopy is also trivial. This brings us to y's and 6's Toda [ l ] showed that yt is a permanent cycle for p 2 7 (1.3.18), but left open the case p = 5. In Section 7.5 we will make calculations to show that y 3 does not exist. We sketch the argument here. As remarked in Section 4.4, 4.4.22 implies that d33(a1 p:,5) = Pf' (up to a nonzero scalar). Calculations show that a , p'$5 is a linear combination , y2). Hence if the latter can be shown to be a of P : y 3 and p 1 ( a I P 3p4, permanent cycle then we must have d33(y3) = p:8. Each of the factors in the above Massey product is a permanent cycle, so it suffices to show that ~ vanish. ( Our calculathe products a , p3p4E ~ 3 2 3 S( o ) and p4y2 E T ~So)~both tion shows that both of these stems have trivial 5-torsion. To construct 6, one could proceed as in the proof of 5.5.2. For p 2 7 there is a finite complex V ( 3 ) with BP,( V(3)) = BP,/14. According to Toda [l] it is a ring spectrum for p I1 1. Hence there is a self-map realizing multiplication by u4 iff there is a corresponding element in T*(V(3)). We will show

216

5. The Chromatic Spectral Sequence

(5.6.13) that the group Ext2p-1.2'p"+p-2) ( BP,/Z,) is nonzero for all p 2 3, so it is possible that d2p-l(u,) # 0. The following result was proved in Miller, Ravenel, and Wilson [ l ] as theorem 8.6.

5.6.1 Theorem. Let m 2 0, p Y s, s 2 1 , l s j Ia*,, (wherea,,, is as in 5.4.1) for s > 1 and 1 5j5 p m for s = 1. Then a IP s p m / J # 0 in Ext3(BP,) iflone of the following conditions holds

+

(i) j = 1 and either s - 1 mod ( p ) or s = - I mod ( P"'+~). (ii) j = 1 a n d s = p - 1 . (iii) j > l + a 2 , , - ~ , ~ J - l ~ - l . In case (ii), we have a I P p -= l -yl and f o r m 2 I , 2a1P(p-l)pm = -yp*"/p"-,p'". The only linear relations among these classes are a1 P s p " p t 2

= SaI Psp2-I

9

and a IP s p 2 m + 2 / 2 + 0 2 , m + , = 2sal p s p 2 m + 2 - p m

for m 2 1.

This result implies that some of these products vanish and therefore certain Massey products (A1.4.1) are defined. For example, a1&,p-l , p m = 0 if t > 1 and p m + 2Y t so we have Massey products such as (PZp-l,a l ra , ) represented up to nonzero scalar multiplication by u:pt,

pu:+p

+ V:P-'t:-2Uq-1ujtl PVl

This product has order p but many others do not. For example, a 1pplz= 0 and (PPl2,a I ,a l ) is represented by

which has order p 2 and p ( P p / 2 , a l , a , ) = a lup ~ pto nonzero scalar multiplication. Similarly, one can show ~1Pp2=P(Pp~/2,*Ir~1)=P2(Pp2/3,Q * I ,1 % ,> .

The following results were 2.8(c) amd 8.17 in Miller, Ravenel, and Wilson r11.

5.6.2 Theorem. With notation as in 5.6.1, i f a l P s p m , j # 0 in Ext3(BP,), then it is divisible by at least p ' whenever O < i 5 m and j 5 a2,m-,. H 5.6.3 Theorem. With notation as above and with tprime top, a , s p k / k + l s a , j 3 , p ~ ~ j - sin p ~Ext3(BP*). +1

P,p~g/j=

217

6. Elements in Exe and Beyond

Now we consider products of elements in Ext2, which are studied in Oka and Shimomura [6].

5.6.4 Theorem. For p

23

we have @,P,

= stP,Pj

in Ext4 for i +j

Prooj To compute PsPl we need the mod I2 reduction of computed in 5.1.20. Hence we find PrP, is represented by

=s

+ t.

PI, which was

-tv;+'-'bl0+( :)U;+'2ko PUL

Now let

A routine computation gives

and hence

&PI

st

is represented by - 2

~

vi+t-2ko and the result follows. PVI

The analogous result in homotopy for p 2 5 was first proved by Toda [6]. The next three results are 6.1, A, and B of Oka and Shimomura [6].

5.6.5 Theorem. For p

(i) (ii) (iii)

PsPtpklr

= 0 for

PsPtp2/p,2

Ptp2/az,z=

23

the following relations hold in Ext4.

k 2 1 and 1 5 r 5 a2,k- 1.

= Ps+rcp2-p,P1p/p-

-

P 2 P t p k / a , , & -Ps+(tp-l),pk-’-p)

for t, k 2 2 .

ft/2)P,+(tp-l)p*~1-(2p-,)pP2p~/az,z

5.6.6 Theorem. F o r p 2 5 , O < r s p , w i t h r s p - 1 i f f = 1, theelement/3,Ptpl, is trivial in r , ( S 0 ) if one of the following conditions holds. (i) r s p - 2 . (ii) r = p - 1 a n d s f - 1 m o d ( p ) . (iii) r = p - 1 o r p a n d t = O mod ( p ) .

W

5.6.7 Theorem. For p 2 5 , s Z 0 or 1 , t f 0 mod ( p ) , and t 2 2, the elements P s P l P l pand Ps@tp2,p,2 are nontrivial. W Now we will display the obstruction to the existence of V(4), i.e., a nontrivial element in EX^^^-^^^(^^+^-^) (BE',/ Id).This group is isomorphic to the corresponding Ext group for P* = P [ tl , t 2 , . . .I, the dual to the algebra

2 18

5. The Chromatic Spectral Sequence

of Steenrod reduced powers. To compute this Ext we use a method described in Section 3.5. Let P ( l), = P / ( t:*, t ; , 1 3 , . . . ), the dual to the algebra generated by PI and Pp. We will give P* a decreasing filtration so that P(1), is a subalgebra of EoP,. We let t , , t , E Fo, and tp2, t f + l , tlt2 p P ’ - ” l ‘ P - l ) for i r 1. Then as an algebra we have 5.6.8

EOP,

= P ( 1 ) * 0 T(f*+2,0, t,+,.1)0P(t,2),

where i 2 1, t,,, corresponds to tf’, and T denotes the truncated polynomial algebra of height p . Let R denote the tensor product of the second two factors in 5.6.8. Then

P ( l ) , -+ EoP,

5.6.9

-+

R

is an extension of Hopf algebras (A1.1.15) for which there is a CartanEilenberg spectral sequence (A1.3.14) converging to ExtEop*(Z/(p ) , Z/( p ) ) with 5.6.10

E2 = EXfP(1 )*(Z/(P>,EXtR(Z/( P), z/ ( P))).

The filtration of P , gives an SS (A1.3.9) converging to Ext,*(Z/(p), Z/( p ) ) with E2= EXfEoP*(Z/(P),Z/( P ) ).

5.6.11

In the range of dimensions we need to consider, i.e., for t - s 5 2( p4- 11, ExtR is easy to compute. We leave it to the reader to show that it is the cohomology of the differential P ( l),-comodule algebra E(ht2, h219 h3o.

hi3,

h22, h 3 1 , h,o)OP(b12, b21, b3o)

with d ( h 2 , ) = h12hl,,d ( h , , )= h Z l h l 3 and , d ( h 4 0 )= h3,h,,. In our range this cohomology is 5.6-12

E ( h i z , h21,

h30,

h13)/hi3(h12, h21, h30)@P(b12,b21,h o ) ,

where the nontrivial action of P(1) is given by Plh30= h,, ,

PPh2,= h , , ,

and

PPb3,= b,, .

We will not give all of the details of the calculations since our aim is merely to find a generator of EX^$',^(^"^-^). The element in question is 5.6.13

K 3h , I h20hl2h2,h 3 0 .

We leave it to the interested reader to decipher this notation and verify that it is a nontrivial cocycle.

CHAPTER 6

MORAVA STABILIZER ALGEBRAS

In this chapter we develop the theory which is the mainspring of the chromatic SS. Let K ( n ) , = Z / ( p ) [ u , , u,'] have the BP,-module structure obtained by sending all u, with i # n to 0. Then define Z( n ) to be the Hopf algebra K ( n ) , OBp*BP,(BP) OeP,K ( n),. We will describe this explicitly as a K(n),-algebra below. Its relevance to the ANSS is the isomorphism (6.1.1)

ExtBp*(BP)( BP* ~BP*/ i 'I n 3

ExtZ(n )( K ( n )* > K ( n

which is input needed for the CSS machinery described in Section 5.1. In combination with 6.2.4, this is the result promised in 1.4.9. Since Z(n)is much smaller than BP,(BP), this result is a great computational aid. We will prove it along with some generalizations in Section 1, following Miller and Ravenel [5] and Morava [2]. In Section 2 we study X ( n ) , the nth Morava stabilizer algebra. We will show (6.2.5) that it is closely related to the Z/(p)-group algebra of a pro-p-group S, (6.2.3 and 6.2.4). S, is the strict automorphism group [i.e., the group of automorphisms f ( x ) having leading term x] of the height n formal group law F, (see A2.2.17 for a description of the corresponding endomorphism ring). We use general theorems from the cohomology of profinite groups to show Sn is either periodic (if ( p - 1 ) 1 n) or has cohomological dimension n2 (6.2.10). In Section 3 we study this cohomology in more detail. The filtration of 4.3.24 leads to a May SS studied in 6.3.3 and 6.3.4. Then we compute HI (6.3.12) and H 2 (6.3.14) for all n and p . The section concludes with computations of the full cohomology for n = 1 (6.3.21), n = 2 and p > 3 219

220

6. Morava Stabilizer Algebras

(6.3.22), n = 2 and p = 3 (6.3.23), n = 2 and p = 2 (6.3.24), and n = 3, p > 3 (6.3.29). The last two sections concern applications of this theory. In Section 4 we consider certain elements / 3 p ~ / in p ~ Ext*(BP,) for p > 2 analogous to the Kervaire invariant elements p20/21 for p = 2. We show (6.4.1) that these elements are not permanent cycles in the ANSS. A crucial step in the proof uses the fact that SP-] has a subgroup of order p to detect a lot of elements in Ext. Theorem 6.4.1 provides a test that must be passed by any program to prove the Kervaire invariant conjecture: it must not generalize to odd primes! In Section 5 we construct ring spectra T ( n ) satisfying EP,( T ( n ) ) = EP,. t l ,. . . , t,] as comodules. The algebraic properties of these spectra will be exploited in the next chapter. We will show (6.5.5, 6.5.6, 6.5.11, and 6.5.12) that its ANSS E,-term has nice properties.

1. The Change-of-Rings Isomorphism Our first objective is to prove 6.1.1 Theorem (Miller and Ravenel [ 5 ] ) . Let M be a EP,(EP)-comodule ( n ) , . Then there annihilated by I,, = ( p, u , ,. . . , u , - ] ) , and let M = M OBP,K is a natural isomorphism

ExtBp*(B P )( BP* u i M ) = ExtZ(n ( K ( n )* 3

3

fi).

Hereu,'Mdenotesu,'EP, OBP, M, whichisacomodute (even ihough u , ' B P , is not) by 5.1.6. This result can be generalized in two ways. Let E ( n ) , = U , ' B P * / ( U , , + ~i :> O )

and E(n),. E ( n ) , ( E ( n ) ) = E ( n ) , OB, B P , ( B P ) OBP*

It can be shown, using the exact functor theorem of Landweber [3], that E (n), Be, BP,( X ) is a homology theory on X represented by a spectrum E ( n ) with n * ( E ( n ) ) =E ( n ) , , and with E ( n ) , ( E ( n ) ) being the object definedabove. We can generalize6.1.1 byreplacingE(n) with E ( n ) , ( E ( n ) ) and relaxing the hypothesis on M to the condition that it be Z,-nil, i.e., that each element (but not necessarily the entire comodule) be annihilated by some power of Z,. For example, N " of Section 5.1 is [,,-nil. Then we have

221

1. The Change-of-Rings Isomorphism

6.1.2 Theorem (Miller and Ravenel [ 5 ] ) . Let M be Zn-nil with M Oe, E ( n ) , . Then there is a natural isomorphism

EXtgp*(BP)(BP*,~ , ' M ) = E x t E f n ) * i E i n ) ) ( E ( n ) , ,

G=

M)-

There is another variation due to Morava [ 2 ] . Regard BP, as a 2/2( p n - 1)-graded object and consider the homomorphism 8: BP, + Z / ( p ) given by e ( u , ) = 1 and 8 ( v i )= 0 for i # n. Let I c BP, be ker 19 and let V, and VT, denote the I-adic completions of BP, and BP,(BP). Let E, = V , / ( v,,+~:i > 0) and EH, = E, Ov, VT, Ov, E,. 6.1.3 Theorem (Morava [ 2 ] ) . With notation as above there is a natural isomorphism Extv-r,( Ve, M ) EXtEH,(Eo,

a)

where M is a VTrcomodule and M

=M

OvaE,.

W

Of these three results only 6.1.1 is relevant to our purposes so we will not prove the others in detail. However, Morava's proof is more illuminating than that of Miller and Ravenel [S] so we will sketch it first. Morava's argument rests on careful analysis of the functors represented by the Hopf algebroids VT, and EH,. First we need some general nonsense. Recall that a groupoid is a small category in which every morphism is invertible. Recall that a Hopf algebroid ( A , r) over K is a cogroupoid object in the category of commutative K-algebras; i.e., it represents a covariant groupoid-valued functor. Let a, @: G + H be maps (functors) from the groupoid G to the groupoid H. Since G is a category it has a set of objects, Ob(G),and a set of morphisms, Mor(G), and similarly for H.

-

6.1.4 Definition. The functors a, /3 : G + H are equivalent if there is a map 8: O b ( G )-+ Mor(H) such that f o r any morphism g : g , + g , in G the diagram I.(Sl)

I

O(g,)

P(g,)

a(g),

ff(g2)

Ng,)

i

P(gJ

commutes. Two maps of Hopf algebroids a, b: (A, r)-+ ( B ,Z ) are naturally equivalent if the corresponding natural transformations of groupoid-valued functors are naturally equivalent in the above sense. Two Hopf algebroids (A, r) and (B, Z) are equivalent if there are maps f :( A , r)+ (B, X) and h :( B, Z) + (A,I')such that hf and f h are naturally equivalent to the appropriate identity maps. Now we will show that a Hopf algebroid equivalence induces an isomorphism of certain Ext groups. Given a map f :(A, r)+ (B, I;) and a

222

6. Morava Stabilizer Algebras

left r-comodule M, define ax-comodulef*( M ) to be B OAM with coaction

B @ A M + B @ A r @ A M + B @g I:@ A M = Z

@g

B @ A M.

Let f : (A, r)+ (B, I;) be a Hopf algebroid equivalence. Then there is a natural isomorphism Extr( A, M ) = Extr( B,f*(M ) ) for any rcomodule M.

6.1.5 Lemma.

Pro05 It suffices to show that equivalent maps induce the same homomorphisms of Ext groups. An equivalence between the maps a, b: (A,r)+ ( B ,2 ) is a homomorphism 4: r + B with suitable properties, including @vR = a and +vL= b. Since vRand T~ are related by the conjugation in r, it follows that the two A-module structures on B are isomorphic and that a * ( M ) is naturally isomorphic to b * ( M ) . We denote them interchangeably by M’. The maps a and b induce maps of cobar complexes (A1.2.11) C,( M) + C,( M’). A tedious routine verification shows that 4 induces the required chain homotopy.

T ,and EHB. Recall that Now we consider the functors represented by I an Artin local ring is a commutative ring with a single maximal ideal satisfying the descending chain condition, i.e., the maximal ideal is nilpotent. If A is such a ring with finite residue field k then it is W(k)-module, where W ( k ) is the Witt ring of A2.2.15. Let Art, denote the category of Z/(2(p” - 1))-graded Artin local rings whose residue field is an F,-algebra. Now let me = ker 8 c BP,. Then BP,/ m i with the cyclic grading is is object BP,.rn;j is an inverse limit of such objects as is VT,. in Art,, so V, =l& For any A E Art,, we can consider Horn‘( VT,, A), the set of continuous ring homomorphisms from VT, to A. It is a groupoid-valued functor on Art, pro-represented by VT,. (We have to say “pro-represented’’ rather than “represented” because VT, is not in Art,.) 6.1.6 Proposition. VT, pro-represents the functor lifts, from Art, to the category of groupoids, defined as follows. Let A E Art, have residue field k The objects in lifts,(A) are p-typical lijiings to A of the FGL over k induced by the composite BP, 8,F, -+ k, and morphisms in lifts,(A) are strict

isomorphisms between such lijtings. 6.1.7 Definition. Let mA c A be the maximal ideal for A E Art,+ Given a homomorphism f:F + G of FGLs over A, let f : F + G denote their reductions mod mA. f is a*-isomorphism i f f ( x )= x. 6.1.8 Lemma. Let F and G be objects in lifts,(A). Then the map Hom( F, G )+ Hom( F, G ) is injectiue.

Pro05 Suppose f = O , i.e., f ( x ) = O m o d mA. We will show that f ( x ) = 0 mod mX implies f ( x )= 0 mod m>+’ for any r > 0, so f ( x ) = 0 since mA is

223

1. The Change-of-Rings Isomorphism

nilpotent. We have G ( f ( x f , f ( y ) ) ~ f ( x ) + f mod (y)

d’

since G(x, y ) = x + y mod ( x , y)'.

Consequently, [ p I G (f ( x ) )= p f ( x ) mod m%= 0 mod m x ' since p

E

mA. On the other hand EPlG(f(X))

=f([PIF(x))

and we know [ p I F ( x ) = x P "mod mA by A2.2.4. Hence f ( [ p I F ( x ) ) = 0 mod mk+’ gives the desired congruence f ( x )= 0 mod mi+'. H Now suppose f , , f i : F + G are *-isomorphisms (6.1.7) as is g: G+F. Then gf, = gf2 by 6.1.8 so f l =f i ; i.e. *-isomorphisms are unique. Hence we can make 6-19 Definition. lifts$(A) is thegroupoid of *-isdmorphism classes of objects in lifts,A. 6.1.10 Lemma.

The functors lifts, and lifts: are naturally equivalent.

B u o j There is an obvious natural transformation a:lifts, + lifts; , and we need to define p : lifts: + lifts,, of each *-isomorphism class. Having done this, a@ will be the identity on lifts; and we will have to prove pa is equivalent (6.1.4) to the identity on lifts,. The construction of p is essentially due to Lubin and Tate [2]. Suppose G, E lifts,(A) is induced by el :BP, + A. Using A2.1.26 and A2.2.5 one can show that there is a unique G2E lifts,(A) *-isomorphic to GI and induced by O2 satisfying e( vnti) = 0 for all i > 0. We leave the details to the interested reader. As remarked above, the *-isomorphism from GI to G2 is unique. The verification that Pa is equivalent to the identity is straightforward. H

To prove 6.1.3, it follows from 6.1.5 and 6.1.10 that it suffices to show EH, pro-represents lifts:. In the proof of 6.1.10 it was claimed that any suitable FGL over A is canonically *-isomorphic to one induced by 8: BP, + A which is such that it factors through E,. In the same way it is clear that the morphism set of lifts;(A) is represented by EH,, so 6.1.3 follows. Now we turn to the proof of 6.1.1. We have a map BP,( B P ) + Z( n ) and we need to show that it satisfies the hypotheses of the general change-of-rings isomorphism theorem A1.3.12, i.e., of A1.1.19. These conditions are 6.1.11 (i) the map

(ii) TU,,,,

r'=BP*(BP)O,,*

K ( n ) , + Z ( n ) is onto and

K ( n ) , is a K(n),-summand of

r'.

224

6. Morava Stabilizer Algebras

Part (i) follows immediately from the definition Z( n ) = K ( n), Oep*I". Part (ii) is more difficult. We prefer to replace it with its conjugate, (ii) K ( n ) , O X ( , , K ( n ) , ( B P ) is a K ( n ) , summand of K ( n ) , B P which is defined to be K ( n ) , OBp*BP,(BP). Let B ( n ) , denote u;'BP,/Z,,. Then the right BP,-module structure on K ( n ) , ( B P ) induces a right B ( n ) , module structure. 6.1.12 Lemma.

There is a map

K(n),BP+Z(n) O K ( " , H n ) , which is an isomorphism of Z( n)-comodules and of B( n),-modules, and which carries 1 to 1.

Proof: Our proof is a counting argument, and in order to meet requirements of connectivity and finiteness, we pass to suitable "valuation rings." Thus let k ( O ) * = Z ( p , c K(O),,

k(n)*=Fp[unlc K ( n ) * ,

n>0,

k ( n ) , B P = k ( n ) * Oep* B P * ( B P ) c K ( n ) * B P , b(n),= k ( n ) , [ u , , u 2 , . . .lc N n ) , , where

Uk = U,lUn+k.

It follows from A2.2.5 that in k(n),BP, 6-1-13

mod

vR(Un+k)'Unt,P"-UV,P*tk

. .. , v ~ ( U n + k - i ) ) .

(v~(V,+i),

Hence vR:BP,+ k ( n ) , B P factors through an algebra map b(n),+ k(n),BP. It is clear from 6.1.11 that as a right b(n),-module, k ( n ) , B P is free on generators t" = t p l t ; Z . . where 0 s a , < p " and all but finitely many aiare 0; in particular, it is of finite type over b( n), . Now define

-

o(n)= k(n)*BP@,(n,* k ( n ) * c z(n);

by the above remarks a ( n ) = k ( n ) , [ t , , t z , ...I/( tkP"--V,PC~'tk: k z l ) as an algebra. ( k ( n), , a(n)) is clearly a sub-Hopf algebroid of ( K (n),, Z( n)), so a(n ) is a Hopf algebra over the principal ideal domain k ( n), . The natural map B P , ( B P ) -* a(n)makes BP,( B P ) a left a(n)-comodule, and this induces a left o(n)-comodule structure on k(n),BP. We will show that the latter is an extended left a(n)-comodule. Define a b( n),-linear map f : k ( n ) , B P + b( n), by 1

0

if a = ( O , O , . . .) otherwise.

225

1. The Change-of-Rings Isomorphism

Then f satisfies the equations fijR

f@b(,,)*

= i d : b(n),+ b( n) , ,

k(n),= e:a(n)+k(n),.

Now let f be the a(n)-comodule map lifting f : 6.1.14

' 'd n )

k( n )* BP

@k(n)*

k(n)*BP

]r(n)Bf dn)@k(?d*

b(n),

Since (CIfjR(x) = 1 0 i j R ( x ) , $ is b( n),-linear, so f is too. We claim f is an isomorphism. Since both sides are free of finite type over b ( n ) , it suffices to prove that f@,,(,,)* k ( n ) is an isomorphism. But 6.1.14 is then reduced to

44,-

A

d n ) Ol;(,,)*d n )

so the claim follows from unitarity of A. Now the map K ( n ) , Ok(,,), satisfies the requirements of the lemma. 6.1.15 Corollary. j j R : B ( n ) , + K(n),O,,,,

K ( n ) , B P i s a n isomorphism of

B( n),-modules. Proo$ The natural isomorphism

is B(n),-linear and carries 1 to 1. Hence

B(n),

TK ( n ) ,

OL(n)K(n),BP

commutes, and i j R is an isomorphism.

H

Hence 6.1.11(ii) follows from the fact that K ( n ) , is a summand of x ( n ) , and 6.1.1 is proved. From the proof of 6.1.12 we get an explicit description of Z ( n ) , namely

226

6. Moravn Stabilizer Algebras

6.1.16 Corollary.

As an algebra

" u,P'ti: i > 0). z ( n ) = K ( n ) , [ t , , t 2 , . . . ] / ( v , t ~Its coproduct is inherited from BP,(BP), i.e., a suitable reduction of 4.3.13 holds.

2. The Structure of Z ( n ) To study X( n ) it is convenient to pass to the corresponding object graded over Z / 2 ( p n - 1 ) . Make F, a K(n),-module by sending u, to 1, and let S(n)=2(n)OK(,,*FvForaX(n)-comodule m let &?=MOK(,)*Fp,which is easily seen to be an S(n)-comodule. The categories of X(n)- and S ( n ) comodules are equivalent and we have 6.2.1 Proposition. For a

X( n)-comodule M,

Extxcn,(K(n)*, M ) Fp Exts(n)(Fp,MIWe will see below (6.2.5) that if we regard S ( n ) and I\;r as graded merely over Z/(2), there is a way to recover the grading over Z/2( p" - 1). If M is concentrated in even dimensions (which it is in most applications) then we can regard fi and S(n)as ungraded objects. Our first major result is that S(n)OF,n (ungraded) is the continuous linear dual of the F,n-group algebra of a certain profinite group S, to be defined presently. 6.2.2 Definition. The topological linear dual S(n)* of S ( n ) is as follows. [ I n Ravenel [ 5 ] S ( n)* and S ( n ) are denoted by S ( n ) a n d S ( n ) , ,respectively.] be the sub-Hopf algebra of S ( n ) generated by { t , ..., t,}. I t Let S(n)(,, is a vector space of rank p"' and S(n)=l&S(n),,,. Then S(n)*= lim Hom(S(n),,,,F,), equipped with the inverse limit topology. The product and coproduct in S( n ) give maps of S ( n)* to and from the completed tensor product S ( n ) * 6 S(n)*=!imHom(S(n)(,,OS(n),,F,).

c

To define the group S, recall the 2,-algebra En of A2.2.16, the endomorphism ring of a height n formal group law. It is a free Z,-algebra of rank n2 generated by w and S, where o is a primitive ( p " - 1)th root of units, So = w p S , and S" = p . S , c E is the group of units congruent to 1 mod ( S ) , the maximal ideal in E n . S, is a profinite group, so its group algebra F,n[S,] has a topology and is a profinite Hopf algebra. S, is also a p-adic Lie group; such groups are studied by Lazard [ 4 ] .

= F,[S,] as projnite Hopf algebras, where q = p", S, is as above, and we disregard the grading on S ( n ) * . 6.2.3 Theorem. S(n)*@F,

227

2. The Structure of P(n)

ProoJ: First we will show S(n)*OF, is a group algebra. According to Sweedler [l], Proposition 3.2.1, a cocommutative Hopf algebra is a group algebra iff it has a basis of group-like elements, i.e., of elements x satisfying Ax = XOX.This is equivalent to the existence of a dual basis of idempotent elements { y } satisfying y f = y i and y,yj=O for i # j . Since S ( n ) O F , is a tensor product of algebras of the form R = Fq[t]/( t 4 - t ) , it suffices to find such a basis for R. Let a E F," be a generator and let ri={

C

-

(a't)'

for O < i < q ,

O 2 let c ( x )= exp(px)l, where I is the n x n identity matrix and x E Z, ; for p = 2 let c ( x )= X I for x E Zg . 6.2.7 Theorem.

Let S!, = ker d.

( a ) I f p > 2 a n d p i n then S,,=Z,OSi. ( b ) I f p = 2 and n is odd then S,,I-S~OZ;. Proof: In both cases one sees that im c lies in the center of S,, (in fact im c is the center of S,) and is therefore a normal subgroup. The composition dc is multiplication by n which is an isomorphism for p i n, so we have the desired splitting. We now describe an analogous splitting for S ( n ) . Let A* = Fp[Zp] for p > 2 and A* = F,[Z,"] for p = 2. Let A, be the continuous linear dual of A. 6.2.8 Proposition. As an algebra A = FP[u I, uz, . ..I/( ui - up). The product A is given by 1" A(ui) = 1" u i O u j irO

CD.

i,jzO

where uo= 1 and G is the formal group law with X P’

log,( X ) = 1 7 .

P

Proof: Since A = F,[S,], this follows immediately from 6.2.3. We can define Hopf algebra homomorphisms c* : S ( n )O F, + A @ F, and d,: AOF, + S(n)OF, dual to the group homomorphisms c and d defined above. 6.2.9 Theorem. There exist maps c* :S( n ) + A and d, :A + S( n ) corresponding to those dejined above, and for p ) n, S ( n ) = A O B , where BOF, is the continuous linear dual of F,[S!,], where S!, is dejined in 6.2.7.

230

6. Mornvn Stabilizer Algebras

ProoJ: We can define c* explicitly by c*ti =

ui/,

I

if n i otherwise.

It is straightforward to check that this is a homomorphism corresponding to the c* defined above. In lieu of defining d , explicitly we observe that the determinant of Eizo tiS', where ti E W(F,) and ti = t : , is a power series in p whose coefficients are polynomials in the ti over Z,. It follows that d , can be defined over F,. The splitting then follows as in 6.2.7. W Our next result concerns the size of Exts(,,(Fp, F,), which we abbreviate by H * ( S ( n ) ) .

6.2.10 Theorem. ( a ) H * ( S ( n ) ) isJinitely generated as an algebra. ( b ) If ( p - l ) t n , then H ' ( S ( n ) ) = O for i > n 2 and H ' ( S ( n ) ) = Hn2-i ( S ( n ) ) for 0 5 i In 2 , i.e., H * ( S( n ) ) satisfies Poincare' duality. ( c ) I f ( p - 1 ) 1 n, t h e n H * ( S ( n ) )isperiodic, i.e., thereissomexe H ' ( S ( n ) ) such that H * ( S ( n ) ) is a j n i t e l y generatedfree module over F , [ x ] . We will prove 6.2.10(a) below as a consequence of the open subgroup theorem (6.3.6), which states that every sufficiently small open subgroup of S, has the same cohomology as Z,"'. Then (c) and the statement in (b) of finite cohomological dimension are equivalent to saying that the Krull dimension of H * ( S ( n ) ) is 1 or 0, respectively. Recall that the Krull dimension of a Noetherian ring R is the largest d such that there is an ascending chain po c p , . * c pd of nonunit prime ideals in R. Roughly speaking, d is the number of generators of the largest polynomial algebra contained in R. Thus d = 0 iff every element in R is nilpotent, which in view of (a) implies (b). If d = 1 then every element in R has a power Z [ x ] for a fixed X E R, so (a) implies (c). The following result helps determine the Krull dimension.

6.2.11 Theorem (Quillen [3]). For a projnite group G the Krull dimension of H:( G; F,) is the rank of a maximal elementary abelian sub-p-group of G, i.e., subgroup isomorphic to ( z / ( p ) l d .

w

To determine the maximal elementary abelian subgroup of S,, we use the fact that D, = En O Q is a division algebra over Q, (A2.2.16), so if G c S, is abelian, then the Q,-vector space in 0, spanned by the elements of G is a subfield K c 0,. Hence the elements of G are all roots of unity, G is cyclic, and the Krull dimension is 0 or 1.

6.2.12. A degree m extension K of Q , embeds in D, i$m

I n.

ProoJ: See Serre [l, p. 1381 or Cassels and Frohlich [ 1, p. 2021.

W

231

3. The Cohomology of X(n)

By 6.2.11 H * ( S ( n ) )has Krull dimension 1 iff S,, contains pth roots of unity. Since the field K obtained by adjoining such roots to Q, has degree p - 1,6.2.12 gives 6.2.10(c) and the finite cohomological dimension statement in (b). For, the rest of (b) we rely on theorem V.2.5.8 of Lazard [4], which says that if S,, (being an analytic pro-p-group of dimension n') has finite cohomological dimension, then that dimension is n2 and Poincari duality is satisfied.

3. The Cohomology of X ( n ) In this section we will use an SS (A1.3.9) based on the filtration of Z( n ) induced by the one on BP,( B P ) / I , , given in 4.3.24. We have

63.1 Theorem. DeJine integers d,,, by d , ,=

1"

max( i, pdn,#-,,)

if i s 0 for i > 0.

Then there is a unique increasing Jiltration of the Hopf algebra S(n) with deg t:'=d,,,, for O s j < m . W

6.3.2 Theorem. Let E o S (n ) denote the associated bigraded Hopf algebra. Its algebra structure is E o S ( n ) = T ( t,,J:I > 0 , E~ Z / ( n ) ) , where T ( .) denotes the truncated polynomial algebra of height p on the indicated elements and t , , corresponds to t f ' . The coproduct is induced by the one given in 4.3.34. Explicitly, let m = p n / ( p - 1). Then

'('I,,)

=

I

1

if

i<m,

if

i = m,

t , , J @ l + l O t , , , + b ; - n , , + . - l ( t , , J . . - t , - 2 , , J ) if

i>m,

Orksi

c

tk,~@t~-k,k+,

tlS,Ofl-~,k+,+b,-,,J+,-l

0 5k sI

where to,J= 1 and

6,, corresponds to the bI,Jof 4.3.14.

As in the case of the Steenrod algebra, the dual object EoS(n)* is primitively generated and is the universal enveloping algebra of a restricted Lie algebra L ( n ) . L ( n ) has basis {x~,,:i> 0, j E Z / ( n ) } ,where x , , is ~ dual to tB3J'

232

6. Morava Stabilizer Algebras

6.3.3 Theorem. with bracket

EoS( n ) is the restricted enveloping algebra on primitives x ~ , ,

r'

I+J

[XI,,, X k J l =

n + k J- % + l X , + k l

for i + k 5 m, otherwise,

where m is the largest integer nor exceeding p n / ( p - l), and 6: t mod ( n ) and 6: = 0 otherwise. The restriction 6 is given by

=1

iff s =

Let L ( n ) be the Lie algebra without restriction with basis xi,j and bracket as above. We now recall the main results of May [2]. 6.3.4 Theorem.

There are spectral sequences

( a ) EZ= H * ( L ( n ) ) O P ( b , j ) ~ H * ( E o S ( n ) ) , ( b ) E2= H * ( E , S ( n ) ) * H * ( S ( n ) ) , where bi,jE H23pdrE,S(n ) with internal degree 2pJ+'( p i - 1) and P ( * ) is the polynomial algebra on the indicated generators. W Now let L ( n , k) be the quotient of L ( n ) obtained by setting x , , =~O for i > k. Then our first result is

The E2-termof the first May SS [ 6 . 3 4 a ) ] may be replaced by H * ( L ( n , m ) ) O P ( b i , , :i s m - n ) , where m = [ p n / ( p - l ) ] as before.

6.3.5 Theorem.

Proof: By 6.3.3 t(n ) is the product of L( n, m ) and an abelian Lie algebra, so H * ( L ( n ) ) = H * ( L ( m , n ) ) O E ( h i , Ji :> m ) , where E ( .) denotes the exterior algebra on the indicated generators and hi,jE H ' L ( n ) is the element corresponding to xi.j. It also follows from 6.3.4 that the appropriate differential will send to -b,-,,j-, for i > m. It follows that the entire spectral sequence decomposes as a tensor product of two spectral sequences, one with the &-term indicated in the statement of the theorem, and the other having E 2 = E ( h , , ) O P ( 6 i - n , , )with i > rn and E,=Fp. W If n < p - 1 then 6.3.5 gives an SS whose &-term is H * ( L ( n, n ) ) , showing that H * ( S ( n ) )has cohomological dimension n 2 as claimed in 6.2.10(b). In Ravenel [6] we claimed erroneously that the SS of 6.3.4(b) collapses for n < p - 1. The argument given there is incorrect. For example, we have reason to believe that for p = 11, n = 9 the element (hlOh2,.. . h70) (h28h37. . hT3)supports a differential that hits a nonzero multiple of h10h20(h18h27. * . h63)(h21h31 . . . h6,). We know of no counterexample for smaller n or p.

233

3. The Cohomology of X ( n )

Now we will prove 6.2.10(a), i.e., that H * ( S ( n ) )is finitely generated as an algebra. For motivation, the following is a special case of a result in Lazard [4]. 6.3.6 Open Subgroup Theorem. Every suficiently small open subgroup of S,, is cohomologically abelian in the sense that it has the same cohomology as Z;', i.e., an exterior algebra on n2 generators. We will give a Hopf algebra theoretic proof of this for a cofinal set of open subgroups, namely the subgroups of elements in E,, congruent to 1 modulo (Si) for various i > 0. The corresponding quotient group (which is finite) is dual the subalgebra of S ( n ) generated by { t k :k < i}. Hence the ith subgroup is dual to S ( n ) / (t k : k < i ) , which we denote by S( n, i ) . The filtration of 6.3.1 induces one on S ( n, i ) and analogs of the succeeding four theorems hold for it. 6.3.7 Theorem. If i > m / 2 [where m = p n / ( p - l ) ] then H*(S(n, i ) ) = E(hkj:i s k < i + n , j E Z / ( n ) ) . Proof: In the analog of 6.3.3 we have i, k > m / 2 so i + k > m so the Lie algebra is abelian. We also see that the restriction 5 is injective, so the SS of 6.3.5 has the E2-term claimed to be H*( S ( n, i ) ) .This S S collapses because h,, corresponds to t:' E S ( n, i), which is primitive for each k and j . Proof of 6.2.10(a). Let A ( i ) be the Hopf algebra corresponding to the quotient of S,, by the ith congruence subgroup, so we have a Hopf algebra extension (Al.1.15)

A( i) + S ( n ) + S ( n, i ) . The corresponding CESS (A1.3.14) has E, = ExtA(JFp, H * ( S ( n , i))) and converges to H * ( S ( n ) ) with d,: E.:'+ Esfr**-'+'. Each E,-term is finitely generated since A( i ) and H*( S ( n, i ) ) are finite-dimensional for i > m / 2 . Moreover, E,,2 = Em, so Emand H * ( S ( n ) ) are finitely generated. Now we continue with the computation of H * ( S ( n ) ) .Theorem 6.3.5 indicates the necessity of computing H * ( L( n, k ) ) for k 5 m, and this may be done with the Koszul complex, i.e., 6.3.8 Theorem. H * ( L ( n , k ) ) for k s m is the cohomology of the exterior complex E ( h i , j )on one-dimensional generators h,,, with i s k and j E Z / ( n ) , with coboundary

The element h,,, corresponds to the element x,,, and therefore has jiltration degree i and internal degree 2p'( p' - 1).

234

6. Morava Stabilizer Algebras

Proof. This follows from standard facts about the cohomology of Lie algebras (Cartan and Eilenberg [l], XII, Section 71). Since L( n, k ) is nilpotent its cohomology can be computed with a sequence of change-of-rings spectral sequences analogous to A1.3.14. 6.3.9 Theorem.

There are spectral sequences with Ez = E ( hk,,) O H*(L( n, k - 1 ) ) JH * ( L( n, k ) )

and E3 = E m .

Proof. The S S is that of Hochschild-Serre (see Cartan and Eilenberg [l, pp. 349-3511) for the extension of Lie algebras

A ( n , k ) + L(n, k ) + L(n, k - 1 ) where A ( n , k ) is the abelian Lie algebra on x k j . Hence H * ( A ( n , k ) ) = E ( h k j ) . The &-term, H * ( L , ( n , k - l ) , H * ( A ( n , k ) ) is isomorphic to the indicated tensor product since the extension is central. For the second statement, recall that the spectral sequence can be constructed by filtering the complex of 6.3.8 in the obvious way. Inspection of this filtered complex shows that E3 = Em. In addition to the spectral sequence of 6.3.4(a), there is an alternative method of computing H*EoS( n ) . Define i(n, k ) for k 5 m to be the quotient of P E o S ( n ) by the restricted sub-Lie algebra generated by the elements xi., for k < i 5 m, and define F ( n, k ) to be the kernel of the extension

O-. F( n, k ) + i(n, k ) + L( n, k - 1) + 0. Let H * i ( n, k ) denote the cohomology of the restricted enveloping algebra of i ( n , k ) . Then we have 6.3.10 Theorem.

There are change-of-rings spectral sequences converging to

H * i ( n , k ) with E , = H * ( F ( n , k ) ) O H * ( ~ ( nk -, 1 ) ) where

H * ( F ( n, k ) ) =

{

E (h k , j ) E(h,,)OP(b,,)

for k > m - n for k s m - n

and H * ( i ( n, m ) ) = H * ( E o S ( n ) ) .

ProoJ: Again the spectral sequence is that given in theorem XV1.6.1 of Cartan and Eileinberg [l]. As before, the extension is cocentral, so the &-term is the indicated tensor product. The structure of H * ( F ( n ,k ) ) follows from 6.3.3 and the last statement is a consequence of 6.3.5. H

235

3. The Cohomology of I ( n )

We begin the computation of HI( S ( n ) ) with:

6.3.11 Lemma. H ' ( E o S ( n ) )is generated by

ln= C h,,]

pn =C h2,,]

and

i

for p = 2 ;

i

and for n > 1, hl,jfor each j

EZ/(n).

Proof: By 6.3.4(a) and 6.3.5 H'(E,S(n)) = H ' ( L ( n , m ) ) . The indicated elements are nontrivial cycles by 6.3.8. It follows from 6.3.3 that L(n, m ) can have no other generators since [x,,,, X ~ - ~ , ~=+ xi,j J - 8:+l~i,j+l.

In order to pass to H ' ( S ( n ) )we need to produce primitive elements in S ( n ) , corresponding to 5, and pn (the primitive t f ' corresponds to hl,j). We will do this with the help of the determinant of a certain matrix. Recall from (6.2.3) that S(n)OF,n was isomorphic to the dual group ring of S,, which has a certain faithful representation over W ( F , n ) (6.2.6). The determinant of this representation gave a homomorphism of S ( n ) into Z i , the multiplicative group of units in the p-adic integers. We will see that in HI this map gives us l,, and p.. More precisely, let M = ( m i , ] ) be the n by n matrix over Z p [ t , ,t 2 , . . . I / ( t l - l f " ) given by

C mi,, =

pkt[:+j-i

for

i l j

c pktkp,ij-i

for

k>j

kz0

k i l

where to= 1. Now define T, E S ( n ) , to be the mod ( p ) reduction of p-'(det M - 1) and for p = 2 define UnE S ( n ) , to be the mod (2) reduction of b(det M 2 - 1). Then we have

6.3.12 Theorem. The elements T,, E S ( n ) , and forp = 2 U,, E S ( n ) , areprimitive and represent the elements 5, and pn + l,,E H i (S ( n ) ) , respectively. Hence H ' ( S (n ) ) is generated by these elements and for n > 1 by the hi,jforj E Z / ( n ) . Proof: The statement that T, and U, are primitive follows from 6.2.6. That they represent I , and pn + 5, follows from the fact that

T,=C

ti'

m o d ( t , , t2 ,..., t,,-')

j

and

U n = ~ t ~ ~ + t ' nm' o d ( t , , t , ,..., t,-,). j

236

6. Morava Stabilizer Algebras

Examples. TI=tl,

U1=t1+t2,

u2= t,+

t:+

tlt:+

T2=t2+t:-t:+P,

t:t,+ t*+ t:t:t2

+tit:,

and

T,= t,+t.;+t;2+f:+-p+p2-

t 1 t 2P - f P t1P Z2 - f P 2 1f

2 '

Moreira [ 1,3] has found primitive elements in B P , ( B P ) / I,, which reduce to our T,,. The following result is a corollary of 6.2.7. I f p 1 n, then H * ( S ( n)) decomposes as a tensorproduct of an appropriate subalgebra with E ( C n )for p > 2 and P(l,,)OE(p,,) for p=2. a 6.3.13 Proposition.

We now turn to the computation of H2(S( n)) for n > 2. We will compute all of H * S ( ( n ) ) for n = 2 below.

Let n > 2 For p = 2 , H 2 ( S ( n ) )is generated as a vector space by the elements (a) s Z , P n S n , S n , lnhl,j, ~ n h i , ]and , hl,ihl,jfori#j*1, whereh,,,h,,j=h,.,h,,i and h:,i # 0. ( b ) For p > 2, H 2 (S ( n)) is generated by the elements

6.3.14 Theorem.

Snhl.i, bl,i, g, = (hl,i, hl,t+I hl,i), 9

k, = (hl,t+lyhl,i+i hI.i), 7

and hl,ihl,jfor i # j * 1, where hl,ihl,J+ hl,jhl,i= 0. Both statements require a sequence of lemmas. We treat the case p 6.3.15 Lemma.

= 2 Erst.

Let p = 2 and n > 2.

( a ) H 1 ( L ( n , 2 ) )is generated by h,,;for i E Z / ( n ) . ( b ) H 2 ( L ( n ,2 ) ) is generated by the elements hl,ihl,,for i # j f 1, gi, k,, = (h,.i, hl,i+l,hl,i+2).The latter elements are represented by hIsih2,,, and hl.i+lh2,i, and hl.ih,i+l+ h2.ihl,i+2, respectively. + e3,ih,,ihl,i+3 = 0 and these are the only relations ( c ) e,,ihl.i+l= hl,ie3,i+l among the elements hl.ie,,j. Roo$ We use the SS of 6.3.9 with E2 = E ( hl,i,h2,i)and d2(h2,,)= hl,ihl,,+l. All three statements can be verified by inspection. 6.3.16 Lemma.

Let p

= 2,

n > 2, and 2 < k 5 2n.

( a ) HI(L( n, k ) ) is generated by the elements hl,i, along with &,for k 2 n and p,, for k = 2n. L( n, k ) ) is generated by products of elements in H I ( L( n, k ) ) ( b ) H2( subject to hl.ihl,i+l= 0, along with

237

3. The Cohornology of X(n)

gi = ( h 1 , i , h l , i , h l , t + l ) , ai

ki

= (hl,i,

= ( h l , i >h l , i + l , h l , i + 2 , hl,i+1),

h l , i + l , hl,i+l),

and

ek+l,i= ( h l , i , h l , i + l * * h l , i + k ) . The last two families of elements can be represented by h3,ihl,i+l+ h2,ih2.i+l and Ish s , i h k + l - s , i + , , respectively. = 0 and no other relations hold amongprod( c ) hl,,ek+l,i+l+ ek+l,ihl,i+l+k ucts of the ek+l,iwith elements of HI. 9

1

9

Proof: Again we use 6.3.9 and argue by induction on k, using 6.3.15 to start the induction. We have E2 = E ( h k i ) O H * ( L ( n ,k - 1)) with d2(hki)= eki. The existence of the ai follows from the relation e3,ihl,i+l= O in H ~ ( L 2)) ( ~ and , that of e k t 1 . i from hl,ieki+l+ ekihl,i+k= 0 in H ’ ( L ( ~ k, - 1)). The relation (c) for k < 2 is formal; it follows from a Massey product identity A1.4.6 or can be verified by direct calculation in the complex of 6.3.8. No combination of these products can be in the image of d2 for degree reasons. H 6.3.17.

Let p

=2

PnSn,Pnhl,i,Snhl,i,

and n > 2. Then H 2 (E , S ( n ) ) is generated by the elements h , , i h , , , f o r i f j ~ l , a i , a n d h : j bi,jfor = l ~ i ~ n , j ~ Z / ( n ) .

Proof: We use the modified first May SS of 6.3.5. We have m = 2n and H 2 ( L ( n ,m ) ) is given by 6.3.16. By easy direct computation one sees that d2(gi)= bl,ihl,i+land d2(ki)= hl,ibl,l+l.We will show that d2(e2n+l,i) = h1,ibn.i + hl,i+nbn,i-l* A(t2n+l)=C t,O t!;+l-j+bn+l,n-l modulo terms of lower filtration by 4.3.15. Then by 4.3.22

d ( bn+l,n- I 1 = t l 0 b,n + bn,n- I 0 tl modulo terms of lower filtration and the nontriviality of d2(e2n+l,i) follows.

Proof of 6.3.14(a). We now consider the second May SS [6.3.4(b)]. By 4.3.22 we have d2(bi,j) = hl,j+lbi-l,j+l + hl,i+jbi-l.j# 0 for i > 1. The remaining elements of H 2 E o S ( n ) survive either for degree reasons or by 6.3.12. For p > 2 we need an analogous sequence of lemmas. We leave the proofs to the reader. 6.3.18 Lemma.

Let n > 2 and p > 2.

H ’ ( L ( n ,2)) is generated by hl,i. , ~ hl,,hl,i+l= O ) . ( b ) H 2 ( L ( n ,2 ) ) is generated by the elements ~ J I ~ (with gi = h l , i h 2 , , 9 ki = h l , i + l h 2 , i and e3.i = hl.ih2,i+lh2,ihl.i+2. (a)

238

6. Morava Stabilizer Algebras

are hl,,el,r+l - e3,,hl,,+, = ( c ) The only relations among the elements hl,ze3,1

rn

0.

6.3.19 Lemma.

Let n > 2, p > 2, and 2 < k 5 m. Then

k r n, l,,. ( b ) H 2 ( L ( n ,k ) ) is generated by hl,lhl,l(with hl,lhl,l+l=O), g,, h,, ( a ) H ' ( L ( n , k ) ) is generated by h,,, and, for

ek+l,l

hl,hk+l-].t+]r

= O 2. Then H 2 (EoS( n ) ) is generated by the elements b,,, for i Im - n and by the elements of H 2 (L( n, m ) ) . rn Proof of 6.3.24(b). Again we look at the spectral sequence of 6.3.4(b). B y arguments similar to those for p = 2 one can show that d,(b,,)

= hl,z+]b,-l,]- hI,]+lb8-1]+1

for i > 1

and where

ds(ern+l,t) = h l . m + l + i - n b r n - - n , r - ~ - h l . , b , - " , j

s=l+ ~ n - ( ~ - l ) m ,

and the remaining elements of H 2 ( E o S ( n ) )survive as before. Now we will compute H * ( S ( n ) ) at all primes for n s: 2 and at p > 3 for n=3. 6.3.21 Theorem. ( a ) H*(S(l))= P ( h , o ) O ~ ( pf,o) r p = 2 ;

( b ) H * ( S ( 1 ) ) =E(hI0) f o r p > 2 [note that S ( 1) is commutatzue and that

L1= hlo].

Proox This follows immediately from 6.3.3, 6.3.5 and routine calculation. rn 6.3.22 Theorem. For p > 3, H * ( S ( 2 ) ) is the tensor product of E ( 1 2 ) with the subalgebra with basis ( 1 , h l o ,h , , , go, g l , gob,,} where gt = (hl,,, hl,,+l,h1.A h,ogl= gohi1 hiago= hligi = 0, 9

and hl ohll= h:,= h i , =O. In particular, the Poincare' series is ( 1+ t ) 2 (1 + t + t 2 ) .

239

3. The Cohomology of L(n)

Pro05 The computation of H*(L(2,2)) by 6.3.8 or 6.3.9 is elementary, and there are no algebra extension problems for the SSs of 6.3.9 or 6.3.4(b). We will now compute H*(S(2)) for p =3. First we need some notation. Let R = E ( h i o , h i i ) @ P ( b i o b11)/'1 ~ where I = ( h i o h i i , b:o+ b;i, hlobio-hiibii, hiibio+hiobii),

and define a class L E H2S(2) as a matric Massey product (A1.4.2)

That this class is well defined will become evident in the proof of the following result. 6.3.23 Theorem. For p = 3 , H*(S(2)) is isomorphic as an algebra to E ( 1 2 ,t ) @ R, where R and 5 are as defined above, and is as defined in 6.3.12. In particular, the Poincare' series is

c2

( 1 + t)'( 1 + t 2 ) / ( 1 - t ) ,

ProoJ Our basic tools are the spectral sequences of 6.3.10 and some Massey product identities from A.1.4. We have H*( i ( 2 , 1)) = E ( h 1 o ,h l l ) @ P ( b l ob, , , ) , and an SS converging to H * ( i ( 2 , 2 ) ) with E 2 = E(L2, 77)OH*(E(2, l ) ) , where c2= h20+h21 and 77 = hzl -h20, d2(L2)= 0, d 2 ( v )= h,,hl,, and E, = E,. Hence E, is a free module over E ( L 2 ) @ P ( b l Ob,l l )on generators 1, ho,h i ,go, g , , and h o g l = hogo2 where g, = @ I , , , h1,,+,, This determines the additive structure of H*( L(2,2)), but there are some nontrivial extensions in the multiplicative structure. We know by 6.3.13 that we can factor out E ( 5 , ) , and we can write b,,, as the Massey product -(h,.,, hi.,, hl,,). Then by A1.4.6 we have hl,,gi = 2 -bl,ihl,i+l,g = - b l , i g i + l , b l , i b l , i + These l. facts along with the usual h?,,= hlohIl = 0 determine H*(L(2,2)) as an algebra. Next we have a spectral sequence converging ,to H*(L(2,3)) = H * ( EOS(2)) (by 6.3.5) with E2= E(h30, h 3 1 ) @ H X ( L ( 22)), , d2(h3,i)= gi - b l , i + l ,and E3 = Em.The computation of E, is essenjially routine and there are no ambiguities in the algebra structure of H*(L(2,3)). The SS of 6.3.4(b) collapses by easy calculation, and the only ambiguity in the multiplicative structure of H*( S(2)) is the value of c in the expression t2= cb,ob,,. We will show below that c = 0. The computation of H * ( i ( 2 , 3 ) ) is clarified by the following construction. The subring of H * ( t ( 2 , 2 ) ) generated by b l o ,go, g , , and b,, can be mapped I

240

6. Morava Stabilizer Algebras

isomorphically to the subring of F3[so, sl] generated by -s& sis,, s,s:, and -s;, respectively. Multiplication of these elements in H*(i ( 2 , 2 ) ) by h i , , corresponds to multiplication of the corresponding polynomials by s,. At this point it is convenient to tensor with F, and perform a change of basis. Let x = xo+ is,, y = so- is,, where i 2 = -1. Then the elements x3, x2y, xv', and y 3 correspond to -blo+ibll, - b l o + i g o + g , - i b , l , -blo-ig,+g,+ib,,, and -bl0- ib,,,respectively. In the spectral sequence for H * ( L ( 2 , 3 ) 0F9L the elements d2(h3,+ i h 3 , ) and d2(h30- ih3,) correspond to ix'y and -iy2x, respectively, and the element 5 is represented by h ~ o h 3 0 + h 3 1 h 1 1Over . F,, 5 is the ordinary Massey product (u, u, u, u ) = (u, u, u, u ) , where u = h i o + i h , , and v = hI0- ih,,. By A1.4.5, 6 can be rewritten as

The appropriate change of coordinates in each matrix yields the expression indicated preceding the statement of the theorem. With these observations in mind it is easy to see that H * f ( 2 , 3 ) = H*EoS(2) has the indicated structure. It remains to be shown that t 2 = 0 in H*S(2). For degree reasons , . will construct a Hopf algebra T and a map we have ~ 2 = c b , o b , We f: T + S(2)@F9such that f"(6) = 0 and f " ( b l 0 b , , f) 0. Recall (6.2.3) that S(2)@Fo=F,[S2], where S2 is a certain compact 3-adic Lie group. S, is the group of proper (congruent to 1 modulo the maximal ideal) units of the noncommutative degree 4 extension E2 of Z3obtained by adjoining i and S with S i = -is, i 2 = -1, and S 2 = 3 . S2 has elements of order 3, e.g., -1/2+[(1+ i ) a / 4 ] S , so we have a map f:F,[Z/(3)]+ S(2)@F, and dually a map f*: S(2)0F, + F,[ z]/ ( z3- z), where z is primitive and f,( t , ) = ( 1 + i ) z . Hence f,( t , - i t : ) = 0, so f*(5) = 0. On the other hand, it is easy to check that f*(blob,,) # 0 , so c = 0. H We now turn to the case n = p = 2. We will only compute EoH*( S(2)), so there will be some ambiguity in the multiplicative structure of H*(S(2)). In order to state our result we need to define some classes. Recall (6.3.12) that H ' ( S ( 2 ) ) is the F,-vector space generated by hlo, h , , , l2and p2. Let a 0 E (52,

h a , h,,),

P E ( h a , 5 2 , 5;, h,),

g = ( h , h2, h, h2),

where h = h , o + h , , , ~ = ( x , h , h 2 ) f o r x = 5 2a, o , 5 : , a n d a o f 2 (more precise definitions of a. and P will be given in the proof). 6.3.24 Theorem. E o H * ( S ( 2 ) ) f o r p= 2 i s a f r e e m o d u l e o u e r P ( g ) @ E ( p , ) on 20 generators: 1, h10, h l l , h;o,J;l, P, P h o , P h , , , Ph:,, P h L Ph:,, C2, ao, C,; a052,f 2 , Go, a , ~ , , where ( Y ~ H'(S(2)) E and has Jiltrution degree 4, P E H3(S(2))and has Jiltration degree 8, g E H4(S(2)) and has

57,

G",

24 1

3. The Cohomology of z ( n )

Jiltration degree 8 , and the cohomological and Jiltration degrees of 2 exceed those of x by 2 and 4, respectively. Moreover h:, = h:, , a ; = f : , and all other products are zero. The Poincare' series is (1 + t ) ' ( 1- t 5 ) / (1 - t )'( 1 + t 2 ) .

ProoJ: We will use the same notation for corresponding classes in the various cohomology groups we will be considering along the way. Again our basic tool is 6.3.10. It follows from 6.3.5 that H*(E0S(2))is the cohomology of the complex P(hin,

h i i , 4'2,

h20)@E(h309h i

, ~

2

h40) ,

with d(h,.i)=d(4'2)=d(p2)=0, and This fact will enable us to solve the algebra extension problems in the spectral sequences of 6.3.10. For H*( L(2,2))we have a spectral sequence with E2= P(hlo, h l l , 4'2, h20) with d2(L2)= 0 and d2(h20)= h l o h l l . It follows easily that

H*(Q2,2)) = P ( h I 0 ,

hll, 52.

b*O)/~~IO~,~)

where b2n= G o = ( h i o , h,i,h,o, h i ) . For H*( i ( 2 , 3 ) ) we have a spectral sequence with E2=

E(h,o, h31)0H*(i(2,2))

and d2(h3,i)= h,&. Let *i

= hl,i+lk,i+ 4'2h2.i E (4’2

3

hl,i, hl,i+I).

Then H*(L"(2,3)) as a module over H*(L1(2,2)) is generated by 1, a o ,and a 1with 4'2h1,i

= lz(ao+a1 +

4’3= hl,i*i = 4'2hl,i+lat= 0

and a;= l:b20, a:=

5:(4':+ b20),

a o a i = 5%ao+ b2o)

The PoincarC series for H * ( i ( 2 , 3 ) ) is ( 1 1- t + t 2 ) / (1 - t ) 2 . For H*(i(2,4)) we have a spectral sequence with E , = E(h40, ~ 2 ) @ H * ( i ( 3)), 2, d 2 ( p 2 )= 0, and d2(h40) = ao+ a I . Define

p E H 3 ( i ( 2 ,4)) by

~ = h 4 0 ( ~ O + ~ 1 + 4 ' ~ ) + 4 ' 2 h 3 0 h 3 1 E ( h 1 04~'2 ,

4’22, h l l ) .

242

6. Morava Stabilizer Algebras

Then H * ( i ( 2 , 4 ) ) is a free module over E ( p 2 ) O P ( b 2 , on ) generators 1, hi,, , 5 2 , S:, (YO, ( ~ ~ P,5 and ~ . phi,, , where t>0. As a module over H * ( i ( 2 , 3 ) ) O E ( p 2 )it is generated by 1 and P, with ( ( ~ ~ + ( ~ ~ ) 1 = 5 : ( 1 ) = a0& 1) = 0. To solve the algebra extension problem we observe that /3c2= 0 for degree reasons; Pa, = P(12,A,,, h,,,,,) = (P, C2, ~ 1 , , ) ~ 1 , ~=+01 since (P, 12,h,,,) = 0 for degree reasons; and E ( p 2 ) splits off multiplicatively by the remarks at the beginning of the proof. This completes the computation of H*(E0S(2)). Its PoincarC series is ( 1 + tI2/(l - t ) 2 . We now use the second May S S [6.3.4(b)] to pass to E0H*(S(2)). H*(E0S(2)) is generated as an algebra by the elements hlo, h,,, 5 2 , p2, a o , b2,, and P. The first four of these are permanent cycles by 6.3.12. By direct computation in the cobar resolution we have

6.3.25

d ( t , + t l t 3 = c 2 8t , ,

so the Massey product for a0 is defined in H"(S(2)) and the permanent cycle. We also have

d ( r20t2+

t,@

t : t 2 + tl t 2 @ t : ) = t , @ t l @ tl

+ t i @ t:@

cyo

is a

t:,

so d2(b20) = hio+ h i , . Inspection of the E3 term shows that b:,= (h, h2, h, h 2 ) , (where h = hlo+ h , , ) is a permanent cycle for degree reasons. We now show that p = (h,,, L2, 5:. h l l ) is a permanent cycle by showing

that its Massey product expression is defined in Eo( H*( S(2)). The products h1052and are zero by 6.3.25 and we have

&,,

6.3.26

d ( F3 0?:

+ T2f30t: + T20t4+ T 2 0ti+ T20t:( 1+ t2 + t i ) )

= T20TZOT2,

where ?,= f 3 + f , t : and T 2 = t 2 + f : + f : , so l:=O in H*(S(2)). Inspection of H3(E0S(2))shows there are no elements of internal degree 2 or 4 and filtration degree >7, so the triple products (h,,, c2, 5;) and (12,l:,h l l ) must vanish and P is a permanent cycle. Now the E3 term is a free module over E ( p , ) @ P ( bzo) on 20 generators: 1, h l 0 , h l , , h:o, G I , h:o=h:,, P, PhIO, k%,, @ G I , m : o , ph:o, 5 2 , (Yo, 55, ( ~ ~&b2,,, 5 ~ aObZO, , Lib,,, 52aOb20. The last four in the list now have Massey product expressions (L2, h, h2), (ao,h, h2), (li,h, h'), and ( a o ,L,h, h2), respectively. These elements have to be permanent cycles for degree reasons, so E3 = Em, and we have determined E0H*S(2). We now describe an alternative method of computing H*(S(2)0F4), which is quicker than the previous one, but yields less information about the multiplicative structure. By 6.3.4, this group is isomorphic to H*( S2 ;F4), the continuous cohomology of certain 2-adic Lie group with trivial

243

3. The Cohomology of P{n)

coefficients in F4, S2 is the group of units in the degree 4 extension E2 of Z2 obtained by adjoining w and S with w z + w + 1 = 0, S 2 = 2, and Sw = w2S. Let Q denote the quaternion group, i.e., the multiplicative group (with 8 elements) of quaternionic integers of modulus 1. 6.3.27 Proposition.

There is a split short exact sequence of groups 1 + G-

6.3.28

i

S+

Q + 1.

The corresponding extension of dual group algebras over F4 is Q* .&S ( 2 ) ,

G,

where Q*=FF,[x,y]/(x4-x,y2-y) and G*=S(2),.(rl, t 2 + w t : ) as algebras where j , ( x ) = t l , j , ( y ) = or2+W 2 t : , and W is the residue class of w.

Proof: The splitting follows the theory of division algebras over local fields (Cassels and Frohlich [ l , pp. 137-1381) which implies that E 2 @ Q 2isomorphic to the 2-adic quaternions. We leave the remaining details to the reader. W 6.3.29.

( a ) H * ( Q ; F2) = P(hlO,hI, g ) / ( h ohll, h:o+h:,). ( b ) H * ( G ; F2) = E(52,P2, h30, h31)-

Proof: Part (a) is an easy calculation with the change-to-rings spectral sequence (A1.3.14) for F2[x]/(x4+x)+ Q*+F2[y]/(y2+y). For (b) the filtration of S(2), induces one on G,. It is easy to see that EoG, is cocommutative and the result follows with no difficulty. 1 6.3.30 Proposition. In the CESS f o r 6.3.28, E3 = E, and we get the same additive structure f o r H*S(2) as in 6.3.24. Proof: We can take H * ( G ) O H * ( Q ) as our El-term. Each term is a free module over E ( p z ) @ P ( g ) .We leave the evaluation of the differentials to the reader. 1 Finally, we consider the case n = 3 and p'5. We will not make any attempt to describe the multiplicative structure as it is quite tedious and of little interest. An explict basis of E ° F f * ( S ( 3 ) )will be given in the proof, from which the multiplication can be read off by the interested reader. It seems unlikely that there are any nontrivial multiplicative extensions. 6.3.31 Theorem For p 2 5 , H*S(3) has the following Poincare' series: ( 1 t)'( 1 t + 6 t 2 + 3 t 3 + 6 t 4 + t 5 + t 6 ) .

+

+

Proof: We use the spectral sequences of 6.3.9 to compute H*(L(3,2)) and ) iEZ/(3), H*(L(3,3)). For the former the E2-term is E ( h l , i ) O E ( h 2 . iwith

244

6. Morava Stabilizer Algebras

d 2 ( h l , i ) = 0 and d2(hl,i)= hi,ihi,i+,.The Poincart series for H * ( L ( 3 , 2 ) ) is (1 t)2(1 t)'( 1 t 5 t2+ t 3 + t*) and it is generated as a vector space by the following elements and t6eir Poincart duals: 1, hi,i, gi = hl.,h2,,, k, = h2,ihl,i+i, e3,i = hl,ih2i+l+ h,ihl,i+2 (where Zi e3,i= O), gihl,,+l= hi$, = hl,ih2,ih1,i+I7and h1,ie3.i 5 gihl,i+2 = hl.ih2,ihi.i+2. For H*(L(3,3)) we have E , = 1 5 ( h ~ , ~ ) @ H * ( L ( 3 , 2 with ) ) d2h3.8 = e3,i, so d2Z h3,i= 0. H * L ( 3 , 3 ) has the indicated PoincarC series and is a free module over E ( & ) , where 6, = Z h3,i, on the following 38 elements and the duals of their products with

+

+

++

c3:

1, hl,i, gi,

kr7 b,.i+2 = hl,ih3.i + h,ih2,i+2+ h3,ihl.t, hl,ih2,ihZ,i+l+ hl,ihl,r+ih3,ir

gihi,i+l = hi,iki, hl,,hz,ihz,i+z, hi,ih2,ih3,r,

hl,ih2,i+2h3,1+ly

C (hl.ih2,1+1- hl,i+ih2,i+2)hl,t, hi,,kih,.j I

(where hl.iki C j h3,j is divisible by hi,ih20h21h22.

c3), and

hl,i+2hl,ih2,i(h3,i+h,,,+,)*

4. The Odd Primary Kervaire Invariant Elements The object of this section is to apply the machinery above to show that E Ext2 (see 5.1.19) is not a permanent cycle for the ANSS element p > 2 and i > 0. This holds for the corresponding ASS element 6, (4.3.2) for p > 3 and i > 0; by 5.4.6 we know maps to b,. The latter corresponds to the secondary cohomology operation associated with the Adem relation P(p-l)p'Pp' =. . . . The analogous relation for p = 2 is Sq2'Sq2'= * . . , which leads to the element h : , which is related to the Kervaire invariant by Browder's theorem, hence the title of the section. To stress this analogy we will denote @ p ~ l pby ~ 0,. We know b y direct calculation (e.g., 4.4.20) that Oo is a permanent cycle corresponding to the first element in cokerJ. By Toda's theorem (4.4.22) we know 8, is not a permanent cycle; instead we have d2p-1(8,) = a,@: (up to nonzero scalar multiplication) and this is the first nontrivial differential in the ANSS. Our main result is @pg,pl

pptlpt

6.4.1 Odd Primary Kervaire Invariant Theorem. I n the ANSS for p > 2 d2p-1( O,+,) 3 a,%: mod ker %$ ( u p to nonzero scalar multiplication) where a, = p ( p ' - l ) / ( p - 1) a n d a,%Pis nonzero modulo this indeterminacy. 1 Our corresponding result about the ASS fails for p permanent cycle even though b, is not.

= 3,

where b2 is a

6.4.2 Theorem. In the ASS for p z 5 b, is not a permanent cycle for irl.

245

4. The Odd Primary Kenaire lnvariant Elements

From 6.4.1 we can derive the nonexistence of certain finite complexes which would be useful for constructing homotopy elements with Novikov filtration 2.

6.4.3 Theorem. There is no connective spectrum X such that B P , ( X ) = BP,/(P,

d ,uzp')

for i > O a n d p > 2 .

Proof: Using methods developed by Smith [ 11, one can show that such an X must be an 8-cell complex and that there must be cofibrations Y f , Y'-x, V(0) V(O)+ y, (iii) p J ' ( P - l ) v (0) V(O)+ Y ' ,

(i) (ii)

Z 2 P 7 p2-1) p-1)

pP’(

where V (0) is the mod ( p ) Moore spectrum, g and g' induce multiplication by up' in BP,( V(0)) = B P , / ( p ) , and f induces multiplication by uf in

BP,( Y )= BP,( Y ' )= B P , / ( p , u f ) .

-

V(0) and the maps g, g' certainly exist; e.g., Smith showed that there is a map f f

:c 2 ' p - l )

V(0)

V(0)

which includes multiplication by u l , Hence LY induces multiplication by u f ' , but it may not be the only map that does so. Hence we have to show that the existence off leads to a contradiction. Consider the composite s 2 p ~ ( p 2 - ~ ) j ,S ~ p ~ ( p ~ - y ~ f) y kf , ,s z + w ( p - ~ ) 1

where j is the inclusion of the bottom cell and k is the collapse onto the ~ be top cell. We will show that the resulting element in 7 ~ & a + l ( ~ - ~ ) - would detected in the Novikov spectral sequence by O,, thus contradicting 6.4.1. The cofibrations (ii) and (iii) induce the following short exact sequence of BP* modules 0 ZZP'( p-1) B P * / ( P ) B P * / ( P ) + BP*/(P, vP')+O, +

and the cofibration

so6so+V(0) induces 0- B P , ,

BP,+ B P , / ( p ) + O .

Hence we get connecting homomorphisms 8, : Exto(B P , / ( p, u p ' ) ) + Ext'(BP,/( p ) )

246

6. Morava Stabilizer Algebras

and 6,: Ext'(BP,/( p)) + Ext2(BP,). The element f j n 2~p y p 2 - I ) ( Y') is detected by U;'E Exto(BP,/(p, u p ' ) ) . We know (5.1.19)that = 6,6,( uf’) E Ext2(BP,)

H

detects the element kfje

The statement in 6.4.1 that a,B; is nonzero modulo the indeterminacy is a corollary of the following result, which relies heavily on the results of the previous three sections. 6.4.4 Detection Theorem. In the ANSS E,-term for p > 2 let 6' be a monomial in the 6,. Then each 6' and a,6 is nontrivial. H

We are not asserting that these monomials are linearly independent, which indeed they are not. Certain relations among them will be used below to prove 6.4.1.Assuming 6.4.4,we have

Proofof6.4.l. We begin with a computation in Ext(BP,/(p)). We use the symbol Oi to denote the mod p reduction of the 6, defined above in Ext( BP,). We also let h, denote the element -[ tf']. In the cobar construction we have

so 6.4.5

ul6,=-h0h,.

May [5] developed a general theory of Steenrod operations which is applicable to this Ext group (see A1.5). His operations are similar to the classical ones in ordinary cohomology, except for the fact that Po# 1 . Rather wehavePo(hl)=h,+, a ndPo(@ ,)=el+,. WealsohavepP'(h,)= 6,,pPo(6,)= 0, pPo(uv,) = 0, PI( 6 , )= 6 ; and the Cartan formula implies that Pp'(6 ; ' ) = 6;'". Applying pP" to (6.4.5)gives 6.4.6

O=Ooh2-h16,.

If we apply the operation Pp'-'Pp'-2 . . . PI to (6.4.5)we get 6.4.7

hl+,ep'= hZ+,e:'.

Now associated with the short exact sequence O+BP,- P B P , + B P , / ( p ) + O there is a connecting homomorphism 6: Ext"*( B P , / ( p ) ) + Ext"+'**(BP,)

241

4. The Odd Primary Kervaire Invariant Elements

with 6(h,+,)= 8,. Applying 6 to 6.4.7 gives 6.4.8

E EXtgp*gp*(BP,,

8,8:' =

Bp,).

We can now prove the theorem by induction on i, using 4.4.22 to start the induction. We have for i > 0

4,- 1(

Of+

1

) 0;'

=

4,- L ( 4 f l 0,"')

=

4-,(~'m

= d,,-,(e,)ep'

= ho0~-,f3f"

mod ker

8gt-l

= ho(ef-lep'-')p = ha( e,e:'-')" =hOe:8:' so

d2p-l(8,+l) = hoe:

mod ker

192.

We now turn to the proof of 6.4.4. We map Ext(BP,) to Ext(v,'BP,/I,,) with n = p - 1. By 6.1.1 this group is isomorphic to Extz(,,,(K(n),, K ( n ) , ) , which is essentially the cohomology of the profinite group S,, by 6.2.4. By 6.2.12 S,, has a subgroup of order p since the field K obtained by adjoining pth roots of unity to Q, has degree p - 1. We will show that the elements of 6.4.4 have nontrivial images under the resulting map to the cohomology of Z / ( p ) . In other words, we will consider the composite

BP, ( BP ) + X( n ) + S( n ) @ Fpn + C, where C is the linear dual of the group ring F,n[Z/(p)]. 6.4.9 Lemma.

Let C be as above. As a Hopf algebra

C =F,~[t]/(t" -t)

with

A t = t 0 1 +lot.

Proof: As a Hopf algebra we have F,n[Z/( p ) ] = F,n[u]/(uP - 1 ) with Au = u @ u , where u corresponds to a generator of the group Z/(p). We define an element t E C by its Kronecker pairing ( u ' , t ) = i. Since the product in C is dual to the coproduct in the group algebra, we have ( u ' , t k ) = ( A ( u l ) ,t O f k - l ) = ( u 't,) ( u', t k - ' )

so by induction on k 6.4.10

We also have ( u ' , 1 ) = 1.

( u ' , t k )= i k .

248

6. Morava Stabilizer Algebras

We show that (1, f, t 2 , . . . , t p - ' } is a basis for C by relating it to the dual basis of the group algebra. Define x, E C by

C (it)k

x,=

OO

with e, E W(Fpm) and ern= e,. Recalling that Sp-' l=xP=l+pe,S+(e,S)P

and ( e , S )P =- e l( P ' - l ) / ( P - l ) S P

so it follows that +e p - l ) / ( P - l )

-=0

=p,

we compute

mod(S)'Cp mod (S)'+P

mod(p).

249

4. The Odd Primary Kervaire Invariant Elements

[Remember that tl(x) is the mod ( p ) reduction of e l . ] Clearly, one solution to this equation is el = 0 mod ( p ) and hence e l = 0. We exclude this possibility by showing that it implies that x = 1. Suppose inductively that e, = 0 for i < k. Then X 1-k ekSkmod (Skf')and X' 1 +pekSk mod (sk'p)SO f?k 0 mod ( p ) . Since e [ "- e, = 0, this implies ek = 0. Hence, f is a map of Hopf algebras, f(t l ) primitive, so f ( t,) = ct where CE is nonzero. Now recall that Fptj

Extc (F,., F,. ) = H*(Z/ ( p ) ; Fpn)= E ( h )0P ( b ) , where E ( ) and P ( ) denote exterior and polynomial algebras over F,., respectively, h = [ t] E H I , and

Let f

* denote the composition

Ext( BP,)

+ Ext( v,'BP,/Z,,)

-

5ExtZ(,,,(K(n),, K(n))+Extc(F,n, F p n )

5 H * ( Z / ( p ) ; Fpm).

Then it follows that f * ( h o )= -ch and f * ( b , )= - c P ' + ' band 6.4.4 is proved. Note that the scalar c must satisfy 1 + C ' ~ " - ~ ) / ' ~ - I = ) 0. Since cpp-'-I = 1, the equation is equivalent to 1 + = 0. It follows that c = w ' ~ - ' ) ' ~ for some generator w so Fir-!, so c is not contained in any proper subfield of F,P-~.Hence tensoring with this field is essential to the construction of the detecting map f: Now we examine the corresponding situation in the ASS. The relations used to prove 6.4.1 (apart from the assertion of nontriviality) are also valid here, but the machinery used to prove 6.4.4 is, of course, not available. Indeed the monomials vanish in some cases. The following result was first proved by May [l]. 6.4.11 Proposition. F o r p = 3 , hob: = O in ExtA,(Z/(3),Z/(3)); i.e., b2 cannot support the expected nontrivial differential.

ProoJ: We use a certain Massey product i#entity (A1.4.6) and very simple facts about ExtA,(Z/(3), Z/(3)) to show hob:=O. We have hob: = -ho(h,

By (6.4.7) h , b l = hzbo,

2

hl

,h 1 N l = -(ho, h,

9

h1)hIbl.

SO

hob:=-(ho, h i , hilhzbo=-(hi, ho, hi)hzbo=-hi(ho,

hi,

hJbo.

The element (ho, h, , h,) is represented in the cobar construction by 6: I c,, which is the coboundary of &, so hob: = 0.

,$:I&+

250

6. Morava Stabilizer Algebras

The case of b2 at p = 3 is rather peculiar. One can show in the ANSS that d5(P7)= f c t & / 3 . (This follows from the facts that dS(P4)= fa1P:/33,3r P: = * P # 7 , P ~ = *P~ 1 ~~6 / 3and ~, /3$3 ~ = * & p a / 3 . We leave the details to the reader.) Hence P9/9fP7 is a permanent cycle mapping to b,. The elements P7 and ( ~ I p : / 3= f a & P 6 / 3 correspond to ASS elements in filtrations 7 and 9 which are linked by a differential. We do not know the fate of the b, at p = 3 for i > 2. To prove 6.4.2 we will need two lemmas. 6.4.12 Lemma.

For p 2 3

+

>, (i) Ext27"'+2(BP,) is generated by the [( i 3)/2] elements Pn,,Jlp4+~ where j = 1,2,. ..,[(i+3)/2], ak,=(p'+2+p'+3-2J)/(p+1), and [(i+3)/2] is the largest integer s(i + 3)/2. Each of these elements has order p. (ii) Each of these elements except / 3 p ~ + 1 , p * + ~ reduces to zero in

Ext2*qp"2( BP*/ Z3). 6.4.13 Lemma. For p 2 5 , any element of Ext29qp'i2( BP,) (for i z 0) which maps to b,+l in the Adams E,-term supports a nontrivial diflerential d2p-1.

We have seen above that 6.4.13 is false for p = 3. Theorem 6.4.2 follows immediately from 6.4.1 3 because a permanent cycle in the ASS of filtration 2 must correspond to one in the ANSS of filtration 5 2 . By sparseness (4.4.2) the Novikov filtration must also be 2, but 6.4.13 says that no elment in Ext2(BP,) mapping to b, for i 2 1 can be a permanent cycle.

fioofof6.4.12. Part (i) can be read off from the description of ExtZ.*(BP*) given in 5.4.5. To prove (ii) we recall the definition of the elements in question. We have short exact sequences of BP,( BP)-comodules 6.4.14

0 -,BP,

6.4.15

O - , BP,/(p) o

-,BP,

-f

BP,/( p ) + 0. C B P , / ( ~-),BP ,/(~ , u y ' + J - 2 ~ ) - , ~ .

Let So and Sl denote the respective connecting homomorphisms. Then we E Extip+BP( BP,, BP,/( p, u : ' + ~ - ~ ' ) ) and ~ , , , J ~ ' + ~ - ~ =J 6&( u ~ l . ~The ) . have (i.e., the above element for j = 1) can be shown to be b,,, element as follows. The right unit formula 4.3.21 gives U

~

.

J

p p l + l / p l + l

6.4.16

vR(v 2 )= u,+ S , ( U 2 p ' + ' ) = t:

r+2

v1r; - o f t , -

upl+2-p'+l

mod ( p ) , tP'+l

1

25 1

4. The Odd Primary Kervaire lnvariaot Elements

and

sO(tf'") = b,+l. Moreover 6.4.16 implies that in Ext(BP,/(p)), rpJ, so u1 P Up'tfJ+l -u, t]. ,+I

p'+2-

t p i + L

pIf2-1

This element is the mod ( p ) reduction of p-'-260(vf'+z)and is therefore in ker a0. Hence 6,S,(uf'=I)= 6,( t =b,+l. This definition of differs from that of 5.4.5, where for i > 0 it is 2+ ] p'-l defined to be 8 0 8 1 ( o ~ 2 - u P 2 ~ ' u ~1 . In principle one can compute this element explicitly in the cobar complex (A1.2.11) and reduce mod 13,but that would be very messy. A much easier method can be devised using Yoneda's interpretation of elements in Ext groups as equivalence classes of exact sequences (see, for example, chapter IV of Hilton and Stammbach [l]) as in 5.1.20(b). Consider the following diagram. PI+')

p p i + ~ / p ~ + l

6.4.17

o --,

BP,

.P'+l

P BP,

"

,B P , / ( p )

P2

BP,/(p,

-

0

The top row is obtained by splicing 6.4.14 and 6.4.15 and it corresponds to an element in Ext2(BP,/( p , vf'ti-z'), BP,). Composing this element with

u2.1E Exto(BP,/( p , ~ f ' + ' - ~ ~ ) ) gives p a Z , , / p l + 3 - 2 ~ . We let p , be the standard surjection. It follows from Yoneda's result that if we choose BP,BP-comodules MI and M 2 . and comodule maps p 2 and p 3 such that the diagram commutes and the bottom row is exact, then the latter will determine the element of ExtLJ*BP(BP*/(p,vr1+3-2J), BP,/(P, 4, 0,)) which, when composed with ~ 9 . 1will , give the mod I3 reduction of P a , . J l p ' + 3 - 2 i . We choose MI = BP,/( p 2 , pol, u:, pv,) and M2 = BP,/( p, and let p 2 and p 3 be the standard surjections. It is easy to check that M , and M2 are comodules over BP,(BP), i.e., that the corresponding ideals in BP, are invariant. (The ideal used to define MI is simply I : + llZ3.) Moreover, the resulting diagram has the desired properties. The resulting bottom row of 6.4.17 is the splice of the two following short exact sequences. 6.4.18 0

+

BP,/ (P,u1, v2)

P

BP,/ ( P 2 , P V I

9

PU2 9

u:,

+

BP,/( P, d , 0, +

252

6. Morava Stabilizer Algebras

6.4.19 P"'-."

o-+ BP,/(P, 4)-L-+ B f , / ( p , u : + ~ ' + ' - ~ ' ) + BP,/

( p , u ~ ' + " ~ ' )-0.

Let Sh, 8: denote the corresponding connecting homomorphisms. The elements we are interested in then are S A S ~ ( U $ . J ) . To compute S ; ( u f n . ~we ) use the formula d ( v:) = ( uz+ u l u l f ~ -uptl)" - u i , implied by 6.4.16, in the cobar construction for El’,/( p , ~ : f ~ ' + ' - ~).' Recall that a , j = ( Pi + 2 + p i + 3 - 2 J ) /p( + 1) 1Pj 5 [ ( i + 3 ) / 2 ] . Hence a J,J.= pi+3-2jmod ( p1+4--2')and d ( vg8.1)= pz+3-Z~

a;(u;i.i)

where bi,j = For j = 1, bi.l = 0 and

= u,b1~[t,

fpi+4-2J], so

~ : Z . J Z ) ~ ' + ' - ~ ' [

I,

( ~ ~ + ~ - p ~ + ~ - ~1).J ) / ( p +

s ; S ; ( V g ~ . l )= -

c Ock 1, bi,Jis divisible by p and d (

t?'

U $ J )

T ( I P - ~ ) P ' ]=

-b., + I ,

= 0 mod ( p 2 , pul, u:) and

v,bd.jd(tP""-") = 0 mod ( pu,),

so S : V $ J E Ext'(BP,/(p, u : ) ) pulls back in 6.4.17 to an element of Ext'( Bf,/( p2, pul, pu,, u 2 ) )

and

S~S:(U$.J)= 0,

completing the proof. Proof of 6.4.13. Any element of Ext2*qP'+Z( BP,) can be written uniquely as cbicl+ x where x is in the subgroup generated by the elements for j > 1. In 5.4.6, we showed that x maps to zero in the classical Adams E,-term. Hence it suffices to show that no such x can have the property /3a,.J,pl+3-?~

d,p-t(x)

=dZp-l(bi+l).

By 5.5.2 for p'5 there is an 8-cell spectrum V(2) = M ( p , u l , u2) with BP,( V ( 2 ) )= BP,/( p , uI, u 2 ) , and a map f : So+ V(2) inducing a surjection in B P homology. f also induces the standard map

f, : Ext( BP,)

+ Ext( BP,/

13).

Lemma 6.4.12 asserts that f , ( / 3 a , . , , p + - 2 J ) = O for j > 1, so f,(d2p-l(x)) =O where x is as above. However, 6.4.1 and the proof of 6.4.4 show that 9 * ( d 2 p - l ( b , + l )#) 0,

where g, is induced by the obvious map g: BP,+ U ~ ! ~ B P , / I ~ - , .

253

5. The Spectra T(n)

Since g factors through BP,/Z3, this shows that f , ( d 2 p - l ( b i + l # ) )0, completing the proof.

5. The Spectra T(n) In this section will we construct certain spectra T ( n ) and study the corresponding CSS. T ( n ) satisfies

B P , ( T ( n ) ) = BP,[t,, t 2 , . . . , t , l c B P , ( B P , ) as a comodule algebra. These are used in Chapter 7 in a computation of the ANSS E,-term. We will see there that the ANSS for T ( n ) is easy to compute through a range of dimensions that grows rapidly with n, and here we will show that its CSS is very regular. To construct the T ( n ) recall that BU = SlSU by Bott periodicity, so we have maps R S U ( m )-+ BU for each m. Let X ( m ) be the Thom spectrum of the corresponding vector bundle over nSU( m ) .An easy calculation shows that H , ( X ( m ) ) = Z[bl, b,, . . . , b,,-,] = H,( M U ) . Our first result is

,

6.5.1 Splitting Theorem. For any prime p, X ( m ) ( is equivalent to a wedge of suspensions of T ( n ) with n chosen so thatp" Im 0.

7.1.6 Definition.

if

We will see below that a wisely chosen complex C n , k can streamline the calculation considerably. To recognize such a complex we need to know more about Ext over T ( n + k + 1). Our next goal is to compute Ext'. The following generalization of the Morava-Landweber theorem (4.3.2) is straightforward. 7.1.7 Proposition.

Ext&,,,(BP,, B P , / I m ) = A ( n + m - l ) / l , .

W

260

7. Computing Stable Homotopy Groups with the ANSS

Combining 6.5.11 and 7.1.3 we get 7.1.8 Theorem.

Unless n = 1 and p = 2 (which is handled in 5.2.6) Exti(,,(BP,, BP,) is t h e A ( n - 1)-modulegenerated bytheelementsa(u',/ip) f o r i > 0 where a is the connecting homomorphism for the SES

O-+BP,+Mo-+N'+O. The following result, combined with 7.1.7 and 7.1.8, gives Ext through a modest range. Unless p t s [ u : ( = 2 ( p n + -' f ) .

7.1.9 Theorem.

=2

and n = 1, Ext:;,, )( BP, , BP,)

= Ofor s 2 2

and

Proof: Let D: = BP,. p - ' u , : i 2 n ] and consider the SES O+BP,+ D:+ D:/BP,+o. We will show that 0: is a weak injective (7.1.6) with Exto= A ( n - l ) , and that D:/ BP, is a weak injective in the indicated range of dimensions. The result will then follow from 7.1.4 (with k =a) or A1.3.2. To analyze we filter it by powers of p . We have

1 L + l , . .. I

W ( P )= A ( n - l ) l ( P ) r f - ' ~ , ,. =A ( n- I)/( p = U n ) / ( p ,V",

",, %+I,.

'

. . ).

This is clearly a weak injective with Exto= A( n - 1)/( p ) , so the claim about D: follows. In our range of dimensions, D",/P, is the submodule of N' generated by u t l ip' for 0 < i < p . We have

Again, filtering by powers of p gives subquotients isomorphic in this range 1 to D : / ( p ) , so the result follows.

IU:,~~.

Now we will extend our range to We will construct a complex C,, of G(n)-comodules as in 7.1.4 satisfying 7.1.10

(a) C', is weak injective with Exto= Exti-(,,,(BP,, BP,) for i = O , 1; (b) C', = 0 for i 2 3 ; and (c) the differential induces a trivial map in Ext so the SS of 7.1.4 collapses from E , giving

Ext&,,,(A(n),C',) = ExtFT:,(BP,,

BP,).

26 1

1. The Method of Infinite Descent

These properties of C, are valid only in dimensions less than luO.

267

2. The Complex C,.,

Then Ci is a weak injective over G ( l ) with

Ext&i)(A(l), C : )= Exti!yi)(BP*, BP,). ProoJ We have to show that C : is indeed a G(1)-comodule with the appropriate Exto. Then the result will follow from 7.2.3 once we have = computed its PoincarC series. Consider the G( 1)-comodule 6’ BP, Or(z)M' where M' is the chromatic comodule of 5.1.5. Each element of C : is primitive over r ( 2 ) so C : is an A(1)-submodule of GI. To see that it is a subcomodule let = im d , c C : be the A(1)-submodule

ei

Vf

generated by 7for i > 0, and let P and

so C : is a subcomodule of

c:= Ci/c;. Then c: is clearly a comodule

GI. Now

Ext&)(A(l), C : )c Ext'&I)(A(l),G')= E X ~ & ~ ) ( B,PM, I ) . This group is known by 5.2.6 to be isomorphic to Ext:(,)(BP,, BP,) in positive dimensions. Since every element of this group is in C : (7.1.8) we have the desired isomorphism. is x/ ( 1 - x)' while that of is readily seen The PoincarC series for to be

c:

XP

(l-x)(l-xP)

xp3 + ( l - x ) (XP2l - x P ) +-(1-x)

so C: has the series required by 7.2.3.

2

W

Now the SS we are studying, i.e., that converging to Extr(')(BP,, BP,) with El = E x ~ ~ ( ~ ) ( Cl,,), A ( ~ must ) , have E:*' = Ext&Z)(BP,,BP,), the A(1)module generated by v i / ip for i > 0. We can conclude 7.2.7 Proposition. IfC1,, is as above then the image of E ~ f g ( ~ ) ( A (C2 :) , , ) v; in Ext'&2,0)(A(2),C:,,)is the A(l)-rnoduZe generated by 7for i > O . IPV 1

Moreover, there is an exact sequence

Ext&z)(A(2), C:,I)+ ExtO,(,)(A(2),C : , , ) ExtO,(z,(A(2),C:) += 0 +

and an isomorphism Ext&,,)(A(2),C : , *=Ext&(A(2), ) where C : is as in 7.1.12.

C:) for all s > 0,

268

7. Computing Stable Homotopy Groups with the ANSS

We wish to compare this result with 7.2.5 by computing the Poincart series for this group. The elements of order p m are spanned by

and the corresponding series is xPm(

1- X P m - ' )

(1 - x P m ) ( 1 - xP"'+P'"-' Multiplying this by

-x)*

1 for rn = 1,2,3 gives the first three terms in 7.2.5. (1 - X'+P)

3. The Homotopy Groups of a Complex with p Cells In this section we will be limited to a lower range of dimensions than before; i.e., we will be computing certain Ext'.' for t < p 3 q , where q = 2p - 2 and p > 2. These restrictions will be in force throughout this section. In 7.1.13 we found Extrc2, in this range [in fact for t < ( p 3 + p 2 + p ) q ] . From 7.1.3 we know this is isomorphic to EXtgp*(BP)( BP,, BP,[ t , ] ) . Moreover BP,[ t l ] = BP,( T ( 1)) by 6.5.1. Alternatively, T (1 ) can be constructed directly as follows. Let Sy + BU be a map corresponding to a generator of .nq(BU)= Z. Use the loop space structure of BU to extend this map to asq+', thereby defining a stable vector bundle over the latter. The resulting Thom spectrum, when localized at p , is T(1). This construction has one shortcoming, namely that f l S q + ' , unlike RSU(p), is not a double loop space so it is not obvious that the corresponding Thom spectrum is homotopy commutative; it clearly is an associative ring spectrum. Corollary 7.1.3 shows that Ext&=O for t < p 3 q and s > 2 p - 2 , so the ANSS for T ( l ) collapses without extensions by sparseness (4.4.2) and we can read off .n,( T ( 1)) in this range of dimensions. The objectof this section is to compute .nR*(X) in the same range, where X = T (1)'p-')4. To do this we will use the complex C,,' constructed in the previous section. Since C:,' and Cf,l are weak injectives, we have Ext&,,,,,(A(2),C : , , )= Ext"B+P:(BP,(BP,, BP,) in our range, so it suffices to analyze the left-hand Ext group. To get at it we use the Hopf algebroid extension (A1.1.15) G(1)-+ G(1, I ) + G(2),

which leads to

3. The Homotopy Groups of a Complex with p Cells

269

7.3.1 Proposition. There is a CESS (A1.3.14) converging to ExtG(',')(A(2),M ) with E>"= Ext&,)(A(l)),E ~ t k ( ~ ) ( A ( M 2 ) ,) for any G(1, I)-comodule M. W The group E X ~ ~ ( ~ , ( C:,,), A ( ~ )which , we denote by E, is described in 7.2.7, which says there is an SES of bigraded G(1)-comodules

7.3.2

O+ B + E + F+O,

where B = i m ExtGc2,(A(2),C : , , ) is the A(1)-module generated by { u;/ ipu; : i > 0 ) concentrated in degree 0 (note that for i 2 p 2 these elements are out of our range) and F = E x ~ ~ ( ~ ) ( C:) A ( ~as) given , by 7.1.10 and 7.1.13, which says F o = E ~ t g ( ~ , ( A (C2 :) ), is the Z/(p)-vector space generated by { u;u:/pu, : i 2 0, j > 0} and

7.3.3

F = Fo@ E ( h,,)@ P ( b,o)

7.3.4 Definition. For a right BP,(BP)-comodule M let r, : M + Z'¶M be the group homomorphism deJined by +( rn)= Zr,( m ) @t i . . , where the other terms involve tk for k > 1. These operations are similarly defned for G(1)and G(1, 1)-comodules. Ler X " c G(1), Y" c G(1, l ) , and T" c BP,(BP) denote the submodule generated by { t i : 0 5 i 5 n}. A G( 1)-comodule M is kTfree if the comodule tensor product M @ X P k - ' is a weak G ( l ) injective. Our ultimate goal in this section is to compute Extgp*(Bp)( BP,, T P - ' )for which we can use the SS associated with the complex C,,,O Y p p 'The . main difficulty here is computing ExtG(,,,,(A(2),C:,,O Y p - ' ) which , we do in C:,,O Y p 2 - ' )using a CESS two stages. First we compute ExtG11,11(A(2), with &-term ExtG(,)(A(l),E @ X P- I ) , then we use another SS (to be described below) to get the Ext group for C&O Yp-' from that for C:,,O yp2-1

T O compute E X ~ ~ ( ' ) ( A (X I )P, ' - ' O E ) we will construct an SES of bigraded G ( 1)-comodules

7.3.5

O+E+E+E+O

with E and ?!J 2-free (7.3.4). Hence the above Ext group will vanish in all degrees and the CESS will collapse.

7.3.6 Lemma. Let M be a G( 1)-comodule and let L c M be the subcomodule r),,p2 ker r,. Then Ext&l)(A(l),MC3Xp'-') is isomorphic to L as a group. ProoJ: Let m E L. Then one easily sees that Z o c l c p 2 r , ( m ) Ot l E M O X P 2 - ' is primitive. Now we show that any primitive x E M 0Xp2-' must have this form. Write x = X o S I S p k x , O t : and assume inductively that x, = ( - l ) ' r n ( x o )

270

7. Computing Stable Homotopy Groups with the ANSS

for i < m. Then we have

Collecting terms where the exponent of t , is zero gives

which gives x , = ( - l ) i r m ( x o ) . NOWwe will construct the SES 7.3.5. We obtain

so we have ?I =

from E by adjoining

{usi;o }

< i <j 0 E ( h,o)B P ( b20).Now we have a commuPVl u2 tative diagram with exact rows and columns 0 0

B-

0-

E-

0-B-E-F-0

F-

0

1 1

E=E

I 1 0

0

where fi is the evident extension of F by l?. We will show that B, and hence are 2-free using 7.3.6 and the 7.2.3. For B we have 0 if p t j = '2 i - j ' if j = p j f

fi, E,

[(r,

'J(%)

iPU 1

p 1pu;-J'

so by 7.3.7 Ext&,,(A( l ) , B O X p 2 - ' )is isomorphic to the kernel of multiplication by u f . To compute the PoincarC series for this group, note that the

271

3. The Homotopy Groups of a Complex with p Cells

subgroup of exponent p is generated by

which has series (1 - X P 2 ) (1 -x”) ’

xp

(1 - X P + ‘ )

the elements of order p2 are spanned by

so the corresponding series is

-X P )

XP2( 1

(1 -x ) ( 1

~

,P+P2).

Hence B O X P 2 - ’will be a weak injective by 7.2.3 if we can show that its series is ( 1 - X I - ’ times the sum of the two above. But from 7.2.5 we know the series for B is ( 1 -x”)( 1 - X I + P ) ’ ( 1 - x ) ( 1 - X P 2 ) ( 1 - X P + P Z )

while that for X P 2 - lis

(1- x P 2 ) ~

u-4

so the result follows.

Next we deal with = FoO E ( A,)@ P(b,,), where Po is the vector space xP+P2 spanned by ,,yV;’+J : i, j z 0 . Its series is ForO O ,

1 si s p ;

E Y by PJ 2

h , , T for

P

i 2 E;O= b,,E;-'-'. Next observe that Ypp' is a finitely generated A(2)-module with Horn,(,,( Y P - 'A(2)) , = X(l-p'q Yp-' as G(1, 1)-comodules. It follows that EXtG(l,I)(A(2),Yppl@c;,,) = EXtG(I,l)(Z(lpP)q YP-',c;,,),which iS a module over the algebra ExtG(l,l,( Y p - ' ,Y p - ' ) .This contains the elements

274

7. Computing Stable Homotopy Groups with the ANSS

h,, and b,, represented by the exact sequences

o-,

and

o-,

yp-I

~

yP-l+

y2P-I+xPqyP-I+o

yPZ-l&

I P Z 4 yp2-1+

XP24 y p - 1

j

0

respectively. Hence we have 7.3.10 Proposition. If x E E;" is a permanent cycle in the SS of 7.3.8 then so are the elements bl,x and h,, b i , x for all i 2 0. Moreover multiplication by h,, and b,, commutes with diflerentials.

Now we can state our main computational result. 7.3.11 Theorem. I n the SS of 7.3.8, the following diflerentials (along with those implied by 7.3.10) occur up to a nonzero scalar. All other diflerentials are trivial.

( a ) ddY,b;O) = nY,h,lb,,b;,'. ( b, d3(Ylh20b;0)= nYhZ0hl lbl lb;;

( c ) d3(u,bm)= ( n + i + l)u,h,,bllb:il for n 2 1. ( d ) ~ p - 2 , - 3 ( u p l , h , , b ~ ~ 2=- Jbpp'-' ) Pp,+p~p-, for 0 5 J 5 p - 3 . ( e ) d2(h,,u,)=(i+l)b,,P,+2 and d3(hzob:ou,)= ( i +j + l ) h l ,b,, h20b&lu, for j 2 1. (f) ~ 2 p - 2 ~ - 2 ~ ~ p r + ~ ~ l l ~ Z O ~ ~ ~ ~ ~h 1~1 bP-'-' ~ 1~ 1 ~ p ~ + for p / ~O p -s ~j -s l p-2. rn

Now we assumefor simplicity that p

= 5, the

generalization to an arbitrary

odd prime being straightforward. We digress to describe a simpler SS which serves as a paradigm for the one in question. Let P ( 1 ) = Z/(5)[ t , , t 2 ] / ( tY5, t : ) with the evident Hopf algebra structure; it is isomorphic to the dual of the algebra generated by the Steenrod reduced powers PI and P'. Let p" be the Z/(S)-vector space spanned by { t ; : 0 5 i 5 n } for n < 25. We have

v24)

E(h20)@P(b20) EXtp(i)(Z/(S), = E x t ~ ( i ) ( z / ( 5 z/(5)) ), where Q(1) = Z / ( 5 ) [ t z ] / [t : ] . The LES

oj

v4-,

v24&x5qY24&x25qp24+.

..

leads to an SS converging to Extp(,)(Z/(5),p4)with E2= E(hl,, h2o)OP(b,,, b20) with h,,E E?, b l l E E:.', hzo€E2', bzoE E Y , a n d d,: E?"+ Es+rvu-r+l . This SS is analogous to that of 7.3.8. 7.3.12 Theorem.

The diflerentials in the above SS are as follows:

( a ) d d b i ~=) ihllbl1b&i1; ( 6 ) d,(h,,b:b+4) = b:,b:b

, 2 )

275

3. The Homotopy Groups of a Complex with p Cells

These differentials commute with multiplication by hzo, h , , and b,, , and all other differentials are triuial. Consequently Extp(l)(Z/(5),Y 4 )is a free module over P(b:,)@E(h,,) on { b f , :O ~ i ~ 4 ) u { h I , bO~s i,1:3 ) with theproduct o f h , , and hllbzo being b;,. H Intuitively, the elements hllbi0 are Massey products (h,,, . . . ,h l l , b:,) with i + 1 factors of h,, . Hence the multiplicative extension h , , * h,, b:, = b;, follows from a Massey product identity. hll(hl1, h ' , , hll, h,, b:,)=(h,,, h ' , , hll, h

l , h11P;l.

Since F4 is not a comodule algebra, Extp(,)(Z/(5),F4)is not a ring, but we can get products by pulling back to ExtP(,)( F4,F4),which has a Yoneda product. We prefer to think of Massey products such as h,,b2, in the following way. b2,e E ~ t i ( , ~ ( Z / ( 5YZ4) ) , pulls back to an element in Ext&,,(Z/(5), F5), and h,, E ExtP'"(Z/(5), F4) pulls back to Ext;,,,( Y', Y 4 ) .The choices made in the pullbacks are equivalent to the indeterminacy in the Massey product, which in this case is trivial. Now let R"' = Ext>t;,,(Z/(S),F4) as described above. Theorem 7.3.11 implies that ExtG(,,,)(A(2),C : , , @Y 4 ) ,after filtering to get rid of the elements of order 25, can be described as a direct sum of variously displaced copies of R, more precisely 7.3.13 Corollary. There is a (nonsplit) SES

O+J+Ext(C:,,@ Y P - ' ) + K + O where J is the Z/( p)-uector space generated by { p i: i = 0 , l mod ( p ) ) , K = K , 0 K 2 where Kl = R 0{ yf: t 2 2 ) and K = Rs+2ic2T'+qi(p2-1) . Let i = pj + k with 0 5 k 5 p - 1. Then the ith copy of R contains Pitz and ppj+p/p-k for 0 s k s p -3, f i p j + p / ( l , z ) and fipj+p/2- for k = p - 2 , and upj+p-lfor k = p-1. > f

air,

To visualize K 2 , imagine a phantom generator in bidegree

(-2i-2, qi(p2- I)), supporting a copy of R for each i 2 0; K 2 consists of the portion of this object having nonnegative bidegree. One can easily construct P( I )-comodules U, satisfying

v4)= Ext",;:;(Z/(S),

EX~;(~)(Z/(S), Ui@

p).

Corollary 7.3.13 suggests that Y 4 0C?,, can be filtered in such a way that there is an SES O + E 0 ( Y 4 0 C : , , ) +G(1,l ) U p ( , )~ + G ( 1 , U P p (V'+O, ,) where G ( l , 1 ) = G(1,1)/(5,ul, u2), is a direct sum of copies of P(1) and F 4 0Ui, and V' = Y 4 0 { r , : t 2 2). Such a description of C,: would

276

7. Computing Stable Homotopy Groups with the ANSS

lead to a much more direct proof of 7.3.13, but we have been unable to construct the necessary isomorphism. Apparently it does not exist before one tensors with Y4. The proof of 7.3.12 is a model for the proof of 7.3.11 and we give the former now for p = 5. We use the P(1) analogs of the following elementary facts which the reader can easily verify. 7.3.14 Proposition.

( a ) There are pairing Y' 0YJ + Y'+.' which induce pairings in Ext groups. ( b ) The element h,, E Ext( Y4, Y 4 ) corresponding to O-+ Y4+ p+ Z5¶Y4+0 can be pulled back from a similar element in Ext'( Y51-5,Y4) for O;,)(Z/(5), Y4). Hence we can use the pairing to pull bS0h2, back to Y5i+4and argue as before. Alternatively, h,, can be realized as an element in Ext#,( Y,),and there is a Yoneda pairing

v4,

E ~ f p ( i ) ( Z l ( S Y4)OExtp,,,( ), Y4, Y4) + Extp(i)(Z/(S),Y4)

For 7.3.12(b) we use the SES O +

Fly+ Y24+Z20qY4-+0 to get

b:, = 0 and b:, = 0 by 7.3.14(c). in which b:, maps to b:, by 7.3.14(d), so ill This forces dy(h,, b$) = b:, and bi0 must be a permanent cycle. Multiplication by it gives the differentials of 7.3.12 for larger powers of b20. This completes the proof of 7.3.12. Until further notice we will abbreviate ExtG(1,1)(A(2)r D:O Y " ) by Ext( Y " ) .

In order to prove 7.3.11 we need to generalize 7.3.14(d) to the elements ui and uih2,. Let i' = 5j + k with - 1 5 k 5 3. Then an easy calculation shows

From the SES

o-+ y 5 m - l +

~ 2 4 %

~5rnqy24-5m

-0

we see that r,,(u,) is the obstruction to pulling u, back to Ext( Y5"-'). Moreover if k # -1 the obstruction P51+5,5-kpulls back to Exto( Y"), which means that u, pulls back to Ext( Y 5 k + 5The ) . element U ~ Exto( E Y 2 4 )corre, this calculation verifies 7.3.14(d). sponds to bzoE Ext&l,l)(A(2),Y 2 4 ) so Using the pairings of 7.3.14(a) we get a similar statement for b;,,u,. If k = -1, u, pulls back to Ext( Y"),where we denote it by p 5 , + 5 / 6 . Combining these we get 7.3.15 Lemma. With notation as above the element biou, E Ext( Y 2 4 )pulls back to an element in Ext( Y5k+5"+9 ) which projects to by, p51+5/5-k under the map y 5 k + S n + 9 + C S q ( k + n + l ) Y".

The elements u,h2, behave a little differently. A low-dimensional calculav2 v: tion shows hzopulls back to an element in Ext( Y4) corresponding to ---. 5 9'

278

7. Computing Stable Homotopy Group with the ANSS

4

there is some indeterminacy generated by 7 ,but it will not affect our 5

2 4 calculations. Applying r4 gives -= 2a2,which is a nontrivial obstruction 5 to pulling back to Y3. 4 the obstruction to pulling Hence for 01 i I3 uih2,pulls back to Y 5 i +and back to Y5i+3is a nonzero multiple of c x Z p 5 / 5 - 1 = ~ ~ l P 5 / 4 -We ~ . want to compute the image under Y5i+4-rX5iqY4, which must be an element in Ext 1 , ( 3 6 + i ) 4( Y4) whose image under r4 is a,p5/4-i.From 7.3.8(a) we see that this element must be h,, p 5 / 4 - # , 2 . and the image under Since u4 pulls back to Y", h20~4pulls back to the map Y'-rZ5'Y4 lies in EX^'*^^^( Y4) which vanishes by 7.3.9, so ~4h2o pulls back to Y4. The uih2, for i z 5 behave similarly and we get

7.3.16 Lemma. Let i = 5 j + k with -1 5 k 5 3 as above. For k f -1, h20uib;0 pulls back to Ext( Y 5 k + 5 n and + 9 )projects to h l lby, p5j+5,4--152 under the map ySk+5n+9+ s S q ( k + n + l ) Y4.Fork = -1, h,,u5,-, pulls back to Y4 so b~oh20u5,-l pulls back to Y4+'" where it projects to by, h2,3u5j-1. To get information about the SS from the previous two lemmas we have 7.3.17 Lemma. In the SS of 7.3.8

) 0 5 i I3 , then ( a ) i f x E E, corresponds to an element in Ext( Y 5 i + 4for h,,x is a permanent cycle; ( b ) if it corresponds to one in Ext( YSi+') for 0 5 i I3 projecting under the map r, : Ysi+9-r X5qY5i+4 to an element corresponding to y E E2 , then h, I y is the target of a diferential and ( c ) i f x E E2 corresponds to an element in Ext( Y24)projecting under the map r2, : YZ4+ XZoqY4 to an element corresponding to z then b, ,z is the target of a diferential. Root (a) The SES 0-r Y4+ Y5i+9-r15qY5i+4-r0has a connecting homomorphism sending x E Ext( Ysi+4)to h,,x E Ext( Y"). (b) The above SES induces Ext( Y5i+9)-r Ext( Y5'+4)-% Ext( Y4) in which x maps to y which is therefore annihilated by h l , . (c) The SES 0-r Y1'+ Y2"+XzoqP-rO induces Ext(YZ4)-r Ext( YI9)in which x maps to z which is therefore annihilated Ext( Y") by h;, and b,, [see 4.14(d)]. We proved 7.3.12 by arguments similar to the above along with the observation that b:, is a permanent cycle because there is no possible target

3. The Homotopy Groups of a Complex with p Cells

279

for a differential. In proving 7.3.11 we need to show that the same is true of the elements b;ilui and h20b&1uiwith p l ( i + j ) ; this involves some bookkeeping to be described in 7.3.19. In applying 7.3.17(b) and (c) one would like to conclude that d , ( x ) = hlly and d , ( h , , x ) = bllz, respectively; indeed this is the case in all of our examples. However, this does not follow immediately because y and z are not necessarily uniquely determined by x since some choices are made in the construction. Equivalently, h,,y and bl,z could conceivably be killed by an earlier differential. Hence some care must be taken to verify the stated differentials. We can, however, give a painless proof of 7.3.11(a) and (b), i.e., of the differentials involving the y's, by showing that each yk for k 2 2 induces a map of the S S of 7.3.12 into that of 7.3.8. Let K c C?,,be the submodule spanned by uiv: {p: i,j, k r O , k ? l + i - j

It is easily seen to be submodule. Define an SES 7.3.18

O+K+k+K+O

by letting k have the same description as K without the condition j r 0 . Then K is spanned by

An easy calculation gives

) , is simply { yk : k r 2) tensored with Hence the S S for E X ~ ~ ( ~ , , ) (KA@( ~Y") the one in 7.3.12. Now Y k E Ext&1,1,(A(2),K),which maps via 7.3.18 to Ext&,,,,(A(2), K ) and hence to ExtL(],JA(2), Df), so 7.3.11(a) and (b) follow. Note that this argument does not prevent a y-related element from being hit by a differential originating elsewhere. To prove 7.3.11(c)-(f) we describe the relation with 7.3.12 in more detail. The elements in 7.3.8 (excluding those 7's) are analogous to those in 7.3.12 as indicated in the following table where 0 Ii I3.

280

7. Computing Stable Homotopy Groups with the ANSS

7.3.8 element

7.3.12 element

U;+2

( i+2)5u1

Hence the differentials in 7.3.11(c), (d), (e), and (f) correspond to those o, in 7.3.12 originating on bS;"+l, b:,h,, , h2,b;2+l, and h l l b 2 0 b ~respectively. The vanishing of each target can be deduced from the previous three lemmas; as an instructive example we will show P3b1, = 0 below. The survival of the elements corresponding to h , I bS0 and h , , h2,bS0 for 0 5 i 5 3 can be deduced from 4.17(a). This accounts for all except the elements corresponding to b;o. These are the only non- y-related elements said to be permanent cycles occurring in ExtS" with t / q divisible by 6 , so it suffices to prove

7.3.19 Lemma. In the SS of 7.3.8 no element of the form u,b&I with i + j divisible by p can support a nontrivial differential hitting a y-related element.

ProoJ: The possible targets are ykb;l and y k h , , h 1 , b ~ ,We . will show that in our range none of these elements have the appropriate bigrading. To be safe we will not assume p = 5 , u,b&' E Ext2J-2,q'p + l ) f p J + l ) = Ext'.' so t / q is divisible by p + 1 and t / q - (s + 2 ) ( p 2- 1)/2 = p 2 j + p ( i +j ) + i -j ( p 2 - 1) = ( p + 1)( z + j )

= 0 mod ( p + p 2 ) . First y k b ; ,E Ext 1 + 2 n . q ( p 2 I k + n ) + ( k - l ) p + ~ ~ 2 ) so t / q E p 2 n - p + k - 2 = k + n - 1 mod ( p + 1). Since k 2 2, this means k + n 2 p + 2, which puts the element out of our range. giving t/qs Next ykh20hlb;, E Ext 3 + 2 n . q ( p 2 ( k + n ) + p ( k + n - 1 ) + m - l ) , p + k - l m o d ( p + l ) so k = 2 . This (s, t ) must also satisfy t - ( s - 3 ) ~ ( p 2 - 1 ) / 2 = 0 m o d ( p Z + p ) . But this quantity is ( 3 + n ) ( l + p ) , so n r p - 3 , which gives t / q = p 3 + 1, which is again out of our range. H Finally, we describe, as a representative example, how to show b , I P3 = 0 . Consider the two maps of SESs,

28 1

4. The “Algorithm” and Computations at p = 3

and

From this we get a commutative diagram Ext’( YI4) I

-

-

E X ~ ’ ( ~ : ~ ~ ~ Y E~ X) ~ ~ ( X ’ ~ ‘ Y ’ ~ )

Ext2(Y‘)

in which the horizontal maps are the connecting homomorphisms and the top “L” is exact. Now r l o ( h 2 0 u = l ) h , , P5,3,2 by 4.16 and r l O P 5 / 3 , 2 = P3 by 4.9. It follows that the image of p3 in E x t 2 ( p ) is trivial. But the bottom composite is multiplication by b,, composed with the inclusion Y4+ p. It follows that b,, p3 must be in the kernel of h , , , and by inspection one sees that EX^',^'^( Y4)= 0, so this indeterminacy is trivial.

4. The “Algorithm” and Computations at p = 3 We begin by recalling the results of the previous sections. We are considering groups ExtS7‘for t < p 3 q (where q = 2p - 2 ) with p > 2. We have a complex C7.l

+

C 3 I +. c . 1

of G(1, 1)-comodules (7.1.1) satisfying the hypotheses of 7.1.4 with C& and C;,, weak injective [i.e., Ext“(Cf,,)= 0 for s > 0, and i = 0, 13 such that we have an isomorphism EXtSG(I,I)(A(2),c:,,,= Ext”B+p:csp,(BP*,BP*),

in our range of dimensions. In Section 3 we considered C:,,@ Yp-’, where

282

7. Computing Stable Homotopy Groups with the ANSS

Yp-' is the submodule of A(2)[tl] generated by t f for i < p . This complex satisfies similar conditions. In 7.3.13 we described Extc(i,i,(A(2),C:.,@ YP-'). Now we need to determine E X ~ ' & ~ . ~ ) (Cf,,@ A ( ~ )Y , P - ' )for i = 0 , 1. Each element of this group can be written as Zocklp-l C k @ t : with ck E c ; , ~ . Moreover such an element is uniquely determined by c,. A routine calculation gives 7.4.1 Proposition. With notation as above Ext&,,,)(A(2), Cf.,@Y P - ' )is generated by elements with the following values of co. (i)

For i = O , cg =

p -Jv:

for O s j s p - 1 .

( i i ) For i = 1, c0=-

U{

for

lsjsp-1

PI and

From 7.4.1 and 7.3.13 we can read off the ANSS &-term for X = T (l ) ( P - ' ) q = s o e4 e z q U u ,. e ( p - i ) q .The ANSS collapses without extensions in our range so we have n*(X). Now we could proceed in two ways. We could compute the ANSS E2-term for So using an SS as in A1.3.2 based on the LES

u,, uu,

-

u,,

--

BP,(ZqX) BP*(XpqX) .., or we could compute P*( So) directly using the ASS based on the homology theory represented by X. For the latter we have a very simple Adams resolution SO'-< spq-2 -< zP4-2x -< ... 0-t BP,+ B P , ( X )

x

x

where ff is the fiber of the map So+ X . This is a skeleton of Z4-'X and Sp4-2is the fiber of the inclusion. Hence we get 7.4.2 Proposition.

Let X be as above.

( a ) There is an SS converging to Extgp*(Bp)(BP*,BP,) with ES.'**= Ext%k*(BP)(BP*, B P , ( X ) ) @ E ( a , ) @ P ( PI) with a l e E:."', P1E Ep*pq*2 1 ,and d, E;W+ E ; - r + l . l , u + r ~

283

4. The "Algorithm" and Computations at p = 3

( b ) There is an SS converging to **(So)with E f.' = E ( a 1 )0 P ( p i ) 0a,(X ) where a l e E i S 4 ,P I € E Y , and d , : E ; ' + E:+r,'+r-'.

.

In both cases the elements in 7.4.1 produce the usual Exto and Ext' or the corresponding elements in r * ( S o ) . We leave the details to the reader. The remaining portion of r * ( S o ) ,commonly known as coker J, is produced in 7.4.2 by ExtG(,,,)(A(2),C:,,@Y p - ' )as described by 7.3.13. This result can be reformulated as follows. 7.4.3 ABC Theorem.

For p > 2 and t < p 3 q

EXtG(I,1)(A(2), c:,,O Y p - ' )= A@ B @ C where A is the Z/( p)-vector space spanned by

B = R @ { Yk and Here R

C"'

I

,

{pi=$:i-O,lmod(p)

Extl,2k(~’-~)-2(P2+P-)).

k 2 2)

RS+21,f+l(p2-1)q

Y p - ' )as described

= Exfp(,)(Z/(p),

in 7.3.12.

The following Fig. 7.4.4 displays this group for p = 5 and 25 5 t / q < 125. Each dot represents a basis element. Vertical lines represent multiplication by 5 and horizontal lines represent the Massey product operation (-, 5, a,),corresponding to multiplication by v , . The diagonal lines correspond to operations (-, h , , , h , , ,. . . ,h,,). b denotes psis and vIIdenotes v;-'(v2t;+ v:t2- vzt:"- v 3 t 3

5a1 The generators to the left of the chart (i.e., with t < 200) are p, for i = 1, 2, 3,4, which would support diagonal lines hitting the generators of EX^^*^'^^+'). We prefer to use the second SS of 7.4.2. Combining 7.4.2 and 7.4.3 we get For p > 2 in dimensions less than p 3 q -3, there is an SS converging to coker J [ i.e., the elements in r * (S o ) having Novikov filtration 2 2 1 with I?:*’ = E ( c r l ) O P ( P I ) @ N 1

1.4.5 Theorem.

where N'

=@u--s=,

N is given in 7.4.3.

.

Extzl,,,(A(2), C:,,O Y p - ' )with a’, p , as in 7.4.2 and

Now we will describe an algorithm (loosely speaking) for computing the desired homotopy groups based on 7.4.5. We will proceed stem b y stem

Ic

I

u)

I

Lo

I

Lo

I

u)

a,

r-

I

a,

cn

*

1

*

I

N

I

N

rr)

0

'u

0 -

0

0

0 a,

0

r-

0 rD

0 d

0

m

285

4. The “Algorithm” and Computations at p = 3

rather than term by term in the SS, i.e., compute all differentials relevant to a given stem and then move on to the next stem. Once a nontrivial differential d , ( x ) = y has been determined, there will be no need to consider the elements P:x and P:y for k > 0 , so we will ignore them. The actual calculation of differentials is not algorithmic, but is done in various ad hoc ways. 7.4.6 Procedure. We make two lists I and 0 (input and output) as follows. I is a basis of E ( a , ) O N as in 7.4.5, listed in order of dimensions. (If the list contains an element of orderp’, we listp times it as well.) 0 is constructed stem by stem inductively. Suppose 0 has been constructed through dimension k - 1 ; for k = 0 0 is empty. Add to 0 the k-dimensional elements in 1. See if any of these support nontrivial differentials in the SS of 7.4.2 and remove from 0 the source and target of each. (One may want to alter the basis in dimension k - 1 by a linear transformation so that each nontrivial target is a basis element.) Then we take the elements in 0 in dimension k - I PI],multiply each by P , ,and add it to 0 in dimension k. This completes the inductive step. The resulting list 0 is a basis for the Em-termof the SS in 7.4.5. We will discuss extension problems in due course. We will now carry out 7.5.6 for p = 3 . We give a basis for N. 7.4.7 Proposition.

For p = 3 N as in 7.4.5 has basis elements in dimensions

indicated below 72 74 78 81

82 83 86 89 90 92

93 96 97 101 104 105 106 107

hZOU2 h20 Y2 h20 P 6 / 3 hll P6/1,2

P2% 774

=6P6/3

P7

3

P9,9

Y2P2

Some comments are in order. The list goes slightly beyond the range of dimensions given in 7.4.4; these additional elements can be obtained by extending the methods of Section 3. Most of the notation is self-explanatory. rli is as in 7.4.4. The notation _nx for an integer n denotes a certain Massey product involving x. 3x and 6x denote h,,x and ( h , , , h , , , x), respectively. Now we turn to the list 0 of 7.4.6, shown in 7.4.8. Elements from N are underlined. A differential is indicated by enclosing the target in square brackets and indicating the source on the right. Hence such pairs are to be

286

7. Computing Stable Homotopy Groups with the ANSS

omitted from the final output. The computation of the differentials will be described below. 7.4.8 Theorem. follows.

With notation as above the list 0 of 7.4.6.for p = 3 is as

93 94 95

96 99 100

101

102

103 104 105 106

[ P I

55

P3lhll P 3 / 1 , 2

alp:

PI P31al h l l P 3 / I , Z [P?P2lll, [zP:1&

2P, P5

Lff1

56 57 59

[fflP:P21ffl77l

92

PlP6/3

[alu3-tP1P6/31hzou,

7.4.9 Remark. In the calculations below we shall make use of Toda brackets (first defined by Toda [5]) and their relation to Massey products. Suppose we have spaces (or spectra) and maps W A X 6Y A 2 with gf and hg null-homotopic. Let f : C W + Y and q: CX + Z be null homotopies. Define a map k:X W + Z by regarding Z W as the union of two copies of CW, and letting the restrictions of k be hf and S(Cf). k is not unique up to homotopy as it depends on the choice of the null homotopies and g.

7

287

4. The “Algorithm” and Computations at p = 3

Two choices o f f differ by a map XW+ Y and similarly for g. Hence we get a certain coset of [Z W, 21 denoted in Toda [5] by {f, g, h } , but here by (f,g, h). Alternatively, let C, be the cofiber of g, K: C, + 2 an extension of h and f : Z W + C, a lifting of XJ Then k is the composite Recall (A1.4.1) that for a differential algebra C with a, b, c E H*C satisfying ab = bc = 0 the Massey product (a, b, c ) is defined in a similar way. The interested reader can formulate the definition of higher matric Toda brackets, but any such map can be given as the composite of two maps to and from a suitable auxiliary spectrum (such as C,). For example, given

v.

xr 0

XI&.

. . Lx,

satisfying suitable conditions with each X i a sphere, the resulting n-fold Toda bracket is a composite Xfl-’Xo+= Y - , X f l ,where Y is a complex with ( n - 1) cells. The relation between Toda brackets and Massey products and their behavior in the ASS is studied by Kochman [2,4,5]. The basic idea of Kochman [4] is to show that the ASS arises from a filtered complex, so the SS results of A1.4 apply. Given Kochman’s work we will use Toda brackets and Massey products interchangeably. 7.4.10 Remark.

In the following discussions we will not attempt to keep track of nonzero scalars mod ( p ) . For p = 3 this means that a should appear in front of every symbol in an equation. The reader does not have the right to sue for improper coefficients. Now we provide a running commentary on this list. The notation 2x denotes the Massey product ( a l ,a1, x). If d , ( y ) = a I x then a , y represents zx. Also note that a , Z x = *Pix. In the 33-stem we have the Toda differential of 4.4.22. The element a l/?3,3

*

01u:

is a permanent cycle giving ZP:. The coboundary of 7 gives 9u1

7.4.1 1

(a19 Q I ,

P:,

=( P 2 9 3 ,

PI).

The differentials shown in the 41-, 48-, 52-, and 55-stems can be computed algebraically; i.e., they correspond to relations in Ext. The elements a,P 3 / 2 , P 1 P 3 / 2 and , PIP3are the coboundaries of

u: 9U1

u:t: -+-9Vl

u:t2

30:’

and

u:v:t:

9u:

u:t, 30:’

= Z/(9). respectively. We also have 3 ( 2 @ 3 / 2 ) = alps, i.e., 7rd2(So) For the differential in the 56-stem we claim ( Y , ~ , = * / ~ forcing ~P~~~, d,( v,)= f/3:P2 in the ANSS. The claim could be verified by direct calculation, but the following indirect argument is easier. P2/33/3must be nonzero

288

7. Computing Stable Homotopy Groups with the ANSS

and hence a multiple of a 1T~ because a 1p: p2 # 0 in Ext and must be killed by a differential. However, we will need the direct calculation in the future, so we record it now for general p . Consider the element U:-1u3(t2-t:+P)-

v~(t3-tlt~-f2t~z+t:+P+~2)+u;+~-~(t~+*-flf2) PUI

Straightforward calculation shows the coboundary is

which gives the desired result since the third term represents qi.The second term is nonzero in our range only in the case i = p = 3, where we have fa173 = P4bll + P l 1 3 b l oThis .

SO

773

element is also the coboundary of

v;t: ?+9Ul

v:t: 3V:'

= 0.

For the differentials in the 57- and 60-stems we claim P: = *P:P4* /3$3/31 in Ext. This must be a permanent cycle since p2 is. It is straightforward that d5(P:/J = in the ANSS, so we get d5(p4)= *ZP:. Then p:= a,2/3:=0 in r , ( S o ) , so dg(aIP4)=/3:. To verify our claim that P:=f/3:P4fflI/3$3, it suffices to compute in Ext(BP,/12). The mod Z2reductions of pz,p4,and /33,3 are 02blo* ko, u:b,o, and b,, ,respectively, where k,, = ( h , , , h , , hll). A Massey product manipulation shows ki= blob:, and the result follows. Now we will show

*z@:

,

7.4.13

x68=(aI,

p3/2,

/3d.

We can do this calculation in Ext and work mod 1, i.e., in Extp, and it suffices to show that the indicate product is nonzero. We have (h10, h l O h l 2 , ( h l l , h l l ,

h o ) ) = ( h * o ,h o .

=(h,o,

h l 2 ( h l l , h l l , h10))

hlO,(hI2, hll, ~ 1 I ) h O )

= bIO(hl2,

h l l , hll)

=(blohl2,

h l l , ~ l l ) = ~ b l , ~ hl lll ,, h l l )

= b:, # 0.

289

5. Computations for p = 5

This element satisfies p I x 6 8= p:. To show (YIx68= 0, consider the coboundary of

Next we show that there is a nontrivial extension in the 75-stem. We have

For the differential in the 77-stem note that a,p5 is the coboundary

4%v:v3 of -+.-. 32);' 3 v , This brings us to the 88-stem, where we need to show p:x6* = 0. Since x68=((Yl,a , , p : , pI) we can show p : X 6 , = 0 . There is no element in the 99-stem other than p,v3to kill it, so the differential follows. The differential in the 89-stem is similar to that in the 41-stem. The one in the 92-stem follows from 7.4.12. v;.

In the 95-stem alh,,-y2is the coboundary of -. The differential in the 3 V I v2 105-stem is a special case of 6.4.1. The others are straightforward. The resulting homotopy groups are shown in Table A3.4.

5. Computations for p = 5

We will apply the results and techniques of Section 4 to compute up to the 1000-stem for p = 5 . Naturally the lists I and 0 are quite long. The length of 0, i.e., the number of additive generators in coker J through dimension k, appears to be roughly a quadratic function of k in our range. The conventions of 7.4.9 and 7.4.10 are still in effect. The highlight of the 5-primary calculation is the following result

i7

7.5.1 Theorem. Forp = 5 , p # 0 and there are ANSS diferentials d33(y 3 )= p;'. Consequently the Smith-Toda complex V ( 3 ) does not exist, and V(2) is not a ring spectrum. W

290

7. Computing Stable Homotopy Group with the ANSS +I-

p T 2 - p # 0 and p: -0. Moreover ( y 3 , y 2 , . . . ,yz) = p(12p-1)(p-1"2 where y 2 appears in the bracket ( p - 5 ) / 2 2-

7.5.2 Conjecture. For p'7, times.

We will prove 7.5.1 modulo certain calculations to be carried out below. First we give a classical argument due to Toda for pf2-pt'= 0. We know a1pf = 0 from Toda [ 2 , 3 ] . It follows by bracket manipulations that wi = ( a l , a l .,. . , a l r P F )is defined with ( i + l ) factors a 1and 1 5 i s p - 2 . The corresponding ANSS element is Now since PI =(a1,. . . ,al>with p factors we have [using A1.4.6(c)] &,Wp-,

= (a1. .

. a1)pf2-2P

= pf2-2p+l-

Hence pf2-'+' is divisible by a1 and is therefore zero. The corresponding with r = 2 p 2 - 4 p + 3 . ANSS differential is d , ( a , p,";;) = py2pp+1 We will give a more geometric translation of this argument for p = 5. Let Xi= T(l)" = So e4 .. eiq.The Toda bracket definition of pi means there is a diagram

ue, ua, ua,

7.5.3.

-

s31

x,

where the cofiber of the top map is X4. From a I p:

=0

we get a diagram

Xl9OX

1

s190

so

We smash this with itself three times and use the fact that X, is a retract of X : to get

Combining this with 7.5.3 we get

so p:'

= 0. The calculation below shows that 2p:fi14, and plX7619 where

cyi

x761=(aIp3,p4,

is a linear combination of p: y3, Y2)EEXt7'768.

5. Computations for p

29 1

=5

Each factor of x 7 6 1 is a permanent cycle, so X761 can fail to be one only if one of the products a, P3P4 and P4y2 is nonzero in homotopy. But these products lie in stems 323 and 619 which are trivial, so x 7 6 1 is a permanent cycle, as is 3P:P14. Since /3'$J =@:I, we must have d33(y3)= /3:8 as claimed. The nonexistence of y3 as a homotopy element shows the Smith-Toda complex V(3) [satisfying BP,( V(3)) = BP,/14] cannot exist for p = 5. If one computes the ANSS for V(2) through dimension 248, one finds that u3 E Exto is a permanent cycle; i.e., u3 is realized by a map S248 f ,V(2). If V(2) were a ring spectrum we could use the multiplication to extend f to a self-map with cofiber V(3), giving a contradiction. Now we proceed with the calculation for p = 5. 7.5.4 Theorem.

For p = 5 N as in 7.4.5 has basis elements in dimensions indicated below, with notation as in 7.4.7.

38 86 134 182 198 206 214 222 230

P I

P2 P3

P4 P s , s = b11 Ps/4 Ps/3 P5/2

Ps Ps/1,2

245 253 261 269 277 278 324 325 326 372 373 374 396 404 412 420 422

h20bIl

430 u4 437 Y2 438 P l 0 / 5 443 h,,b:, 446 P10/4 451 h 2 0 b 1 1 P S / 4 454 P10/3 459 h20b1 I P S / 3 462 P10/2 469 77s 470 P I 0

h20/35/4

hzoPs/3 h 1 1 P 5 / 1.2 771

= hllb20

P 6

b20P2 772 P7

b20 P3 773

Ps P:/5 PS/SP5/4

P:/4 b20 P4 P9

P10/1,2

477 476 484 485 493 501 509 515 516 517 518 523 562 563 564

hZ0u, 5Y2

6Y2

hzoP1o/s h20P10/4

h20P10/3 hl, P l O / l , Z

b20771 h2077s 776 PI1

P2Y2

b:oP* b20772 b20 P 7

565 566 594 602 610 612 613 614 620 628 635 636 641 644 649 652 659 660 662 667 670 675 678 682 683 685 686

777 Pi2

P!/5

PS/SP5/4

b:OP3

b20Ps 778

PI3

620~3 b11~4 bllY2 blIPlO/S h20b:1 PS/4PlO/S h20b:1

PS/4

PlO/SP5/3

h20b20P8 b20P9 PI4

h20b20u3 ug

h2ObllU4 PlS/S

h2ob11~2 h20b11 P l 0 / 5

Y3 P1S/4

292

7. Computing Stable Homotopy Groups with the ANSS

691 h20b11 P l 0 / 4 694 P l 5 / 3 699 771 P 9 702 P l 5 / 2 706 h2ObI I b20u3 707 b2075 709 7710 710 P I 5

818 826 833 834 839 842 g49 850 852 g53 854 857 860 865 868 873 876 880 881 883 884 889 8g2 896 897 899 900 902 907 910 915 918 923

P I S / 1,2

714 717 724 725 732 733 741 749 753 754 755 756 757 758 761 771 792 800 802 803 804 805 806 810

hllb20Y2 h20~9 hllY3 h2oPlsls h20Y3 h20

P 15/4

h20P15/3

hllPl5ll.2 b:0771 h2ob20775 b20776

h207710 7711

PI0

h20hllY2 P2Y3 P:/5 GOP2

b:OP7 b2077, b20P12 7712

Pi7

bgou2

b,,b2ou, b: u4 GIY2

b:IPlo/s

h2,btI b:l

&0/4

h20b:0P7

b:oP8 b20P13

TI3 P I S

h20b;ou2 b2O~g h20biihou3 b, u9 h2,b:, w4 P:ols

h2ob:i

~2

h20b:I bllY3

Pl0/5

P514P1515 Pl0/4

h20b:I P:0/4

h20hllb~ou2 hl,b:,u3 h20b20P13

b20P14

PI9 h20b20U8 ~

1

4

h20b11u9 P2O/S

h20bIl P l 5 / s

926 P 2 0 / 4 930 h 2 0 b l l Y 3 931 h 2 0 b l , P l 5 / 4 933 Y4 934 P20/3 939 V I P 1 4 942 P 2 0 / 2 944 h20hllb:OU3 945 b:0775 946 h o h i ib2ou8 947 b2n7710 949 7715

950

P20 P20/1,2

952 957 962 965 972 973 980 981 989 992 993 994 995 996 997 998

h11b:OY2 h20U14

hllb20Y3 h20P20/5 hl I

y4

hmP20/4 h20Y4

h20P20/3 hll PlO/l,*

h*ob:o775 b:0776

h2Ob207771" b207711

h207715 7716

P2l P25/25

999 G o P 2 Y z 1000 b&u, H

Now we will describe the list 0, i.e., the analog of 7.4.8. The notation of that result is still in force, and we assume the reader is familiar with techniques used there. We will not comment on differentials with an obvious 3-primary analog, in particular on those following from 7.4.12. Many differentials we encounter are periodic under u2 or u ; . Since the list 0 is quite long, we will give it in six installments, pausing for comments and proofs when appropriate. 7.5.5 Theorem.

For p

=5

the list 0 ( 7.4.6) is asfollows. (First installment)

5. Computations for p = 5

293

294

7. Computing Stable Homotopy Groups with the ANSS

7.5.6 Remark. The SSs of 7.4.2 have some useful multiplicative structure even though X (the complex with p cells) is not a ring spectrum and its BP-homology is not a comodule algebra. Let X , = T(1)Iq, the iq-skeleton of T(l), which is a complex with i + 1 cells, e.g., X = X P - , . Then .rr*(X) is filtered by the images of .rr*(X,) for i ~p - 1. One has maps X, A X, + X,,, inducing pairings F, 0 F, + F,+, for i + j Ip - 1. SS differentials always lower this filtration degree and respect this pairing. The filtration can be dualized as follows. A map S" + X , is dual to a map X"-lqX, + So since OX, = X-'qX, for i s p - 1 . An element in T , , , ( X ~ - ~is) in F, iff the diagram

S"

-

XD-I

can be completed. The pairing X,A X, + X i + jdualizes to DXi+j+ DXi A OX,. If a E T,(X,-,) is in F, and p E 7r,,(XP-,)is in F, with i + j = p , then we get a map Xm'"DXp + So. If this map is trivial on the bottom cell then it factors through Xrn+"DXp-, = X m + n - q ( p - l ) X P - , . This factorization will often lead to a differential in our SS. For the differentials in dimensions 323, 371, and 419 recall (4.3.22) that there is an element b , , ~C(BP,/Z2) with d ( b 2 , )= ( b l o ~ f ? 2 ) - ( f ~ ~ bSince ll). bIo and fy2 are both cycles there is a y e C ( B P , / I , ) such that d ( y ) = ( bIoI tf2) - ( $1 blo). Hence the coboundary of

is

where the second term is nonzero only if i = -2 mod ( p ) . This gives 7.5.7

for

a1 p, pi =

izp,

i+-2mod(p)

Remember (7.4.10) we are not keeping track of nonzero scalar coefficients. The differentials in question follow. Next we show that there is a nontrivial group extension in the 427-stem, similar to that for p = 3 in the 75-stem. We want to prove a , p:p2p4 = 52x412. Since alpzp4=p12p5/2 we need to look at p;zp5/2. We have P:2@5/2=P:(al, a l , P 5 / 2 ) = P ; ( a l ,

= a1P5/3PS/4=al(al,5, =aI(aI,

5, x 4 1 2 ) = x 4 1 2 ( a l ,

ff39p5/4)=(p:,a l r a 3 ) p S / 4 p5/4)P5/4=aI(aI*

alr 5)'52X412.

5, P:/4)

295

7.5.5 (Second installment) 430 P:P, 437 2P9

296

7. Computing Stable Homotopy Groups with the ANSS

For the differential in the 514-stem, note that in the corresponding SS for Ext,(Z/(p), Z / ( p ) ) the image of b20vl kills that of P I j y 2 ,so the target in our SS of b2,,v1 is p,5y2 plus some multiple of P : P l o / 5 . On the other hand, we have

and the result follows. The relation in the 524-stem follows from 7.4.12. The differential in dimension 561 is hzo times that in the 514-stem. The one in dimension 562 comes from a relation in Ext, i.e., P:P2P10/5=P:P7Ps,s= P I P 6 a 1 ~ 1 = 0 since a I PIP6 = 0. Theorem 7.5.4 shows that there is no element in dimension 601 to give this relation, so we must have ~ 1 P 2 P 1 0as , 5 indicated. More generally, we have in Ext for 1 < i < p and j > 1

In some cases this result along with inspection of

1 implies PIPippjlp= 0.

5. Computations for p = 5

297

7.5.5 (Third installment)

The differential in the 609 stem is an Ext relation derived as follows. Since x602 is divisible in Ext by P5/4 we have d(h20b:l)=PIx602, so d ( ~ , h ~ ~ b=: a1 , ) /31x602. On the other hand, whenever crlx = n , y = 0,2xy = 0, e.g., 2( pi p6)’= 2 p : =~0, ~ forcing ~ the image of bgOP3to contain a nonzero multiple of 2p:plI. Similarly

In many cases (such as k = 12) inspection of 8 (7.5.4) shows 2P:Pk = 0. To get the other term we compute modulo filtration 2 in our SS (7.4.2), i.e., mod PlopThen we get ( h i , , h , , , b:,) is killed by b:, in Ext(BP,/12), so P3b& V:V:

kills ( P3, h , , , hll)b:, and the coboundary of - shows ( P3, h l l , hll) = 25v: P5/4a1-

There is a nontrivial group extension in dimension 617 similar to the one in the 427-stem. We have

298

7. Computing Stable Homotopy Groups with the ANSS

so

=(I?:> (YI,%)X.?O4 so the result follows. We also have

c&%02

= a,(@,,5, ~

P I PjPlO/S.

In the 627-stem we have an Ext relation @

7.55 (Fourth instalhent)

635 636 637 639 640

642

6443 644 646 647 648 65 1 652 655

658 659

L p4P10/5 =

&P5JS=0*

~

so ~( a ,2, 5 , )x602) =

5. Computations for p = 5

299

300

7. Computing Stable Homotopy Groups with the ANSS

7.5.11 Proposition.

Let x be an element satisfying p x = 0, ( a lP I ,p , x) = 0,

and alx # 0. Then Pplp-lx= P7-l 2 px. For the differentials in dimensions 666 and 673 it suffices to show We have P4pIO/S=P9P5/5=(P9, f f l P:> so 2 P 4 P 1 0 / 5 = (P9, 2P3. Then P:2p4P10/5=0.

9

7

P:2P4Pl0/5=(P:P9, % 2 P 3 =(ffIPlP6,P4,2P:)=o.

The differential on -y3 is explained in 7.5.1. Recall that the key point was that a l P : / s in Ext is a linear combination of the three elements 3P;P,,,

5. Computations for p = 5

30 1

and P:y3. In our setting this relation is given by the differential on b:,Pz, whose target is some linear combination of the four elements (including a IP&5) in question. This target is difficult to compute precisely, but it suffices to show that it includes a nontrivial multiple of Knowing then that 3P:Pl4 and PIX761 are permanent cycles and ( Y 1 P : / 5 is not, we can conclude that the linear combination also includes P : y 3 and that the latter is not a permanent cycle in the ANSS. To make this calculation we map to the S S going from Extp(l,(Z/(p),P ( 0 ) ) (this is the, R of 7.3.13 and 7.4.4) to Extp(,,(Z/(p),Z/(p)). The elements 3P:PI4, y3, and x761 all have trivial images, while b:,P2 and a Ip&5 do not, and it suffices to show that a 1p:15= blob:, vanishes in Extp(l).h l l b l l is killed by b20, so (b;l, h l l , h,,, h l l ) is killed by b& so we have P1x761

Given this situation the target of the differential from 4p:', is the same as 3 P f X 6 9 2 , and a , P : / 5 is 4P:x692, which accounts for the indicated differentials in dimensions 791 and 799. The differential in the 752-stem can be recovered from the corresponding S S for Extp. The images of v l and y2 are the Massey products ( l ~ ,h,~_b), and (h12,h, _b) where b and _b denote the matrices

(h12b10v1,h, _b)=(hllbllvl, h, _b)=o respectively. Then we have P1q1yZ= since h I 1 v 1= 0.

302

7. Computing Stable Homotopy Groups with the ANSS

For the differential in the 838-stem we use the method of 7.5.6. We have maps f :Z190Xl+ So and g: Z6O9X,+ So where f is P: on the bottom cell, and g is (YIx602 on the bottom cell and on the second cell. The smash So which product vanishes on the bottom cell so we have a map CSo7X4+ is P : x ~ ~ ~ + ~on P the : x bottom ~ ~ ~ cell. The second term vanishes because p:&jo2 E ~ 7 9 = 2 0. We have

A routine calculation gives P I X 6 1 7 = 3P:Pll and P:x617 = ( Y I x ; ~ ~ . Our map hence the desired differential. gives o= 4P:x617 = We use a similar argument in the 848-stem. We start with the maps Z316X1+S0 and Z 5 3 1 X 4 + Scarrying 0 PIP6 and a1P2Plo/5 on the bottom cells. The resulting relation is P:X,,~= 0. From 7.5.4 we see that N is vacuous in dimensions 887 and 856, so the indicated differential is the only one which can give this relation. The argument in dimension 849 is similar to that in dimension 609. In dimension 864 we use 7.5.6 again starting with the extensions of j3: and ZP4p10/5 to XI and X,.

5. Computations for p

=5

303

7. Computing Stable Homotopy Groups with the ANSS

304

The element x868 is constructed as follows. There is a commutative diagram

where the cofiber o f f is ESs7X,and g is an extension of 3p:P14. Both f and P9 extend to C860X,.The difference of the composite extensions of Plol5P9 and gf gives xu68 on the top cell. In other words x868 is the Toda bracket for ~ 8 6 7

~ 8 6 0 5438 ~

v

2837xz

+

so.

We will see below that P : g = 0 and PI PIoi5P9= 0 so it follows that P:X,,, is divisible by a 1 and hence trivial. For the relation in dimension 875 we have x761=(cu,P,, P6, y2) and PIP10/5=@;i, f f , P l , P 6 ) so P:x761= Y 2 P I P l O I S .

305

5. Computations for p = 5

For dimension 879 we have, using 7.5.11, y2x404

= y2( P S / 4 9

=(Z-

PI

9

a1

5YZP:,Pl,

P:) = ( 7 2 P 5 / 4 , a,P3=3.

P13 a 1

P:)

sP1~2=3P:0P10/s

so

3P

'Pl0/5 = y2

P1x404

= O*

In dimension 888 we have

P: P 3 P l S / S = P: P8P10/5 = p8P15y2 = P2&5y2= O. For the 896 stem we have ~ 1 ~ ~ P l P 3 P l S , 5 = P 1 P 2 ~ 2 P 2 P 1 5 / 5 = o ,

which (by inspection of 7.5.4) implies 4 ZP, P3 = 0. We are not sure about y4. A possible approach to it is this. Extrapolating 7.5.4 slightly we see that Ext7."16 has two generators, P:y4 and ( y 3 , yl, P3). The latter supports a differential hitting P:5P10/5 = ( Pi8, yl, P2). The same Ext group contains ( y 2 , y 2 ,03),which is a permanent cycle. Hence if it is nonzero it is neither P:y4, in which case y4 is a permanent cycle, or P:?'4+(73, ?'i,P3>, in which case ~ x ( Y ~ ) = P : ~ P ~ o / ~ . In the 992-stem we have P1x954= P4Xj310 so P:xgs4= PI P 4 x S I O = 0. Extrapolating the pattern in 7.5.4 we find that the only element in the appropriate dimension is b:ly2, which kills 3pi5y2.

APPENDIX 1

HOPF ALGEBRAS AND HOPF ALGEBROIDS

Commutative, noncocommutative Hopf algebras, such as the dual of the Steenrod algebra A(3.1 .l), are familiar objects in algebraic topology and the importance of studying them is obvious. Computations with the Adams spectral sequence require the extensive use of homological algebra in the category of A-modules or, equivalently, in the category of A,-comodules. In particular there are several change-of-rings theorems (A1.1.18, A1.1.20, and A1.3.13) which are major labor-saving devices. These results are well known, but detailed proofs (which are provided here) are hard to find. The use of generalized homology theories such as MU- and BP-theory requires a generalization of the definition of a Hopf algebra to that of a Hopf algebroid. This term is due to Haynes Miller and its rationale will be explained below. The dual Steenrod algebra A, is defined over Z/( p ) and ) dual to the product on A. The has a coproduct A:A,+A, O z l ( pA, BP-theoretic analog BP,( BP) has a coproduct A : BP,( B P ) + BP,(BP) O,*(Bp)BP,( BP), but the tensor product is defined with respect to a .ir,(BP)-bimdule structure on BP,(BP); i.e., n , ( B P ) acts differently on the two factors. These actions are defined by two different Z, ,,-algebra maps T ~ vR: , 7r.J B P ) + BP,( BP), known as the left and right units. In the case of the Steenrod algebra one just has a single unit 7: Z/( p) -+ A,. Hence BP,(BP) is not a Hopf algebra, but a more general sort of object of which a Hopf algebra is a special case. The definition of a Hopf algebroid (Al.l.1) would seem rather awkward and unnatural were it not for the following category theoretic observation, due to Miller. A Hopf algebra such as A, is a cogroup object in the category 306

Introduction

307

of graded Z/(p)-algebras. In other words, given any such algebra R, the coproduct A : A, + A, 0A, induces a set map Hom( A,, R ) x Hom(A,, R ) + Hom(A,, R ) which makes Hom(A,, R ) into a group. Now the generalization of Hopf algebras to Hopf algebroids corresponds precisely to that from groups to groupoids. Recall that a group can be thought of as a category with a single object in which every morphism is invertible; the elements in the group are identified with the morphisms in the category. A groupoid is a small category in which every morphism is invertible and a Hopf algebroid is a cogroupoid object in the category of commutative algebras over a commutative ground ring K [Zfp)in the case of BP,(BP)J. The relation between the axioms of a groupoid and the structure of a Hopf algebroid is explained in A l . 1.1. The purpose of this appendix is to generalize the standard tools used in homological computations over a Hopf algebra to the category of comodules over a Hopf algebroid. It also serves as a self-contained (except for Sections 4 and 5) account of the Hopf algebra theory itself. These standard tools include basic definitions (Section l ) , some of which are far from obvious; resolutions and homological functors such as Ext and Cotor (Section 2 ) ; spectral sequences of various sorts (Section 3), including that of Cartan and Eilenberg [ 1, p. 3491; Massey products (Section 4); and algebraic Steenrod operations (Section 5 ) . We will now describe these five sections in more detail. In Section 1 we start by defining Hopf algebroids ( A l . l . l ) , comodules and primitives (A1.1.2), cotensor products (A1.1.4), and maps of Hopf algebroids (A1.1.7). The category of comodules is shown to be abelian (A1.1.3), so we can do homological algebra over it in Section 2 . Three special types of groupoid give three corresponding types of Hopf algebroid. If the groupoid has a single object (or if all morphisms have the same source and target) we get an ordinary Hopf algebra, as remarked above, The opposite extreme is a groupoid with many objects but at most a single morphism between any pair of them. From such groupoids we get unicursal Hopf algebroids (Al.l.11). A third type of groupoid can be constructed from a group action on a set, and a corresponding Hopf algebroid is said to be split (A1.1.22). The most difficult definition of Section 1 (it took us quite a while to formulate) is that of an extension of Hopf algebroids (A1.1.15).An extension of Hopf algebras corresponds to an extension of groups, for which one needs to know what a normal subgroup is. We are indebted to Higgins [ l ] for the definition of a normal subgroupoid. A groupoid C,, is normal in C, if (i) the objects of C, are the same as those of C,, (ii) the morphisms in C,, form a subset of those in C , , and

308

A l . Hopf Algebras and Hopf Algebroids

(iii) if g: X + Y and h : Y + Y are morphisms in C , and C,, respectively, then g - ' h g : X + X is a morphism in C,. This translates to the definition of a normal map of Hopf algebroids (Al.l.10). The quotient groupoid C = C , / C , is the one (i) whose objects are equivalence classes of objects in C, , where two objects are equivalent if there is a morphism between them in C,, and (ii) whose morphisms are equivalence classes of morphisms in C , ,where two morphisms g and g' are equivalent if g ' = hlghz where h , and h2 are morphisms in C,. The other major result of Section 1 is the comodule algebra structure theorem (A1.1.17) and its corollaries, which says that a comodule algebra (i.e., a comodule with a multiplication) which maps surjectively to the Hopf algebroid Z over which it is defined is isomorphic to the tensor product of its primitives with X. This applies in particular to a Hopf algebroid I' mapping onto C (A1.1.19). The special case when C is a Hopf algebra over a field was first proved by Milnor and Moore [3]. In Section 2 we begin our study of homological algebra in the category of comodules over a Hopf algebroid. We show (A1.2.2) that there are enough injectives and define Ext and Cotor (A1.2.3). For our purposes Ext can be regarded as a special case of Cotor (A1.1.6). We find it more convenient here to state and prove our results in terms of Cotor, although no use of it is made in the text. In most cases the translation from Cotor to Ext is obvious and is omitted. After defining these functors we discuss resolutions (A1.2.4, A1.2.10) that can be used to compute them, especially the cobar resolution (A1.2.11). We also define the cup product in Cotor (A1.2.14). In Section 3 we construct some spectral sequences for computing the Cotor and Ext groups we are interested in. First we have the SS associated with an LES of comodules (A1.3.2); the example we have in mind is the chromatic SS of Chapter 5. Next we have the SS associated with a (decreasing or increasing) filtration of a Hopf algebroid (A1.3.9); examples include the classical May SS (3.2.9), the SS of 3.5.2, and the so-called algebraic Novikov

ss (4.4.4).

In A1.3.11 we have an SS associated with a map of Hopf algebroids which computes Cotor over the target in terms of Cotor over the source. When the map is surjective the SS collapses and we get a change-of-rings isomorphism (Al.3.12). We also use this SS to construct a Cartan-Eilenberg SS (A1.3.14 and A1.3.15) for an extension of Hopf algebroids. In Section 4 we discuss Massey products, an essential tool in some of the more intricate calculations in the text. The definitive reference is May [3] and this section is little more than an introduction to that paper. We

309

1. Basic Definitions

refer to it for all the proofs and we describe several examples designed to motivate the more complicated statements therein. The basic definitions of Massey products are given as A1.4.1, A1.4.2, and A1.4.3. The rules for manipulating them are the juggling theorems A1.4.6, Al.4.8, and A1.4.9. Then we discuss the behavior of Massey products in spectral sequences. Theorem A1.4.10 addresses the problem of convergence; A1.4.11 is a Leibnitz formula for differentials on Massey products; and A1.4.12 describes the relation between differentials and extensions. Section 5 treats algebraic Steenrod operations in suitable Cotor groups. These are defined in the cohomology of any cochain complex having certain additional structure and a general account of them is given by May [5]. Our main result (A1.5.1) here (which is also obtained by Bruner et al. [ l ] ) is that the cobar complex (A1.2.11) has the required structure. Then the theory of May [5] gives the operations described in A1.5.2. Our grading of these operations differs from that of other authors including May [5] and Bruner et al. [ 11; Our Pi raises cohomological (as opposed to topological) degree by 2i( p - 1).

1. Basic Definitions A1.1.1 Definition. A Hopf algebroid over a commutative ring K is a cogroupoid object in the category of (graded or bigraded) commutative K algebras, i.e., a pair (A, r) of commutative K-algebras with structure maps such that for any other commutative R-algebra B, the sets Hom(A, B ) and Hom(T, B ) are the objects and morphisms of a groupoid ( a small category in which every morphism is an equivalence). The structure maps are

v,:A+T

lefl unit or target,

‘llR:A+r

right unit or source

A:r+raAr

coproduct or composition,

F:r+A

counit or identity,

c:r+r conjugation or inverse. Here r is a left A-module map via vr and a right A-module via ’llR, r OAr is the usual tensor product of bimodules, and A and E are A-bimodule maps. The defining properties of a groupoid correspond to the following relations among the structure maps: = &vR = lA, the identity map on A. (The source and target of (a) qL an identity morphism are the object on which it is defined.)

310

A l . Hopf Algebras and Hopf Algebroids

(b) (rOE) A = ( & O r )A = I,-. (Composition with the identity leaves a morphism unchanged.) (c) ( r O A ) A = (AOT) A. (Composition of morphisms is associative.) (d) cqR = q L and = q R .(Inverting a morphism interchanges source and target.) (e) cc = I,-. (The inverse of the inverse is the original morphism.) (f) Maps exist which make the following commute

A

L --LA ~

where c . T ( y , @ y 2 )= c ( y , ) y , and r . c ( y , O y 2 )= y , c ( y 2 ) .(Composition of a morphism with its inverse on either side gives an identity morphism.) If our algebras are graded the usual sign conventions are assumed; i.e., commutativity means xy = ( -l)'x'tvtyx,where 1x1 and lyl are the degrees or dimensions of x and y, respectively. A graded Hopf algebroid is connected if the right and left sub-A-modules generated by To are both isomorphic to A. In most cases the algebra A will be understood and the Hopf algebroid will be denoted simply by r. Note that if q R = q L ,then r is a commutative Hopf algebra over A, which is to say a cogroup object in the category of commutative A-algebras. This is the origin of the term Hapf algebroid. More generally i f D c A is the subalgebra on which q R = q L ,then r is also a Hopf algebroid over D. The motivating example of a Hopf algebroid is ( r * ( E ) ,E , ( E ) ) for a suitable spectrum E (see Section 2.2). A1.1.2 Definition. A left r-comodule M is a left A-module M together with a left A-linear map $1 M + r @ A M which is counitary and coassociative, i.e., such that ( & O M ) $ =M (i.e., theidentityon M ) and ( A @ M ) $ = ( r @ $ j $ . A right r-comodule is similarly defined. A n element m f M is primitive i f $(m)=lam. A comodule algebra M is a comodule which is also a commutative associative A-algebra such that the structure map $ is an algebra map. If M and N are left r-comodules, their eomodule tensor product is M O AN with strwcture map being the composite

MON

-

Bu 0 BN

r@M@ N+TOT@MON + T O M O N ,

31 1

1. Basic Definitions

where the second map interchanges the second and third factors and the third map is the multiplication on r. All tensor products are over A using only the left A-module structure on A. A differential comodule C* is a cochain complex in which each C sis a comodule and the coboundary operator is a comodule map.

A1.1.3 Theorem. If f is jlat as an A-module then the category of left r-comodules is abelian (see Hilton and Stammbach [ 11).

Pro05 If O + M ' + M flat over A,

o+

is an SES of A-modules, then since

+ M"+O

@ A

M'+

r OAM + r

@A

r is

M"+O

is also exact. If M is a left r-comodule then a comodule structure on either M' or M" will determine such a structure on the other one. From this fact it follows easily that the kernel or cokernel (as an A-module) of a map of comodules has a unique comodule structure, i.e., that the category has kernels and cokernels. The other defining properties of an abelian category are easily verified.

In view of the above, we assume from now on that

r isJlat over A.

A1.1.4 Definition. Let M and N be right and left r-comodules, respectively. Their cotensor product over r is the K-module dejined by the exact sequence O + M 0,N + M

@A

N *'N-MO*,

MOArOAN,

where I+! denotes the comodule structure maps for both M and N. Note that M Or N is not a comodule or even an A-module but merely a K-module. A left comodule M can be given the structure of a right comodule by the composition M*T@M-

T

MOT-

MOc

MOT,

where T interchanges the two factors and c is the conjugation map (see Al . l . l ) . A right comodule can be converted to a left comodule by a similar

device. With this in mind we have

A1.1.5 Proposition. M Or N

=N

0,M.

The following relates the cotensor product to Hom.

A1.1.6 Lemma. A. Then

Let M and N be left r-comodules with Mprojective over

( a ) HomA(M,A ) is a right r-comodule and ( b ) H o m r ( M , N ) = H o m A ( M , A ) O r N , e.g., Homr(A, N ) = A O r N .

A t . Hopf Algebras and Hopf Algebroids

312

Proof: Let $-: M + r 0,M and GIN:N ture maps: Define

+

r O AN

be the comodule struc-

$f,, , $%:Horn,( M , N ) + Horn,( hf,r @ A N ) by $f,,(f)

=

(r@fMM

and

Jf*N(f) = Jf~f

for f € HomA(M,N ) . Since M is projective we have a canonical isomorphism, Horn,( M , A ) @ A N

Horn,( M , N ) .

Hence for N = A we have $*,:HO~A(M,A)+HO~A(~,A)@A~.

To show that this is a right r-comodule structure we need to show that the following diagram commutes HomA(M, A )

s*,

Horn,( M, r)

313

1. Basic Definitions

and the following diagram commutes

I:* ):,

p

Hom( M, A ) @ N N]

Horn,( M, N )

M,A)Olb,,

Hom(M, A ) @ r @N -HOmA(M,

r@A

N)

The next few definitions and lemmas lead up to that of an extension of Hopf algebroids given in A1.1.15. In A1.3.14 we will derive a corresponding Cartan-Eilenberg spectral sequence.

A1.1.7 Definition. A map of Hopf algebroids f : ( A , r)+ ( B , I;) is a pair of K-algebra maps f , : A -+ B, f 2 :r + I; such that f i E = &fi,

f2vR

f 2 c = cf,

= 7R.h 9

and

f27)L=

vLfl9

Af2 = ( f 2 @f i ) A.

A1.1.8 Lemma. Let f : ( A ,r)+ ( B , r) be a map of Hopf algebroids. Then r @ A B is a right X-comodule and for any left C-comodule N, (r@ A B ) 0,N is a sub-left r-comodule of r On N, where the structure map for the latter is A@ N.

ProoJ: The map (I-@f 2 ) A :r+= r' O AC = (rOAB ) O BC extends uniquely to rOAB, making it a right C-comodule. By definition (rOAB ) OxN is the kernel in the exact sequence 0-

( r @ A

B ) 0-iN + = r @ AN +

r @ A

C@B N

where the right-hand arrow is the difference between ( r O f 2 )A O N and r@ L!,I Since r OAN and r OAC BeN are left r-comodules it suffices to show that the two maps respect the comodule structure. This is clear for r@$,and for ( r Of ) A@ N we need the commutativity of the following diagram, tensored over B with N.

r

(r@,fi)A@B

B

roAc

AB.1

[email protected] B

i-@

( 1’Ofi A 0B

,rOAr@AI;

It follows from the fact that f is a Hopf algebroid map.

A1.1.9 Definition. r f ( A , r ) is a Hopf algebroid the associated H o p f algebra ( A , r') is dejned by r'=r/(vL(a ) - v R ( a ) l aE A ) . ( T h e easy verijication that a Hopf algebra structure is induced on r' is left to the reader.) Note that r' may not b e y a t over A even though r is.

314

At. Hopf Algebras and Hopf Algebroids

A.l.l.10 Definition. A map of Hopf algebroids f : ( A , I-) + ( A , X) is normal iff2 :r + Z is surjective, f,: A + A is the identity, and r Ox.A = A Ox. r in r. A . l . l . l l Definition. A Hopf algebroid (A, U ) is unicursal if it is generated as an algebra by the images of qL and q R , i.e., if U = A O DA where D = A 0 A is a subalgebra of A. ( The reader can verify that the Hopf algebroid structure of U is unique.) This term was taken from page 9 of Higgins [l].

A1.1.12 Lemma. (A, U ) . Then

Let M be a right comodule over a unicursal Hopf algebroid

( a ) M is isomorphic as a comodule to M OAA with structure map M 0q R and ( b ) M = ( M O U A ) O D Aas A-modules. Proof. For m E M let $( m ) = m ' 0 m". Since U is unicursal we can assume that each rn" is in the image of v R .It follows that

($63 U ) $ (m ) = ( M O A ) $ ( r n )= m'O 1 0 m" so each m' is primitive. Let 6 = m'E( m"). Then $( 6 )= m'63 m"= $ ( m ) , so m = 6 since $ is a monomorphism. Hence A4 is generated as an A-module by primitive elements and ( a ) follows. For (b) we have, using (a), ( M 0u A ) OD A = M

0 A

( A 0u A ) OD A = M

0 A

D OD A = M.

A1.1.13 Lemma. Let ( A ,Z) be a Hopf algebroid, (A, Z') the associated Hopf algebra (Al.1.7) D = A O x A , and ( A , U ) the unicursal Hopf algebroid (A1.1.9) with U = A @IDA.Then ( a ) U = C O Z . Aand ( b ) for a left Z-comodule M, A UxrM is a left U-comodule and A UzM = AUu (A0x.M). Proof. By definition, Z' = A OUZ, where the U-module structre on A is given by E : U + A, so we have Z@A Z ' = E @ A A @ , X = Z @ u Z.

By A1.1.3, there is a short exact sequence O+ZOyA+XOuZ

where the last map is induced by A-Z:OqL. An element A ( u ) = cr0 1 in Z 0"Z iff u E U, so (a) follows. For (b) we have AO~M=ACIu(UOxM)

U E

Z has

315

1. Basic Definitions

and

U OrM

=(

A02s I;) O r M

= A 0 2 ,M .

W

The following example may be helpful. Let ( A ,r) = (7.JBP), BP,( B P ) ) (4.1.19),i.e., A = Z ( , , [ u l ,u z , . . .] and r = A [ t l ,t 2 , . . .I where dim ui= dim t, = 2( p i- 1). Let C = A [ f n + 2 , . . .] for some n 2 0. The Hopf algebroid structure on Z is that of the quotient r/ ( tl ,. . . , t n ) . The evident map ( A , r ) + ( A , C )is normal (A1.1.10).D = A 0 2 A is Z(p)[ul,..., u,] and @ = A O2r O2A is D [ t l , . . . , t , ] . ( D , @ )is a sub-Hopf algebroid of ( A ,r) and (0,@) + ( A , r)+ ( A ,2 ) is an extension (A1.1.15below). Al.l.14 Theorem. L e t f : ( A ,r)+ ( A ,Z) be a normal map ofHopfalgebroids and let D = A O , A and @ = A A O , r O , A . Then ( D , @ ) is a sub-Hopf algebroid of ( A ,r). (Note that by A1.1.8,A 0,r and r U 2 A are right and left r-comodules, respectively, so the expressions ( A Uzr) REA and A U, (rU2A ) make sense. It is easy to check, without using the normality of J; that they are equal, so @ is well defined.)

Roo$ By definition an element a E A is in D iff f z q L ( a )= f 2 q R ( a )and is in @ iff (f2@r0f2) A2(y ) = 1 0y 0 1. To see that qR sends D to @, we have for d E D

(fz@r@fi)A277~(d)=1 @ 1 @ f 2 7 ~ ( d ) = 1@1@f27L(d)=

1@VR(d)@1.

The argument for T~ is similar. It is clear that Q, is invariant under the conjugation c. To show that P sends Q, to D we need to show f 2 q R &4) ( = f ~ q ~ ~ for ( 4 +) E @ . But f Z q R ~ ( 4 ) = ~ ~ ~ f ~ ( and ( b ) since A2f2(4)= 10fX4)01 we have Af~(4)=10fz(4)=f2(4)01so f 2 ( 4 ) € D ,and (7)R

-qL)Ef(+)=o-

To define a coproduct on @ we first show that the natural map from @ OD@ to r OAr is monomorphic. This amounts to showing that a4 E @ iff a E D. Now by definition a 4 E @ iff

f z ( a 4 ' ) @ 4 ' ' 0 f 2 ( 4 "=' )1 0 a 4 0 1 =f2r]R

Since

~ E Q we ,

( a) 0 4 01.

have

f2(4’) 0#"Of.( 4'")= 1 04 0 1, so the criterion is f2(

i.e., a E D.

a 101 0 1 =f2vR( a ) 01 01 ,

316

A l . Hopf Algebras and Hopf Algebroids

Now consider the commutative diagram

(A,Z') = (A, 17 where I;' is the Hopf algebra associated to Z (AI.l.9), f is the induced map, U is the unicursal Hopf algebroid ( A l . l . l l ) A O D A , 6 = A UL'r Ox.A, and g will be constructed below. We will see that @ and 6 are both Hopf algebroids. Now the map f' is normal since f is and A UX8A = A, so the statement that 6 is a Hopf algebroid is a special case of the theorem. Hence we have already shown that it has all of the required structure but the coproduct. A = A U z , A Ox.r = Since r O E .A = A U x . r , we have & = A Ox.rUEP A Ox, r. One easily verifies that the image of A: r + r OAr is contained in rOrr and hence in TCl.,T. There A sends & = A D z , T U p . A to A Ox,r Ox.r Ox, A = 6 Ox. 6 c 6 OA6, so 6 is a Hopf algebroid. Since 6 = r 0,. A and U = 2 UxzA [A1.1.13(a)] we can define g to be f 2 U A. It follows from A1.1.13(b) that @ = A0 1I'OI A = A 0 (A Ox, r Ox,A ) 0UA

=AOU6DUA. By A1.1.12(b) we have 6 = A O D @ @ D A , so A @ D 6 OD A @ D @ OD A. The coproduct A sends @ to 6 0u and we have

60"& = 6 0 , . , ( A O U A ) @ A 6

by A1.1.12(a)

= A OD @OD( A O U A ) OD @

= A OD

600,6= OA6

6c

ODA

On D OD 0 OD A

= = AOD @OD@@DA.

Since A is A-bilinear it sends @ to @ O D@ and @ is a Hopf algebroid. Al.1.15 Definition. An extension of Hopf algebroids is a diagram (0,

r)f ,( A ,2 )

-L (A,

where f is normal (Al.i.10) and ( D , @ )is as in Ai.1.14.

H

317

1. Basic Definitions

The extension is cocentral

if the diagram

(where t interchanges factors) commutes up to the usual sign. In particular Z must be cocommutatiue. A nice theory of Hopf algebra extensions is developed by Singer [ 5 ] and in section I1 3 of Singer 161. Note that (as shown in the proof of A1.1.14) if Z is a Hopf algebra then @ = A 0,r = r 0,A. More generally we have A1.1.16 Lemma. comodules.

Roo$

With notation as above, A O r r = @ O D A as right

r-

Using A1.1.12 and A1.1.13 we have

r0, A @ D A = A 0,r 0,. A O u AODA = A O r~ o , . ~

@ O D A = A 0,

= A Ox AOn, = A 0,AO,, A O z T =A

0u A 0,. r

= A O ~ ~ . A1.1.17 Comodule Algebra Structure Theorem. Let ( B , C ) be a graded connected Hopf algebroid, M a graded connected right C -comodule algebra, and C = M OxB. Suppose

( i ) there is a surjective comodule algebra map f:M -+ X and ( i i ) C is a B-module and as such it is a direct summand of M. Then M is isomorphic to C right Z-comodule.

C simultaneously as a left C-module and a

We will prove this after listing some corollaries. If C is a Hopf algebra over a field K then the second hypothesis is trivial so we have the following result, first proved as theorem 4.7 of Milnor and Moore [3]. A1.1.18 Corollary. Let ( K ,C ) be a commutative graded connected Hopf algebra over a3eld K . Let M be a K-algebra and a right Z-comodule and let C = M OxK. If there is a surjection f :M + X which is a homomorphism of

318

A l . Hopf Algebras nod Hopf Algebroids

algebras and C-comodules, then M is isomorphic to C O X simultaneously as a left C-module and as a right Z-comodule. A1.1.19 Corollary. Let f : ( A ,r)+ (B, C ) be a map of graded connected Hopf algebroids (Al.l.7) and let r'=r O AB and C = r' OTB. Suppose ( i ) f;: I"+ Z is onto and ( i i ) C is a B-module and there is a B-linear map g : r'+ C split by the inclusion of C in r'. Then there is a map g': r' + C Os Z dejned by g'( y ) = g( y ' ) 0f;( y " ) which is an isomorphism of C-modules and C-comodules. R

A1.1.20 Corollary. Let K be a field and f :( K , r)+ ( K , C ) a map of graded connected commutative Hopf algebras and let C = OEK. Zf f is surjective then r is isomorphic to C O X simultaneously as a left C-module and as a right C-comodule. In A1.3.12 and A1.3.13 we will give some change-of-rings isomorphisms of Ext groups relevant to the maps in the previous two corollaries.

Proof of A1.1.17. Let i: C + M be the natural inclusion and let g : M + C be a B-linear map such that gi is the identity. Define g': M + C Os 1 to be ( g O Z ) + ;it is a map of 2.-comodules but not necessarily of C-modules and we will show below that it is an isomorphism. Next observe that f B: C + B is onto. In dimension zero it is simply J which is onto by assumption, and it is B-linear and therefore surjective. Let j : B + C be a B-linear splitting off B. Then h = g-'( j O C ) :C + M is a comodule splitting o f f Define i : C O e Z + M by c ( c O u ) = i ( c ) h ( u )for C E Cand ~ E CIt. is clearly a C-linear comodule map and we will show that it is the desired isomorphism. We have $( c 0 a ) = g'( i ( c ) h ( a ) )= g ( i( c )h ( a ' ) ) O u "= C O O

where the second equality holds because i ( c ) is primitive in M and the congruence is modulo elements of lower degree with respect to the following increasing filtration (A1.2.7) on C Oe X. Define F,( C Os C) c C Oe Z to be the sub-K-module generated by elements of the form c O a with dim U S n. It follows that g ' i and hence h are isomorphisms. We still need to show that g' is an isomorphism. To show that it is 1-1, let m 0 a be the leading term (with respect to the above filtration of M OC) of $ ( m ) . It follows from coassociativity that m is primitive, so g ( m )# 0 if m f 0 and ker g' = 0. To show that is onto, note that for any c 0u E C OsZ

319

2. Homological Algebra

we can choose m Ef - ’ ( a )and we have

so coker

i = 0 by standard arguments.

A1.1.21 Definition. A n ideal I c A is invariant if if is a sub-r-comodule, or equivalently if^^( I ) c I T . A1.1.22 Definition. A Hopf algebroid ( A ,r) is split if there is a Hopf algebroid map i : ( K , X ) + ( A , Y ) (A1.1.19) such that i;:E;OA+T is an isomorphism of K-algebras. Note that composing q R : A + T with the inverse of i; defines a left X-comodule structure on A.

2. Homological Algebra Recall (A1.1.3) that the category of comodules over a Hopf algebroid ( A ,r) is abelian provided r is flat over A, which means that we can do homological algebra in it. We want to study the derived functors of Horn and cotensor product (Al.1.4).Derived functors are discussed in most books on homological algebra, e.g., Cartan and Eilenberg [l], Hilton and Stammbach [ 13, and Mac Lane [ 11. In order to define them we must be sure that our category has enough injectives, i.e., that each r-comodule can be embedded in an injective one. This can be seen as follows.

A1.2.1 Definition. Given an A-module N, dejine a comodule structure on r O AN by [email protected] for any comodule M 8:HomA(M, N ) + Homr(M, r @ A N ) is the isomorphism given by 8 ( f )= ( r @ f ) t )for f E HornA(M, N ) . For g E Hornr( M, r @ A N ) , 8-’(g) is given by 8 - ’ ( g ) = ( E 0N ) g . A1.2.2 Lemma. If I is an injective A-module then r O AI is an injective r-comodule. Hence the category of r-comodules has enough injectives.

ProoJ: To show that r @ A I is injective we must show that if M is a subcomodule of N, then a comodule map from M to r OAI extends to N. But Homr( M, r @ A I ) = Horn,( M, I ) which is a subgroup of HornA(N, I ) = Homr(N, r@AI ) since I is injective as an A-module. Hence the existence of enough injectives in the category of A-modules implies the same in the category of r-comodules.

320

A l . Hopf Algebras and Hopf Algebroids

This result allows us to make

A1.2.3 Definition. For left r-comodules M and N , Extk(M, N ) i s the ith right derived functor of Hornl( M, N ) , regarded as a functor of N . For M a right r-comodule, Cotor;-(M, N ) , is the ith right derivedfunctor of M l3, N ( A I . l . 4 ) ,also regarded as a functor of N . The corresponding graded groups will be denoted simply by Ext, ( M , N ) and Cotor, ( M , N ) , respectively. In practice we shall only be concerned with computing these functors when the first variable is projective over A. In that case the two functors are essentially the same by A1.1.6. We shall therefore make most of our arguments in terms of Cotor and list the corresponding statements about Ext as corollaries without proof. Recall that the zeroth right derived functor is naturally equivalent to the functor itself if the latter is left exact. The cotensor product is left exact in the second variable if the first variable is flat as an A-comodule. One knows that right derived functors can be computed using an injective resolution of the second variable. In fact the resolution need only satisfy a weaker condition.

A1.2.4 Lemma.

Let

0 - N + Ro+ R 2 + . . .

be a long exact sequence of left r-comodules such that Cotor:( M , R ' ) = 0 for n > 0. Then Cotor,( M, N ) is the cohomology of the complex

A1.2.5

Cotor:( M, R o ) A Cotor, ( M , R ' ) A . . . .

Roo$ Define comodules N ' inductively by N o = N and N"' quotient in the short exact sequence

is the

O+ N ' + R'+ N'+'+O. These give long exact sequences of Cotor groups which, because of the behavior of Cotor,( M, R ' ) , reduce to four-term sequences O-, Cotor:-( M,N ' )-* Cotor:( M , R ' ) + Cotor:-(

M,

,'+I)

+ Cotor:.(M,

N ' )-, 0

and isomorphisms

A1.2.6

Cotor:( M , N ' + I ) = Cotorf+'(M, N ' )

for

n > 0.

Hence in A1.2.5, ker 6, =CotorF(M, N ' ) while im 8, is the image of so Cotor:( M , R ' ) in Cotor:( M , " + I ) , ker 6,/im 6,. I = Cotor:( M, N ' - ' ) = Cotor;( M , N ) by repeated use of A1.2.6. This quotient by definition is H’of A1.2.5.

a

2. Homologicml Algebrm

32 1

For another proof see A1.3.2. We now introduce a class of comodules which satisfy the Ext condition of A1.2.4 when M is projective over A. A1.2.7 Definition. A n extended I'-comodule is one of the form r OAN where N is an A-module. A relatively injective r-comodule is a direct summand of an extended one.

This terminology comes from relative homological algebra, for which the standard references are Eilenberg and Moore [l] and chapter IX of Mac Lane [l]. Our situation is dual to theirs in the following sense. We have the category r of left (or right) I'-comodules, the category A of A-modules, the forgetful functor G from I’ to A, and a functor F : A + r given by F ( M ) = r @ , M (A1.2.1). Mac Lane [ l ] then defines a resolvent pair to be the above data along with a natural transformation from G F to the identity on A, i.e., natural maps M + r O AM with a certain universal property. We have instead maps E 0M : r O AM + M such that for any A-homomorphism p : C + M where C is a r-comodule there is a unique comodule map a : C + r @ A M such that p = ( E @ M ) a . Thus we have what Mac Lane might call a coresolvent pair. Our F produces relative injectives while his produces relative projectives. This duality is to be expected because the example he had in mind was the category of modules over an algebra, while our category r is more like that of comodules over a coalgebra. The following lemma is comparable to theorem IX.6.1 of Mac Lane [l]. A1.2.8 Lemma.

( a ) If i : M + N is a monomorphism of comodules which is split over A, then any map f from M to a relatively injective comodule S extends to N. (If i is not assumed to be split, then this property would make S injective.) ( b ) I f M is projective as an A-module and S is a relatively injective comodule, then Cotor;.(M, S ) = O for i > O and i f S = r O AN then Cotor:-( M, S ) = M @ A N. Proof: f ) ( r @ j ) $ = g is a co(a) Let j : N + M be a splitting of i. Then (r@ module map from N to r OAS such that gi = $f:M + r OAS. It suffices then to show that S is a direct summand of r OAS, for then g followed by the projection of r O AS onto S will be the desired extension o f f : By definition S is a direct summand of r O AT for some A-module T. Let k : S + r OAT and k - I : r @ A T + S be the splitting maps. Then k - ' ( T @ E 0 T ) ( T @ k )is the projection of r O AS onto S. ( b ) One has an isomorphism 4: M O A N + M UT(rO AN ) given by 4( m @ n ) = $( m ) @ n. Since S is a direct summand of r OAN, it suffices to

322

Al. Hopf Algebras and Hopf Algebroids

replace the former by the latter. Let

O+N+Io+I'+*.

-

be a resolution of N by injective A-modules. Tensoring over A with r gives a resolution of r OAN by injective r-comodules. Cotor,(M, r OAN ) is the cohomology of the resolution cotensored with M, which is isomorphic to M

@A

Io+ M OAz'+. . . .

This complex is acyclic since M is projective over A. Compare the following with theorem IX.4.3 of Mac Lane [I].

A1.2.9 Lemma. ( a ) Let

O+Md-'-

p d ! + ...

O + N d -l -R o+?.!!-

RIdl,.

and

..

be long exact sequences of r-comodules in which each P' and R' is relatively injective and each map is split over A. Then a comodule map f : M + N extends to a map of long exact sequences. ( b ) Applying L 0,( .) (where L is a right r-comodule projective over A ) to the two sequences and taking cohomology gives Cotor, ( L , M ) and Cotor,(L, N ) , respectively. The induced map from the former to the latter depends only on f:

Proof: That the cohomology indicated in (b) is Cotor follows from A1.2.4 and A1.2.8(b). The proof of the other assertions is similar to that of the analogous statements about injective resolutions. Define comodules M' and N ' inductively by M o = M , N o = N, and M'+' and N'+' are the quotients in the short exact sequences O + M ' + p ' + M'+'+O

and O+ N ' + R ' + N ' + ' + o .

These sequences are split over A. Assume inductively that we have a suitable map from M' to N ' . Then Al.2.8(a) gives us f;: P' + R', and this induces a map from M'+' to thereby proving (a). For (b) it suffices to show that the map of long exact sequences is unique up to chain homotopy, i.e., given two sets of maps f;, f::P ' + R ' we need Conto construct h,: P ' + R'-' (with ho=O) such that h,+ld,+dl-,h,=if:. sider the commutative diagram

-

323

2. Homological Algebra

0

M i 4-1

, pi

4 +Mi+'

,0

where gi =fi: -f,!: Pi + R ' and we use the same notation for the map induced from the quotient Mi+'. Assume inductively that hi:Pi + R'-' has been constructed. Projecting it to N ' we get hi:Pi + N' with hidi-, = gi-'. Now we want a map : M i + ' + R' such that &+,di= gi - di-,hi. B y the exactness of the top row, hi+, exists iff (gi-di-,hi)di-l =O. But we have gidi-, - di-'( hidi-,) = gidi-' - d,-lgi-l = 0 , so exists. By A l. l. ll( a ) it extends from Mi+' to Pi+’giving the desired hi+'.

Li

Resolution of the above type serve as a substitute for injective resolutions. Hence we have A1.2.10 Definition. A resolution by relative injectives of a comodule M is a long exact sequence O + M + = R o + R 1 +.. * in which each R' is a relatively injective and each map is split over A. We now give an important example of such a resolution. A1.2.11 Definition. Let M be a left r-comodule. The cobar resolution DF( M ) is deJned by D ; ( M ) = r @ A rOsOAM, where r = ker E, with coboundary ds:D;( M ) + Of;"(M ) given by

-

d s ( y o Oy,O. * y , O m ) S

=

C

(-l)'yoO. . .y i - , O h ( yi)Oyi+,O. am

i=O

+(-l)'+'yoO~ . . y s O + ( m )

for y o € r, y , , . . . , ys E r, and m E M . For a right r-comodule L which is projective over A, the cobar complex C,*(L,M ) is LEIr D F ( M ) , so C;( L, M ) = L OAYo' OAM, where I@"denotes the s,fold tensor product of r over A. Whenever possible the subscript r will be omitted, and CT(A, M ) willbeabbreviatedtoC,*(M).TheelementaOy,O. .O y , O m ~Cr(L, M ) , where a E L, will be denoted by a y , y 2 ( .. 1 y,m. If a = 1 or m = 1, they will be omitted from this notation.

I

A1.2.12 Corollary. H ( C r ( L , M ) )= Cotorr(L, M ) ifL is projective over A, and H ( C,*(M ) )= Ext,(A, M ) . Prooj It suffices by A1.2.9 to show that Dr( M ) = Cr(T, M ) is a resolution of M by relative injectives. It is clear that D;( M ) is a relative injective and

324

A l . Hopf Algebras and Hopf Algebroids

that d" is a comodule map. To show that D J M ) is acyclic we use a contacting homotopy S : D;-(M ) + Dk--'(M ) defined by S ( y y , 1. . y p m )= E ( y)yIy2I . ysm for s > 0 and S( y m ) = 0. Then Sd + dS is the identity on D ; ( M ) for s > 0, and 1 - 4 on D F ( M ) , where c,b( -ym) = E ( y)m'm". Hence im4=M

for for

s>O s=O.

and R

Our next job is to define the external cup product in Cotor, which is a map Cotor,-(MI , NI)OCotor14M 2 , Nz) + Cotord M I OA Mz, N1 OA Nz) (see A1.1.2 for the definition of the comodule tensor product). If MI = M2= M and N, = N2 = N are comodule algebras (Al.l.2) then composing the above with the map in Cotor induced by M OAM + M and N OAN + N gives a product on Cotor,.( M, N ) . Let P t and P ; denote relative injective resolutions of N, and N?, respectively. Then PT OAPT is a resolution of N, OAN2. We have canonical maps Cotor,(M,, N,)OCotorr(M2, N 2 ) + H ( M IOr-P T O M 2 0 r P,*) (with tensor products over K ) and

MI

0 1 . PTO

M2 0 1 . P ; + (MI OA M2) 0,( P fOA P ; ) .

A1.2.13 Definition. The external cup product Cotorr(M,, N,)OCotorr-(Mz, N2)+ Cotorr(MI@ A M2, NI OAN 2 ) and the internal cup product on Cotor,( M, N ) for comodule algebras M and N are induced by the maps described above. Note that A1.2.9(b) implies that these products are independent of the choices made. Since the internal product is the composition of the external product with the products on M and N and since the latter are commutative and associative we have

A1.2.14 Corollary. If M and N are comodule algebras then Cotor,-(M , N ) is a commutative ( i n the graded sense) associative algebra. It is useful to have an explicit pairing on cobar complexes

C I ~ M N,)OC,.(M2, ,, Nz)+ Ci.(MiOMz, NiON2). This can be derived from the definitions by tedious straightforward calculation. To express the result we need some notation. For m, E M2 and n , E N, let m'O'

0.. .Om:"'€ M 2 0 AYo’

325

3. Some Spectral Sequences

and

n : ” @ . . . @ n:‘+I)Eyot @ A N , denote the iterated coproducts. Then the pairing is given by

where S

T=degm,degn,+

x degmy’ s - i +

i=O

S

1 deg yj j=i+l

r+t

i-I

+ i1 = l deg nii’( i - 1 +j 2 = l deg %+s Note that this is natural in all variables in sight. Finally, we have two easy miscellaneous results.

A1.2.16 Proposition. ( a ) If I c A is invariant (A1.2.21) then (A/ I, r/ I r) is a Hopf algebroid. ( b ) If M is a lefi F-comodule annihilated by I as above, then Extr(A, M ) = Extr/,r(A/I, M ) . Proof: Part (a) is straightforward. For (b) observe that the complexes Cr( M) and Cr,rr( M) are identical. H

A1.2.17 Proposition. If (A, r) is splir (A1.1.22) then Extr(A, M ) = Ext,(K, M ) where the left E-comodule structure on the left r-comodule M comes from the isomorphism r OAM = ZO M. Proof: C , ( M ) = C , ( M ) .

H

3. Some Spectral Sequences In this section we describe several spectral sequences (SSs) useful for computing Ext over a Hopf algebroid. The reader is assumed to be familiar with the notion of an SS; the subject is treated in each of the standard references for homological algebra (Cartan and Eilenberg [l], Mac Lane [ 13 and Hilton and Stammbach [ 13) and in Spanier [ 11. The reader is warned that most SSs can be indexed in more than one way. With luck the indexing used in this section will be consistent with that used in the text, but it may

326

A l . Hopf Algebras nod Hopf Algebroids

differ from that appearing elsewhere in the literature and from that used in the next two sections. Suppose we have a long exact sequence of r-comodules O + M + Rn&

A1.3.1 Let S"'

= im

R'

d'

R2+. ' .

d' and So= M so we have short exact sequences O - . S ' ~ R ' + Sb'i + l + ( l

for all i 2 0. Each of these gives us a connecting homomorphism 8': Cotor;:'( L, S ' ) + Cotor;-+'"( L, St-').

Let a(,):Cotor:?'( L, S ' ) + Cotork+'."(L, So) be the composition 6'S2 . . . 6'. Define a decreasing filtration on Cotor>*(L,M ) by F' = im a(,, for is s, where 6(o)is the identity and F' = 0 for i 5 0. A1.3.2 Theorem. Given a long exact sequence of r -comoduies A1.3.1 there is a natural trigraded spectral sequence (E*,**) such that

(a) (b) and d, (c)

En*".'=Cotor+'(L, R " ) ; d,: E A ' E:fr.s-r+l.l is the map induced by d* in A1.3.1; and E F ' is the subquotient F " / F n C 1of Cotor:+'"( L, M ) dejined above. +

Proof: We will give two constructions of this SS. For the first define an exact couple (2.1.6) by

E

= Cotor;"*(

L, R " ) ,

DS,' = Cotor,"*( L, S " ) ,

i, = S*, j , = a*, and k , = b*. Then the associated SS is the one we want. The second construction applies when L is projective over A and is more explicit and helpful in practice; we get the SS from a double complex as described in Cartan and Eilenberg [ 1, section XV.61 or Mac Lane [ 1 , section XI.61. We will use the terminology of the former. Let B".S.*

a;."*

= C",L,

R") (A1.2.11), = (-1)"C:.(d"): B"."-*+B"+lJ.*,

and

a,".V# = ds: B".".*

~

B".s+1.*

(Our dl, d2 correspond to the d,, d2 in Cartan and Eilenberg [l], IV.41.) Then a,n , s + l , * + d n , s + l , * a2"A* = 0 since d " commutes with C : ( d " ) .The

327

3. Some Spectral Sequences

associated complex (BP7*, d ) is defined by BP.* = @ B".'**= @ C;(L, R " )

A1.3.3

n+s=p

n+s=p

+

with a = 8, d2 :Ap3*+ AP+',*. This complex can be filtered in two ways, i.e.,

FPB=

@ @

Br'"*

mrp 4

F;,B = 8 8BP~S.* S?¶

P

and each of these filtrations leads to an SS. In our case the functor C;( L, 00) is exact since r is flat over A, so Hs’* FI,A= C;(t,M). Hence in the second

ss

and

The two SS converge to the same thing, so the first one, which is the one we want, has the desired properties. H A1.3.4 Corollary. The cohomology of the complex E** Cotor;E*(t, M ) .

of A1.3.3 is

Note that A1.2.4 is a special case of A1.3.3 in which the spectral sequence collapses. Next we discuss spectral sequences arising from increasing and decreasing filtrations of r.

A1.3.5 Definition. An increasing filtration on a Hopf algebroid ( A ,r) is an increasing sequence of sub-K-modules K = Fort F I T c F 2 r c . * .

with

r = U FsT such that

(a) FJ. F J - ~ FS+,r, ( b ) c(F,T)c FJ, and (c) A J T C @ ~ + ~ =Fpr ~ oAF q r . A decreasing filtration on ( A ,r) is a decreasing sequence of sub-K-modules

r = F o r I F ' r I F 2 r > * .*

328

A l . Hopf Algebras and Hopf Algebroids

n

with 0 = F T such that conditions similar to ( a ) , ( b ) ,and ( c ) above (with the inclusion signs reversed) are satisfied. A filtered Hopf algebroid ( A ,r) is one equipped with a filtration. Note that afiltration on induces one on A, e.g.,

F,A= ~ L ( A ) n F s T = ~ R ( A ) n FE(F,T). sT= A1.3.6 Definition. Let ( A ,I') be Jiltered as above. The associated graded object E a r (or E a r ) is defined by

or The graded object E i A (or E t A ) is deJined similarly. A1.3.7 Definition. Let M be a r-comodule. A n increasingfiltration on M is an increasing sequence of sub-K-modules

O=F,Mc FzMc * - . such that M = U FsM, F s A . F,M c F,+,M, and

$ ( F , M ) c @ FJOF,M. p+q=s

A decreasing Jiltration on M is similarly defined, as is the associated graded object E i M or E t M . A filtered comodule M is a comodule equipped with a Jiltrution. A1.3.8 Proposition. (EOA, E'T) or ( EoA, EoT) is a graded Hopf algebroid and E o M or EoM is a comodule over it. W Note that i f (A,I') and M are themselves graded than (EOA, E a r ) and EOM are bigraded. We assume from now on that E a r or E0r is flat over EOA or E,A. A1.3.9 Theorem. Let L and M be right and leftfiltered comodules, respectively, over a filtered Hopf algebroid (A,r). Then there is a natural spectral sequence converging to Cotor,( L, M ) such that ( a ) in the increasing case

E ~ * = C o t o r ~ o r ( E OEOM) L, where the second grading comes from the filtration and d,: E:f+ E sr+ l , r - r . 9

( b ) in the decreasing case

EI.* = Cotor&,( EoL, E,M)

329

3. Some Spectral Sequences

Note that our indexing differs from that of Cartan and Eilenberg [ 11 and Mac Lane [l].

Prooj The filtrations on r and M induce one on the cobar complex (A1.1.14) C r M and we have EoCr(L,M ) = C,r(E,L, E o M ) or EnCr(L, M ) = C E o r ( E'L, E ' M ) . The associated spectral sequence is the H one we want. The following is an important example of an increasing filtration. A1.3.10 Example. Let ( K , r)be a Hopf algebra. Let i.e., the quotient in the SES

r be the unit coideal,

0- K&I'+r+O.

The coproduct map A can be iterated by coassociativity to a map A':T+ r O s + 1. Let FsT be the kernel of the composition ~ O S + I TOSCI r This is the filtration of r by powers of the unit coideal. +

Next we treat the SS associated with a map of Hopf algebroids. A1.3.11 Theorem. Let f : ( A , r)-+ ( B ,C ) be a map of Hopf algebroids ( A l . l . l S ) ,M a right r-comodule and N a left Z-comodule.

( a ) C,(TOAB, N ) is a complex of left r-comodules, so B, N ) is a left r-comodule. Cotor,(r ( b ) If M is flat over A, there is a natural SS converging to Cotor,( M O AB, N ) with E;'

= Cotor;-( M, Cotori(T OAB,

N))

and d,: E:'+ E s, + r, f - r+ 1 ( c ) If N is a comodule algebra then so is Cotor,(r OAB, N ) . If M is also a comodule algebra, then the SS is one of algebras. Prooj For (a) we have C i (r OAB, N ) = r O AEBs Oe N with the coboundary d, as given in A1.2.11. We must show that d, commutes with the coproduct on r. For all terms other than the first in the formula for d, this commutativity is clear. For the first term consider the diagram A

-r r

@A

r-

r@A

mar r

ror

*r@AC

330

A l . Hopf Algebras and Hopf Algebroids

The left-hand square commutes by coassociativity and other square commutes trivially. The top composition when tensored over B with X@’ OsN is the first term in d,. Hence the commutativity of the diagram shows that d, is a map of left r-comodules. For (b) consider the double complex

c f ( M , C g ( r @ A B ,N ) ) , which is well defined because of (a). We compare the S S s obtained by filtering by the two degrees. Filtering by the first gives

E , = C : ( M , Cotorx(T OAB, N))

so E2 = Cotor,( M, Cotor,(

r OAB, N ) )

which is the desired SS. Filtering by the second degree gives an SS with

ET'

= Cotorl(M,

Ck(T OAB, N))

= Cotor,-( M,

r

=M

@A

OA

Pt0

ZO' 0B N

= C i ( M OAB,

sN

)

by A1.2.8(b)

N)

E2 = Em= Cotor,( M @ A B, N ) . For (c) note that r OAB as well as N is a Lcomodule algebra. The r-coaction on C,(TOAB, N ) is induced by the map

SO

C(AOB, N ) : c,(r

B, N ) + c,(r =

r

@A

r

B, N )

cX(r@ A B, N ) .

Since the algebra structure on C,( , ) is functorial, C(A@B, N ) induces an algebra map in cohomology and CotorI(TOA B, N ) is a r-comodule algebra. To show that we have an S S of algebras we must define an algebra structure on the double complex used in the proof of (b), which is M Or Dr(T OAB Ox D,( N)). Let fi = r OAB Oz D,( N). We have just seen that it is a r-comodule algebra. Then this algebra structure extends to one on Dr(fi) by A1.2.9 since Dr( fi) OADr( fi) is a relatively injective resolution of fi Oafi. Hence we have maps

M

&(

k)@ h f Or D1-(s)+ M @ A M ur &( fi)@ A &( fi) + M D , D , ( I ? ) O ~ D ~ ( I ? ) +M E I ~ . D , - ( I ? ) ,

which is the desired algebra structure.

331

3. Some Spectral Sequences

Our first application of this SS is a change-of-rings isomorphism that occurs when it collapses. A1.3.12 Change-of-Rings Isomorphism Theorem. Let f : (A, r)-+ (B,I;) be a map of graded connected Hopf algebroids (Al.l. 7 ) satisfying the hypotheses of A1.1.19; let M be a right r-comodule and let N be a left Z-comodule which isflat over B. Then Cotor,( M , (rOAB) 0,N ) = Cotor,( M OAB, N ).

In particular Extr(A,

(r@A

B) CIz N)=EXtx(B, N ) .

ProoJ: By A1.1.19 and A1.2.8(b) we have Cotort(T OAB,N ) = 0

for

s > 0.

A1.3.11(b) gives

Cotor,( M, Cotori(r OAB, N)) = Cotor,( M OAB,N). Since N is flat over B, Cotori(T O AB, N ) = (rOAB)0,N and the result follows. A1.3.13 Corollary. Let K be a field and f:( K , r)-+ ( K , Z) be a surjective map of Hopf algebras. If N is a left Z-comodule then Extr(K, r 0,N ) = Ext,:(K, N ) . Next we will construct a change-of-rings SS for an extension of Hopf algebroids (A1.1.15) similar to that of Cartan and Eilenberg [l, XVI 6.11, which we will refer to as the CESS. A1.3.14 Cartan-Eilenberg Spectral Sequence Theorem. (0,@)

Let

--L(A, r)f ,(A, Z)

be an extension of graded connected Hopf algebroids (A1.1.15). Let M be a right r-comodule and N a lefi r-comodule. ( a ) Cotor,(A, N ) is a left @-comodule. If N is a comodule algebra, then so is this Cotor. ( b ) There is a natural SS converging to Cotor,( M O D A , N) with

E;’ = Cotor$( M , Cotori(A, N ) ) and d,: E;'

-+

E;+r,f--r+l

332

A l . Hopf Algebras and Hopf Algebroids

(c)

If M

and N are comodule algebras, then the SS is one of algebras.

Proof: Applying A1.3.11 to the map i shows that Cotor,(@ @,A, N ) is a left @-comodule algebra and there is an SS converging to Cotor,( M O o A, N ) with

E2= Cotor,( M, Cotor,(@ Oa0, N ) ) . Hence the theorem will follow if we can show that Cotor,-(@O DA, N ) = Cotor,(A, N ) . Now @ O DA = A Ox r by A1.1.16. We can apply A1.3.12 to f and get Cotorr( P 0,r, R 1 = Cotor,( P, R ) for a right Z-comodule P and left r-comodule R. Setting P = A and R = N gives the desired isomorphism Cotor,.(@ @,A, N ) = Cotor,-(A 0,r, N ) = Cotor,(A, N ) . The case M

=D

gives

A1.3.15 Corollary. With notation as above, there is a spectral sequence of algebras converging to Ext,-(A, N ) with E2 = Ext@(0, Ext,(A, N ) ) . Now we will give an alternative formulation of the CESS (A1.3.14) suggested by Adams [12, 2.3.11 which will be needed to apply the results of the next sections on Massey products and Steenrod operations. Using the notation of A1.2.14, we define a decreasing filtration on Cr( M ODA, N ) by saying that m y , 1. . ysn E F’ if i of the y’s are in kerf’.

A1.3.16 Theorem. The SS associated with the above Jiltration of C r ( M O D A , N ) coincides with the CESS of Al.3.14.

ProoJ: The CESS is obtained by filtering the double complex C $ ( M , C,*(X0,A, N ) ) by the first degree. We define a filtration-preserving map 0 from this complex to C,( M @,A, N ) by 0(mO&@.

4 s 0 4 0 y s + 1 0 ?. ,~+~, O n )

= m O i l ( d l ) O .. ~

i l ( A ) i l ~ ( ~ ) O y. s- Y+cl+O, O~n .

Let E: ( M, N ) = Ck( M, Cotor;-(@0, A, N ) )= C&(M , Cotorh(A, N ) ) be the El-term of the CESS and E , ( M , N ) the El-term of the SS in question. It suffices to show that

o*: E,(M, N ) + E ~ ( M N, ) is an isomorphism. First consider the case s = 0. We have

F o / F ’ = C z ( M O D A ,N ) =M

0,C,(A, N )

333

3. Some Spectral Sequences

so this is the target of 8 for s = 0. The source is M O DCr(@O DA, N ) . The argument in the proof of Theorem Al.3.16 showing that Cotor,(@ O DA, N ) = CotorZ(A, N ) shows that our two complexes are equivalent so we have the desired isomorphism for s = 0. For s > 0 we use the following argument due to E. Ossa. The differential d o :E ; ' ( M , N ) + E;'+'

(M, N)

depends only on the Z-comodule structures of M and N. In fact we may define a complex fi,(N) formally by

d:'(N ) = &'(Z,

N).

Then we have

I?;'( M,N ) = M Ox &'( N ) . Observe that &'(

N ) = cgz, N )

Now let G = kerf and

with s + 1 factors. Note that

as left Z-comodules, where the tensor products are over D. Define

ps: G" Usfi?’( N )

-

&'(N)

by

Ps((glO.. .g,)Oa,O- * .Oa,On) = c f ( g ; ) g ; O g , .. .Og,Oa,O. * -Oa,On.

334

At. Hop( Algebras and Hopf Algebroids

Then ps is a map of differential Z-comodules and the diagram

= E $ ‘ ( M O P ” ,N )

&‘(M, N )

p.

e’*‘

I??‘(M@@@”,N )

I?:‘(M, N )

M Ox dl’(N)

p, M Or G‘ [SIX dl‘(N)

commutes. We know that 6”‘ is a chain equivalence so it suffices to show that ps is one by induction on s. To start this induction note that Po is the identity map by definition. Let N ) = Fq+,(r, N) and

mr,

Fs.’(T,N ) = F”’(G,N ) + Fs+l*‘-l(I‘, N) = F‘q‘(r,N ) .

Then Fs3*(r,N) is a Z-comodule subcomplex of CI-(l-,N ) which is invariant under the contraction S(yOyl... ysOfl)=E(y)@y,... ysOn.

Since Ho(F’, *(I , N ) )= 0, the complex F”*(T,N ) is acyclic. Now look at the SES of complexes

0-

FS+I(T,N )

-

Fyr, N )

Fm+l(r, N)

-

Fyr, N )

FS+I(T,N )

-0

J.

G Oxd;( N )

d:+y N )

The connecting homomorphism in cohomology is an isomorphism. We use this for the inductive step. By the inductive hypothesis, the composite

G O ~ ( Gox ” N ) + G U ~ ( Gor ” D;(N)) +

G O r D;( N )

is an equivalence. If we follow it by 4 d$ we get &+,. This completes the inductive step and the proof. H

335

4. Massey Products

A1.3.17 Theorem. Let @ + r + Z be a cocentral extension (A1.1.15)ofHopf algebras over a j e l d K ; M a left @-comodule and N a trivial left r-comodule. Then Ext(K, N ) is trivial as a left @-comodule, so the CESS (A1.3.14) &term is Ext@(M,K)OExt,(K, K ) O N .

Pro08 We show first that the coaction of @ on Ext,(K, N ) is essentially unique and then give an alternative description of it which is clearly trivial when the extension is cocentral. The coaction is defined for any (not necessarily trivial) left r-comodule N. It is natural and determined by its effect when N = r since we can use an injective resolution of N to reduce to this case. Hence any natural @-coaction on Ext,(K, N ) giving the standard coaction on ExtZ(K, r) = @ must be identical to the one defined above. Now we need some results of Singer [5]. Our Hopf algebra extension is a special case of the type he studies. In proposition 2.3 he defines a @-coaction on Z, pn:Z + @ OZ via a sort of coconjugation. Its analog for a group extension N + G + H is the action of H on N by conjugation. This action is trivial when the extension is central, as is Singer’s coaction in the cocentral case. The following argument is due to Singer. Since Z is a @-comodule it is a r-comodule so for any N as above Z O KN is a r-comodule. It follows that the cobar resolution D,N is a differential r-comodule and that Hom,(K, D , N ) is a differential comodule over Horn,( K , r)= @. Hence we have a natural @-coaction on Ext,(K, N ) which is clearly trivial when N has the trivial r-comodule structure and the extension is cocentral. It remains only to show that this @-coaction is identical to the standard one by evaluating it when N = r. In that case we can replace D, N by N, since N is an extended Z-comodule. Hence we have the standard r-coaction on r inducing the standard @-coaction on Horn,( K , r)= @. W

4. Massey Products In this section we give an informal account of Massey products, a useful structure in the Ext over a Hopf algebroid wheich will figure in various computations in the text. A parallel structure in the ASS is discussed in Kochman [4 and 2, section 121. These products were first introduced by Massey [3], but the best account of them is May [3]. We will give little more than an introduction to May’s paper, referring to it for all the proofs and illustrating the more complicated statements with simple examples.

336

A l . Hopf Algebras and Hopf Algebroids

The setting for defining Massey products is a differential graded algebra (DGA) C over a commutative ring K . The relevant example is the cobar complex C,(L, M ) of A1.2.11,where L and M are r-comodule algebras and r is a Hopf algebroid (Al.l.1)over K. The product in this complex is given by A1.2.15. We use the following notation to keep track of signs. For X E C, let X denote (-l)’+deg 'x, where deg x is the total degree of x ; i.e., if C is a complex of graded objects, deg x is the sum of theinternal and cohomological degrees of x. Hence we have d ( x ) = - d ( x ) , ( x y ) = - X j J , and d ( x , v ) = d ( x ) y= Xd(y). Now let a , E H * ( C ) be represented by cocycles a, E C for i = 1, 2, 3. If aiai+,= 0 then there are cochains u, such that d ( u j )= d , a i + l ,and O l a3+ 6 ,u2 is a cocycle. The corresponding class in H * ( C ) is the Massey product ( a 1 , a 2 , a 3 ) . I f a , ~ H s ~ t h e n t h i sa*, ( a a,}fH"-',wheres=Zs,.Unfortul, nately, this triple product is not well defined because the choices made in its construction are not unique. The choices of a, d o not matter but the u, could each be altered by adding a cocycle, which means ( a l ,a > ,a 3 )could be altered by any element of the form x a , + a , y with X E Hsi+r2-1and Y E HSz+"3-'.The group is . called the indeterminacy, denoted by In(a,, a ? ,a3).It may be trivial, in which case ( a , , a ? , a,} is well defined. A1.4.1 Definition. With notation as above, (a,, a2,a3)c H"(C ) is the coset of In(a,, a 2 ,a3) represented by 6,u2+ii,a3. Note that ( a , , a 2 ,a 3 )is only deJined when (Y,a2= E 2 a , = 0. This construction can be generalized in two ways. First the relations aiGi+,= 0 can be replaced by m

1 ((Yl)j(a2),,k=O

for

l s k ~ n

j=l

and n

(&Z),.k((Y,)k=O

for l s j l m .

k=l

Hence the ai become matrices with entries in H * ( C). We will denote the set of matrices with entries in a ring R by MR. For x E MC or M H * ( C), define P by ( f ) j , k = % j , k . As before, let a, E M C represent a; E MH*( C ) and let u1E M C be such that d ( u i )= ci,a,+, . Then u Iand u2 are (1 x n)- and (rn x 1)-matrices, respectively, and 5 , u 2 + 0 , a 3 is a cocycle (not a matrix thereof) that represents the coset ( a , , a z ,a3).

337

4. Massey Products

Note that the matrices a, need not be homogeneous (i.e., their entries need not all have the same degree) in order to yield a homogeneous triple product. In order to multiply two such matrices we require that, in addition to having compatible sizes, the degrees of their entries be such that the entries of the product are all homogeneous. These conditions are easy to work out and are given in 1.1 of May [3]. They hold in all of the applications we will consider and will be tacitly assumed in subsequent dejinitions.

A1.4.2 Definition. With notation as above, the matric Massey product ( a 1a , z , a 3 )is the coset of In(a,, a 2 ,a3)represented the cocycle ii,u,+zi,a,, where In(@,, a 2 ,a3)is the group generated by elements of the form xa3+ a l y where x, y E MH*( C ) have the appropriate form. The second generalization is to higher (than triple) order products. The Massey product ( a l ,a 2 , .. . , a,,) for a iE MH*( C ) is defined when all of the lower products (a,, a;+,,. . . , a,) for 1 Ii <j 5 n and j - i < n - 1 are defined and contain zero. Here the double product ( a , ,a,+,)is understood ~. a,+,,, be a matrix of cocycles representto be the ordinary product ( Y , ( Y ~ +Let with d(ai-l,i+l)= fii-l,iai,i+l. ing ai Since ( ~ , a ;=+0~there are cochains Then the triple product ( a , , is represented by bi-l,i+2= fii-l,i+lai+l,i+l+ u i - l , i a j , i + Since 2. this triple product is assumed to contain zero, the above choices can be made so that there is a matrix of cochains whose coboundary is bi+l,i+2. Then the fourfold product ( a , , a 2 ,a 3 ,a4)is represented by the cocycle E0,3u3,4+ 6 0 , 2 ~ 2 , 4 + i i o , , ~ , . 4 .More generally, we can choose elements a , , and bj,j by induction on j - i satisfying b,,j = & < k < j i i i , k a k , j and d ( a i , j = ) b,., for i - j < n - 1. A1.4.3 Definition. The n-fold Massey product ( a , , a z , . . . , a,) is dejined when all of the lowerproducts ( a , ,. . . , a j )contain zerofor i <j andj - i < n - 1 . It is strictly defined when these lower products also have trivial indeterminacy, e.g., all triple products are strictly dejined. In either case the matrices a,,j chosen above for 0 5 i <j 5 n and j - i < n constitute a defining system for the product in question, which is, modulo indeterminacy ( t o be described below), the class represented by the cocycle

C

&,iai,n,

O*

. . ,a , - 3 ,

Zf ( a l , .. . ,a,-2, an) and (a17 . .

2 ~ ~ a3 43 , . . . ,

is defined, then so is ( a l , .

., an-,,5 n - I a n ) c

*.

*

7

an-39

an-zan-1,an).

. . . a k - 1 , @ k a k + l , a k + 2 , . . an) and ( a , , . . ., a k r & k + l a k + 2 , (f) r f ajui+3, . . . , a,) are strictly defined, then the intersection of the former with minus 9

a ,

the latter is nonempty.

Now we come to some commutativity formulas. For these the DGA C must satisfy certain conditions (e.g., the cup product must be commutative) which always hold in the cobar complex. We must assume (if 2 f 0 in K ) that in each matrix ai the degrees of the entries all have the same parity E~ ; i.e., E~ is 0 if the degrees are all even and 1 if they are all odd. Then we define A1.4.7

s(i,j ) =j - i +

C

(1

+ E ~ ) 1( +

E,)

irksrnsj

and k

t(k)=(l+E1)

1(l+Ej). j=2

The transpose of a matrix a will be denoted by a’. The following result is 3.7 of May [3].

A1.4.8 Second Juggling Theorem. Let ( a l ,. . . , a,) be dejned and assume that either 2 = 0 in K by the degrees of all of the entries of each ai have the

340

A l . Hopf Algebras and Hopf Algebroids

same parity

Then ( ak , . . . , a I) is also dejned and

E~.

( a1 , . . . , a,)' = (

- 1 ) S ( y a : ,

, . . . , a ;).

I

(For the sign see A1.4.7.)

The next result involves more complicated permutations of the factors. In order to ensure that the permuted products make sense we must assume that we have ordinary, as opposed to matric, Massey products. The following result is 3.8 and 3.9 of May [3]. A1.4.9 Third Juggling Theorem. Massey product.

Let ( a l ,. . . , a,) be dejined as an ordinary

( a ) If ((Yk+l,.. . , a,, a I ,. . . , (Yk) is strictly dejned for 1 5 k < n, then k=l

( b ) If ( a ~. .,. , a k ,

ak+I,

. . . ,a n )is strictly de$ned for 1 Ik In then

n ((Yl,.

..

-

(-l)'(k)(a21... , a k ,

01, ( Y k + l , .

k=2

(For the signs see A1.4.7.)

.. ,a n ) .

I

Now we consider the behavior of Massey products in spectral sequences. In the previous section we considered essentially three types: the one associated with a resolution (A1.3.2), the one associated with a filtration (decreasing or increasing) of the Hopf algebroid r (A1.3.9), and the CESS associated with an extension (A1.3.14). In each case the SS arises from a filtration of a suitable complex. In the latter two cases this complex is the cobar complex of A1.2.11 (in the case of the CESS this result is A1.3.16), which is known to be a D G A (A1.2.14) that satisfies the additional hypotheses (not specified here) needed for the commutativity formulas A1.4.8 and A1.4.9. Hence all of the machinery of this section is applicable to those two spectral sequences; its applicability to the resolution SS of A1.3.2 will be discussed as needed in specific cases. To fix notation, suppose that our D G A C is equipped with a decreasing filtration { F P C }which respects the differential and the product. We do not require F°C = C, but only that limp+-mF P C = C and limp+- FPC=O. Hence we can have an increasing filtration { FpC}by defining F,C = F-PC. Then we get a spectral sequence with E ; 4 = FPCP+¶/FP+1CP+q 9

€79

= HP+q(FP/FP+'),

d,: E,P.¶_,E P + r . q - r + l

34 1

4. Massey Products

and E&4

= FPHP+4/FP+2HP+qa

We let E t 2 c E,P.¶ denote the permanent cycles and i : EF:+ E Z 4 and r r : F P C P + '+ EFq the natural surjections. If x E E and y E F P Hp + q projects to i ( x ) E EGq we say that x converges to y . If the entries of a matrix B E MC are all known to survive to E,, we indicate this by writing rr(B)E ME,. In the following discussions a , will denote an element in M E , represented by a, E MC. If a , E M E , , , P I E M H * ( C ) will denote an element to which it converges. Each E, is a DGA in whose cohomology, E,,,, Massey products can be defined. Suppose ( a , ,. . . , a,) is defined in E,+, and that the total bidegree of the a, is (s, t ) , i.e., that the ordinary product a I a Z ... a , (which is of course zero if n 2 3 ) lies in E::, . Then the indexing of d, implies that the Massey product is a subset of E : T ~ ( n ~ 2 ) , r + ( r - ' ) ( n - 2 ) . May's first SS result concerns convergence of Massey products. Suppose that the ordinary triple product (PI,P z , P 3 ) c H * ( C ) is defined and that ( a 1a , z , a 3 )is defined in E,+, . Then one can ask if an element in the latter product is a permanent cycle converging to an element of the former product. Unfortunately, the answer is not always yes. To see how counterexamples can occur, let 6, E E, be such that d,( 6 , )= a p t + , .Let ( p , q ) be the bidegree of one of the 6#.Since ( P I ,p2,p3) is defined we have as before u, E C such that d ( u , ) = a , a , + , .The difficulty is that a,a,+,need nor be a coboundary in F P C ; i.e., it may not be possible to find a U , E F P C . Equivalently, the best possible representative C,E F P C of 12, may have coboundary iiralil- e, with 0 # .rr(e,)E E pfr7q-ft1 for some t > r. Then we have d ( u, - C,)= e, and rr( u, - 6 , )= ~ ( u ,E )E ;;yq+" for some m > 0, so dm+,(rr(u , ) )= r r ( e , ) . In other words, the failure of the Massey product in E,+, to converge as desired is reflected in the presence of a certain higher differential. Thus we can ensure convergence by hypothesizing that all elements in EL;?$m for m 2 0 are permanent cycles. The case m = 0 is included for the following reason (we had m > 0 in the discussion above). We may be able to find a u, E F P C with d ( u , )= &a,+, but with .rr(u,)# u , , so d,( T ( U - G,))= r r ( e , )f 0. In this case we can find a convergent element in the Massey product in E,+r, but it would not be the one we started with. The general convergence result, which is 4.1 and 4.2 of May [3], is

tz

A1.4.10 Convergence Theorem. ( a ) With notation as above let ( a l r . .. ,a,) be dejned in E,,,. Assume that a, E ME,+I,, and a , converges to p, where ( P I , .. . , P,) is strictly dejned in H*( C ) . Assumefurther that if ( p , q ) is the bidegree of an entry of some

342

Al. Hopf Algebras and Hopf Algebroids

a,., ( f o r 1 <j - i < n ) in a defining system f o r ( a , , . . . , a,) then each element

in EFi,"$"' for all m 2 O is a permanent cycle. Then each element of ( a l , . . . , a,) is a permanent cycle converging to an element of ( P I , .. . , pn). ( b ) Suppose all of the above conditions are met except that ( a l , .. . , a,,) is not known to be defined in E,,,. If f o r ( p, q ) U S above every element of ErL2q+mf o r m 2 1 is a permanent cycle then ( a l ,. . . , a,) is strictly dejned so the conclusion above is valid.

The above result does not prevent the product in question from being hit by a higher differential. In this case (PI,. . . ,P,,)projects to a higher filtration. May's next result is a generalized Leibnitz formula which computes the differential on a Massey product in terms of differentials on its factors. The statement is complicated so we first describe the simplest nontrivial situation to which it applies. For this discussion we assume that we are in characteristic 2 so we can ignore signs. Suppose ( a , ,a z , a& is defined in E,+, but that the factors are not necessarily permanent cycles. We wish to compute dr+l of this product. Let a, have bidegree ( p , , q, 1. Then we have u, E FP,+Pf+l-'C with d ( u , )= a,a,+, mod FP,tP~+lf'C. The product is represented by u,a,+ alu2. Now let d ( a , ) = a : and d ( u , ) = a , a , + , + u : . Then we have d ( u , a , + alu2)= u;a,+ ulai + a',u2+a z u i . This expression projects to a permanent cycle which we want to describe as a Massey product in Consider

Since d ( u , ) = a , a , + I + u : is a cycle, we have d ( u : ) ) = d ( a , u , + , ) = a ~ a , + , + a,a:+,, so dr(7r(u:))= d , + l ( a , ) a , + I +a,d,+l(a,+I).It follows that the above product contains 7r(u;a,+ u,a;+a:u,+a,u;)E E,+(. Hence we have shown that

We would like to show more generally that for s > r with d , ( a , )= O for r < t < s, the product is a d,-cycle and d, on it is given by a similar formula. As in A1.4.10, there are potential obstacles which must be excluded by appropriate technical hypotheses which are vacuous when s = r + 1. Let ( p , q ) be the bidegree of some u,. By assumption u : E F p + r + l Cand d ( u : ) = Q:LI,+~+ a,a:+, . Hence 7r(a,a:+,+ a:a,+,)E E $ + r f ' * q - r - F +is2killed by a d,+,-, for r < t s s . If the new product is to be defined this class must in fact be hit by a d, and we can ensure this by requiring E'";?;::: = 0 for r < t < s. We also need to know that the original product is a d,-cycle for r < t < s . This may not be the case if ~ ( u :#)O E Ef+r9q-r+1 for r < f < s, because then we could not get rid of ~ ( u :by ) adding to u, an element in F P + ' Cwith '

343

4. Massey Products

coboundary in F p + r + l C(such a modification of u, would not alter the original Massey product) and the expression for the Massey product's coboundary could have lower filtration than needed. Hence we also require E p+l*q-'+l = 0 for r < t < s. We are now ready to state the general result, which is 4.3 and 4.4 of May ~31. A1.4.11 Theorem (Leibnitz Formula).

( a ) With notation as above let ( a , , a2,.. . , a,,) be defined in E,+l and let s > r be given with d,( a i )= 0 f o r all t < s and 1 5 i 5 n. Assume further that f o r ( p , q ) as in A1.4.10 and f o r each t with r < t < s , EP+f,q-1+1 1

-

-0

and

p + 1 q-1+1-

E , + , L ~ -0

( f o r each t one of these implies the other). Then each element a of the product isfa d,-cycle f o r r < t < s and there are permanent cycles a:E ME,+I,, which survive to d , ( a i ) such thaf ( y l , . . . , y,,) is defined in E,+l and contains an element y which survives to - d s ( a ) , where for

l,a , ) ) = r ( u ~ a , + a , u ; )Here . u: represents the product E H * ( C ) , where P, E H * ( C )is the class represented by a,. The product PIP,+,has filtration greater than the sum of those of p, and PI+,,and the target of the differential +PI(PzP3). represents the associator (PIPz)P3 Next we generalize by replacing r + 1 by some s > r; i.e., we assume that the filtration of PIP,+,exceeds the sum of those of P I and by Y - r. As in the previous result we need to assume

this condition ensures that the triple product is a d,-cycle. The general theorem has some hypotheses which are vacuous for triple products, so in order to illustrate them we must discuss quadruple products, again ignoring signs. Recall the notation used in definition A1.4.3. The elements in the defining system for the product in E,,, have cochain representatives corresponding to the defining system the product would have if it were defined in H * ( C). As above, we denote a,-,., by a,, a , - , ,+, by u,, and also a,_,,,+, by u,. Hence we have d ( a , ) = O , d ( u , ) = a , a , + , + u : , d ( u , ) = a,u,+,+ u,a,+,+ 0:, and the product contains an element a represented by m = a,u,+ u , u , + u,a4, so d ( m ) = a,u;+ u ; u , + u , u : + u ; a 4 . We also have d ( u : ) = O and d ( u : ) = u : a , + z + a , u : + , . We are assuming that ( P I ,p 2 ,P 3 ,p4)is not defined. There are two possible reasons for this. First, the double products PIPt+,may not all vanish. Second, the double products all vanish, in which case u : = O , but the two triple products ( P I ,P , + , ,p,+z>must not both contain zero, so u : # 0. More generally there are n -2 reasons why an n-fold product may fail to be defined. The theorem will express the differential of the n-fold product in E r + , in terms of the highest order subproducts which are defined in H * ( C). We will treat these two cases separately. Let ( p , , 4 , ) be the bidegree of a,. Then the filtrations of u , , u,, and m are, respectively, p , + p I + ' - r , p , + ~ , + ~ + p , + ~ - and 2 r , p1+p2+p3+p4-2r. Suppose the double products do not all vanish. Let s > r be the largest integer such that each u : has filtration 2 s - r + p , + p l + , . We want to give conditions which will ensure that ( a , , a,, a ? ,a4)is a d,-cycle for r < f < s and that the triple product

is defined in E,+, and contains an element which survives to & ( a ) ;note

4. Massey Products

345

that if all goes well this triple product contains an element represented by d ( m ) . These conditions will be similar to those of the Leibnitz formula A1.4.11. Let ( p , q ) be the bidegree of some u,. As before, we ensure that d , ( a )= 0 by requiring E:+'3q-'f', and that the triple product is defined in E,,, by requiring EfZ:YYf+' = 0. The former condition is the same one we made above while discussing the theorem for triple products, but the latter condition is new. Now we treat the case when the double products vanish but the triple products do not. First consider what would happen if the above discussion were applied here. We would have s = co and a would be a permanent cycle provided that EYf'.q-'+l- 0 for all t > r. However, this condition implies that u can be chosen so that v' = 0, i.e., that the triple products vanish. Hence the above discussion is not relevant here. Since u : = 0, the coboundary of the Massey product m is a,v;+ via4. Since d ( u , )= a,u,+,+ u,a,+,+ u : , v: is a cocycle representing an element of (PI,P,+1,Pl+2).Hence if all goes well we will have d , ( a ) = a l r (u ; ) + r(v : ) a 4 ,where s > r is the largest integer such that each u : has filtration at least p , + p , + , + P ~ + ~ s -+2r. To ensure that d , ( a )= 0 for t < s, we require E:+~,~-'+' - 0 for r < t < s as before, where ( p , q ) is the degree of v z .We also need to know that ( a , ,a , + l ar+J , converges to ( P I , Plt2); since the former contains zero, this means that the latter has filtration greater than pI+pl+1+pt+2-r.We get this convergence from A1.4.10, so we must require that if ( p , q ) is the bidegree of r(ul), then each element of E!;,";4:"+' for all m 2 0 is a permanent cycle. Now we state the general result, which is 4.5 and 4.6 of May [3]. A1.4.12 Differentials and Extensions Theorem.

( a ) With notation as above, let ( a l , .. . ,a,)be dejined in E,+, where each a, is a permanent cycle converging to P, E H*( C ) . Let k with 1 Ik In - 2 be is strictly dejined in H * ( C ) and such that if such that each ( P I , .. . , ( p , q ) is the bidegree of an entry of some a,, f o r 1 <j - i Ik in a dejining system f o r ( a l ,. . . , a,,) then each element of E ::;q,+" for all m 2 0 is a permanent cycle. Furthermore, let s > r be such that for each ( p , q ) as above with k <j - i < n and each t with r < t < s, E : + f , q - - L + l - 0 , and, i f j - i > k + l , E -y ,;;; + 1 = 0. Then for each a E ( a 1 ,... , a,), d, ( a )= 0 f o r r < t < s, and there are permanent cycles 6, E M E , + I , , f o r 1 Ii 5 n - k which converge to elements of ( P I ,. . . ,p t + k ) cH * ( C ) such that ( y l , . . . , m - k ) is defined in Eril and contains an element y which suruiues to - d y ( a ) , where

346

A l . Hopf Algebras and Hopf Algebroids

and

( b ) Suppose in addition to the above that each 6, is unique, that each

( a , ,. . . , E l - , , a,, a ) , + k + l , . . . ,a,) isstrictlydejinedin E , + ] ,andthatallMassey

products in sight (exceptpossibly (PI, . . . , P l + k ) ) havezero indeterminacy. Then n-k

d s ( ( a i , .. . , a n ) ) =

C

,=I

( G I , . . . ,G , - I ,

6, a t + k + l , . . . ,a n > .

Note that in (b) the uniqueness of 6, does not make ( P I , .. . ,P , + k ) have zero indeterminacy, but merely indeterminacy in a higher filtration. The theorem does not prevent 6, from being killed by a higher differential. The requirement that E,4;mm;4+m=E , + , + , , , is vacuous for k = 1, e.g., if n = 3 . The condition E;If.!;'+' = 0 is vacuous when k = n -2; both it and E f + t * q - r +l 0 are vacuous when s = r + 1. A1.4.13 Remark. The above result relates differentials to nontrivial extensions in the multiplicative structure (where this is understood to include Massey product structure) since 6, represents ( P I ,. . . ,P , + k ) but has filtration greater than that of (a,,. . . ,a , + k ) . The theorem can be used not only to compute differentials given knowledge of multiplicative extensions, but also vice versa. If d , ( a ) is known, the hypotheses are met, and there are unique 6, which fit into the expression for y, then these 6, necessarily converge to

..

(Pi$.

7

Pi+k)-

5. Algebraic Steenrod Operations In this section we describe operations defined in Cotor,(M, N ) , where

r is a Hopf algebroid over Z/( p ) for p prime and M and N are right and left comodule algebras (A1.1.2) over r. These operations were first introduced by Liulevicius [2], although some of the ideas were implicit in Adams [12]. The most thorough account is in May [S], to which we will refer for most of the proofs. Much of the material presented here will also be found in Bruner et a/. [ 11; we are grateful to its authors for sending us the relevant portion of their manuscript. The construction of these operations is a generalization of Steenrod's original construction (see Steenrod [ 11) of his operations in the mod ( p ) cohomology of a topological space X . We recall his method briefly. Let G = Z/( p ) and let E be a contractible space on which G acts freely with orbit space B. X p denotes the p-fold Cartesian

347

5. Algebraic Steenrod Operations

product of X and X p x G E denotes the orbit space of X p x E where G acts canonically on E and on X p by cyclic permutation of coordinates. Choosing a base point in E gives maps X +. X x B and X +. X x E. Let A : X + X p be the diagonal embedding. Then there is a commutative diagram

x

1

A

X x B-XP

xp

I

xGE

Given x E H * ( X ) [all H * groups are understood to have coefficients in Z/(p)] it can be shown that ~ 0 x 0 .~ X H E * ( X P )pulls back canonically to a class P ~ HE* ( X Px G E ) . We have H ' ( B )= Z / ( p ) generated by e, for each i 2 0 . Hence the image of Px in H * ( X x B ) has the form C l z 0x,Oe, with x, E H*( X ) and xo= x p. These x, are certain scalar multiples of various Steenrod operations on x. If C is a suitable DGA whose cohomology is H * ( X ) and W is a free R-resolution (where R = Z/ ( p ) [ GI) of Z/ ( p ) , then we get a diagram

c

I

-

c,

I

C O RW t - C P O R W where C, is the p-fold tensor power of C, R acts trivially on C and by cyclic permutation on C,, and the top map is the iterated product in C. It is this diagram (with suitable properties) that is essential to defining the operations. The fact that C is associated with a space X is not essential. Any DGA C which admits such a diagram has Steenrod operations in its cohomology. The existence of such a diagram is a strong condition on C ; it requires the product to be homotopy commutative in a very strong sense. If the product is strictly commutative the diagram exists but gives trivial operations. In 11.3of May[5] itisshownthatthecobarcomplex(A1.2.11)C,(M, N ) , for M , N as above and a Hopf algebra, has the requisite properties. The generalization to Hopf algebroids is not obvious so we give a partial proof of it here, referring to Bruner et al. [l] for certain details. We need some notation to state the result. Let C = C,(M, N ) for a Hopf algebroid over K (which need not have characteristic p ) and M, N comodule algebras. Let C, denote the r-fold tensor product of C over K . Let 7~ be a subgroup of the r-fold symmetric group C, and let W be a negatively graded K[~r]-freeresolution of K . Let 7r act on C, by permuting

348

A l . Hopf Algebras and Hopf Algebroids

the factors. We will define a map of complexes 0: WOK,,,

c,+C

with certain properties. We define 0 by reducing to the case M = r, which is easier to handle because the complex D = Cr(T, N ) is a r-comodule with a contracting homotopy. We have C = M Or D and an obvious map j : WOK,,, C,-,Mr01-(WOK,,,Dr),

where the comodule structure on W OK[,,D, is defined by

+ ( w @ d , * . . @ d , ) = d { d S * .. d : O w O d ~ @ .. .d: for w E W, di E D, and C ( d , )= d : O dy, and the comodule structure on M , is defined similarly. Given a suitable map

6:WOKl,,D,+D, we define 0 to be the composite ( p 0 6)j, where p : M r + M is the product.

A1.5.1 Theorem. With notation as above assume W, = K [T ] with generator e,. Then there are maps 0, e' as above with the following properties. (i) The restriction of 0 to e , O C, is the iterated product (A1.2.15) C, + C. ( i i ) 0 is natural in M , N , and r up to chain homotopy. (iii) The analogs of ( i ) and ( i i )for e'characterize it up to chain homotopy. ( i v ) Let A : W -+ W O W be a coassociative diferential coproduct on W which is a K [ ~ ] - m a (where p K ( v ] acts diagonally on W @ W, i.e., given a E T , and w, , w2E W, a ( w ,O w 2 ) = a ( w l )0a ( wz));such coproducts are known to exist. Let p : C O C + C be theproduct of ,412.15. Then the following diagram commutes up to natural chain homotopy. WOK,,, ( C O C ) ,

where T is the evident shuffle map.

W@Pr

WOK,,,

c,

349

5. Algebraic Steenrod Operations

= X p 2 and let r be the split extensions of v P by 7~ = v = Z/( p ) , in which 7~ permutes the factors of u p . Let W, V , and Y be resolutions of K over K [ T T ] , K [ u ] , and K [a], respectively. Let j : r + cr ( r is a p-Sylow subgroup of u)induce a map j : W O V, + Y ( W O V, is a free K [ r ] resolution o f K ) . Then t h e r e i s a m a p w : Y O , , , , Cp2+Csuchthatthefollowingdiagram commutes up to natural homotopy

( v ) Let

7~

where U is the evident shuge.

Proof: The map e" satisfying (i), (ii), and (iii) is constructed in lemma 2.3 of Bruner's chapter in Bruner et al. [l]. In his notation let M = N and K = L = C(A, N ) , which is our D. Thus his map @ is our e". Since e" extends the product on N it satisfies (i). For (ii), naturality in M is obvious since cotensor products are natural and everything in sight is natural in r. For naturality in N consider the (not necessarily commutative) diagram

I: CAT, N )

__$

CAT, N ' ) .

Bruner's result gives a map

WOK,,] C,-(l-,N ) , + C J f ,

"1

extending the map iVr+ N ' . Both the composites in the diagram have the appropriate properties so they are chain homotopic and is natural in fi up to the chain homotopy. For (iv) note that TT acts on ( C O C ) ,= C,, by permutation, so TT is a subgroup of I;?,. The two composites in the diagram satisfy (i) and (ii) as maps from WOK[,,,C,, to C, so they are naturally homotopic by (iii). To prove (v), construct w for the group u in the same way we constructed 0 for the group T.Then the compositions w ( j 0 Cp2)and ( W O 0,) U both satisfy (i) and (ii) for the group 7, so they are naturally homotopic by (iii). With the above result in hand the machinery of May [ 5 ] applies to Cr(M,N ) and we get Steenrod operations in Cotor,(M, N ) when K = Z/(p). Parts (i), (ii), and (iii) guarantee the existence, naturality, and uniqueness of the operations, while (iv) and (v) give the Cartan formula

350

Al. Hopf Algebras and Hopf Algebroids

and Adem relations. These operations have properties similar to those of the topological Steenrod operations with the following three exceptions. First, there is in general no Bockstein operation p. There are operations PP', but they need not be decomposable. Recall that in the classical case /3 was the connecting homomorphism for the SES

o+ C + S @ Z / ( p 2 ) +c+o,

c

where is a DGA which is free over Z , whose cohomology is the integral cohomology of X and which is such that c O Z / ( p ) = C. If C is a cobar complex as above then such a may not exist. For example, it does not is the dual Steenrod algebra, but exist if C = C,*(Z/( p ) , Z / ( p ) ) where if C = CBP*(BP)/,~)(SP*/(P)) we have C = CBP*,BP~BP*). Second, when dealing with bigraded complexes there are at least two possible ways to index the operations; these two coincide in the classical singly graded case. In May [ S ] one has P': Cotor"' Cotor'"''-')( p-')3pf, which means that P' = 0 if either 2i < t or 2i > s + t. (Classically one would always have t = 0.) We prefer to index our P' so that they raise cohomological degree by 2i( p - 1) and are trivial if i < O or 2i > s (in May [ 5 ] such operations are denoted by PI).This means that we must allow i to be a half-integer with P' nontrivial only if 2i = t mod ( 2 ) . (This is not a serious inconvenience because in most of our applications for p > 2 the complex C** will be trivial for odd t.) The Cartan formula and Adem relations below must be read with this in mind. Finally, Po :Cotor'.*' --* Cotors92P'is not the identity as in the classical case. The following is a reindexed form of 11.8 of May [ 51.

c

+*

-$

A1.5.2 Steenrod OperationsTheorem. Let r be a Hopf algebroid over Z/ ( p ) and M and N right and left r-comodule algebras. Denote Cotor:'( M , N ) by H",’. Then there exist natural homomorphisms

sq':Hss1 p1/2:H S . 1

+

~

HSt0' for p = 2, Hl/2+&P'

and pp1/2. H S . 1

+

H1/2+S+l,P'

for p > 2 and q = 2p -2,

all with i z 0 , having the following properties. ( a ) F o r p = 2 , S q ' = O i f i > s . Forp>2, P'12 andPP"*=O $ i > s or 2i f t mod ( 2 ) . ( b ) F o r p = 2 , S q ' ( x ) = x 2 i f i = s . Forp>2 ands+teven, P ' ( X ) = X p $2i=s. ( c ) If there exists a Hopf algebroid f and f-comodule algebras A? and fi all flat over Z , p , with r = r @ Z / ( p ) , M = f i O Z / ( p ) , and N = % O

351

5. Algebraic Steenrod Operations

Z / ( p ) , t h e n P S q i = ( i + 1 ) S q ' + ' f o r p = 2 andforp>2 Pp'isthecomposition of /3 and Pi, where P : HI*'+ H"+'.' is the connecting homomorphism for the SES O+N+ fiOZ/(p2)"+0. (d) Sq'(xy)= 1 Sq'(x)Sq'-'(y) for p = 2 . Oslsi

For p > 2, =

1 Osjsi

~J/2(~)p(i-J)/2 (Y)

and

Similar external Cartan formulas hold. ( e ) The following Adem relations hold. For p = 2 and a < 26,

For p > 2, a 2, be represented by a cochain which is a sum of elements of the form my,I * . I ysn. Then Sqo(x)or P o ( x )is represented by a similar sum of elements of the form m pyfl . y:n p . H The operations also satisfy a certain suspension axiom. Consider the category C of triples ( M , r, N ) with M , r, N as above. A morphism in C consists of maps M + M ' , r + r', and N + N' which respect all the structure in sight. Let C,, i = 1, 2, 3 , be the cobar complexes for three objects in C

352

A l . Hopf Algebras and Hopf Algebroids

and suppose there are morphisms which induce maps

c , L c++ c, such that the composite gf is trivial in positive cohomological degree. Let HT*, i = 1, 2, 3, denote the corresponding Cotor groups. Define a homomorphism (T (the suspension) from kerf * c Ht"~' to H:'/im g* as follows. Given x E kerf *, choose a cocycle a E C1 representing x and a cochain bE C2 such that d ( b )=f ( a ) . Then g( b ) is a cocycle representing u ( x ) . It is routine to verify that a ( x ) is well defined.

A1.5.4 Suspension Lemma. Let u be as above. Then for p > 2, a(P ' ( x ) )= P ' ( u ( x ) )and u ( p P ' ( x ) )= p P ' ( u ( x ) ) and similarly for p = 2. Prooj We show how this statement can be derived from ones proved in May [ 5 ] . Let c C, be the subcomplex of elements of positive cohomological degree. It has the structure necessary for defining Steenrod operations in its cohomology since C, does. Then May's theorem 3.3 applies to

cl

-

c,

f

c,-Lc,

and shows that suspension commutes with the operations in kerf * c H*( C,).We have H *( = H ' ( C,) for s > 1 and a four-term exact sequence

c,)

O + MI Or, N , + MI OA,N , += H ' (

c,) HI( C , ) +

+0

so the result follows.

A1.5.5 Corollary. Let 6 be the connecting homomorphism associated with an SES of commutative associative r-comodule algebras. Then P'6 = 6P' and pP'6 = -8pP' for p > 2 and similarly for p = 2. ( I n this situation the s u b comodule algebra must fail to have a unit.) Let O + MI + M , + M , + 0 be such an SES. Then set N , = N and in the previous lemma. Then 6 is the inverse of u so the result follows. Prooj

r, = r

We need a transgression theorem.

( A ,C) be an extension of A 1 5 6 Corollary. Let (0,@) A ( A ,r) Hopf algebroids over Z / (p ) ( A l . l . 1 5 ) ;let M be a right 0-comodule algebra and N a left r-comodule algebra, both commutative and associative. Then M , A OxN ) to there is a suspension map u from ker i* c Cotor&+'7'( Cotor:'( M 0 A, N)/im f * which commutes with Steenrod operations as in A1.5.4. ProoJ: A Or N is a left @-comodule algebra by A1.3.14(a). We claim the composite 6~ i + & ~ is zero; since @ = A A ~ ~ U ~ A , A O z Z O z A = A A O I A = D , so f i ( 6 ) = 0 . Hence C , ( M , A U , N ) +

353

5. Algebraic Steenrod Operations

C,( M O DA, N ) + C,( M O DA, N ) is zero in positive cohomological degree. Hence the result follows from A1.5.4. W

The following is a reformulation of theorem 3.4 of May [ 5 ] . A1.5.1 Kudo Transgression Theorem. Let CD + r + Z be a cocentral extension (A1.1.15) of Hopf algebras over a field K of characteristic p. I n the CESS (A1.3.14)for Ext,(K, K ) we have E;'= Ext&(K,K)OExt:(K, K ) with d,: E : ' ~E : + r 3 t - r + 1 , Then the transgression d,: Eg3r-1+ E:" commutes with Steenrod operations up to sign as in A1.5.4; e.g., if d , ( x ) = y then d r + 2 s ( p + l , ( P s (= x )P) ' t v ) . Moreouer f o r p > 2 and r - 1 even we have d ( p - l ) r ( X P - l Y=) -PP

(r-!)/Z

(u).

APPENDIX 2

FORMAL GROUP LAWS

In this appendix we will give a self-contained account of the relevant aspects of the theory of commutative one-dimensional formal group laws. This theory was developed by various algebraists for reasons having nothing to do with algebraic topology. The bridge between the two subjects is the famous result of Quillen [2] (4.1.6) which asserts that the Lazard ring L (A2.1.8) over which the universal formal group law is defined is naturally isomorphic to the complex cobordism ring. A most thorough and helpful treatment of this subject is given in Hazewinkel [l]. An account of the Lazard ring is also given in Adams [ S ] , while the classification in characteristic p can also be found in Frohlich [l]. We now outline the main results of Section 1. We define formal group laws (A2.1.1) and homomorphisms between them (A2.1.5) and show that over a field of characteristic 0 every formal group law is isomorphic to the additive one (A2.1.6).The universal formal group law is constructed (A2.1.8) and the structure of the ring L over which it is defined is determined (A2.1.10). This result is originally due to Lazard [l]. Its proof depends on a difficult lemma (A2.1.12) whose proof is postponed to the end of the section. Then we define p-typical formal group laws (A2.1.17 and A2.1.22) and determine the structure of the p-typical analog of the Lazard ring, V (A2.1.24). This result is due to Cartier [l]; Quillen [2] showed that V is naturally isomorphic to .rr,(BP) (4.1.12). Using a point of view due to Landweber [ 13, we determine the structure of algebraic objects LB (A2.1.16) and VT (A2.1.26), which turn out to be isomorphic to M U , ( M U ) (4.1.11) and B P , ( B P ) (4.1.19), respectively. 354

1. Universal Formal Group Laws and Strict lsomorphisms

355

All of the results of this section can be found in Adams [ 5 ] , although our treatment of it differs from his. )] In Section 2 we give the explicit generators of V [i.e., of n T T * ( B Pgiven by Hazewinkel [2] (A2.2.1) and Araki [l] (A2.2.2) and determine the behavior of the right unit T~ on Araki’s generators (A2.2.5). For the Morava theory of Chapter 6 we will need the classification of formal group laws over separably closed fields of characteristic p > O (A2.2.11) originally due to Lazard [2], and a description of the relevant endomorphism rings (A2.2.17 and A2.2.18) originally due to DieudonnC [ 11 and Lubin [ 11.

1. Universal Formal Group Laws and Strict Isomorphisms A2.1.1 Definition. Let R be a commutative ring with unit. A formal group law over R is a power series F ( x , y ) E R [ [ x ,y ] ] satisfying ( i ) F ( x , 0 ) = F ( 0 , x) = x, ( i i ) F ( x , y ) = F ( y , x), and (iii) F ( x , F ( y , z ) ) = F ( F ( x ,y ) z ) .

Strictly speaking, such an object should be called a commutative onedimensional formal group law; we omit the first two adjectives as this is the only type of formal group law we will consider. It is known (Lazard [3]) that (ii) is redundant if R has no nilpotent elements. The reason for this terminology is as follows. Suppose G is a onedimensional commutative Lie group and g : R + U c G is a homomorphism to a neighborhood U of the identity which sends 0 to the identity. Then the group operation G x G + G can be described locally by a real-valued function of two real variables. If the group is analytic then this function has a power series expansion about the origin that satisfies (i)-(iii). These three conditions correspond, respectively, to the identity, commutativity, and associativity axioms of the group. In terms of the power series, the existence of an inverse is automatic, i.e., A2.1.2 Proposition. If F is a formal group law over R then there is a power series i(x) E R [ [ x ] ](culled the formal inverse) such that F(x,z(x)) = O . 1

In the Lie group case this power series must of course converge, but in the formal theory convergence does not concern us. Formal group laws arise in more algebraic situations; i.e., one can extract a formal group law from an elliptic curve defined over R. One can also reverse the procedure

356

A2. Formal Group Laws

and get a group out of a formal group law; if R is a complete local ring then F will converge whenever x and y are in the maximal ideal, so a group structure is defined on the latter which may differ from the usual additive one. Before proceeding further note that A2.1.1(i) implies Proposition.

If F is a formal group law then F ( x , y ) = x + y mod (x, y)'.

H

Examples of Formal Group Laws. F,(x, y ) = x + y, the additive formal group law. F ( x , y ) = x + y + u x y (where u is a unit in R), the multiplicative group law, so named because 1+ u F = (1 + ux)(1 + uy). F ( x , y ) = f x + y ) / ( l +XY). - y 4 + y J 1 - x4)/(1+ x 2 y 2 ) , a formal group law over F(X, y ) = (XJI

zr 1/21. The last example is due to Euler and is the addition formula for the elliptic integral dt

5: 7

7

(see Siege1 [ l , pp. 1-91). These examples will be studied further below (A2.2.9). The astute reader will recognize (c) as the addition formula for the hyperbolic tangent function; i.e., if x = tanh( u ) and y = tanh( u ) then F(x, y) = tanh( u + u ) . Hence we have tanh-'( F(x, y ) ) = tanh-'(x) + tanh-'(y)

or F(x, y) = tanh(tanh-'(x)+ tanh-'(y)), where tanh-'(x) = Zaz0x2'"/(2i+ 1) E ROQ[[x]]. We have a similar situation in (b), i.e., log( 1 + uF) = log( 1+ ux) +log( 1 + uy), where l o g ( l + ~ x ) = (C- l~) i~+ '( ~ u~x ) ' / i g ROQ[[x]]. This means that the formal group laws of ( b ) and (c) are isomorphic over Q to the additive formal group law (a) in the following sense. A2.1.5 Definition. Let F and G be formal group laws. A homomorphism from F to G is a power series f ( x ) E R[[x]] with constant term 0 such that f(F(x, y)) = G (f ( x ) ,f ( y ) ) . It is an isomorphism ifit is invertible, i.e., i f f ' ( 0 ) (the coeficient of x ) is a unit in R, and a strict isomorphism i f f ' ( 0 )= I . A

1. Universal Formal Group Laws and Strict Isomorphisms

357

strict isomorphism from F to the addition formal group law x + y is a logarithm for F, denoted by log,(x). Hence the logarithms for A2.1.4(b) and (c) are Xi,o(-u)i-'xi/i and tanh-'(x), respectively. On the other hand, these formal group laws are not isomorphic to the additive one over Z. To see this for (b), set u = 1. Then F(x, x) = 2 x + x 2 = x2 mod 2, while F,(x, x ) = 2x = 0 mod 2, so the two formal group laws are not isomorphic over Z/(2). The formal group law of (c) is isomorphic to F, over Z(,,, since its logarithm tanh-' x has coefficients in Z ( z ) , but we have F ( F ( x , x ) , x ) = ( 3 x + x 3 ) / ( 1 + 3 x 2 ) = x 3 m o d ( 3 ) while F,(F,(x, x ) , x) = 3x = O mod 3. Similarly, it can be shown that F and F, are distinct at every odd prime (see A2.2.9). A2.1.6 Theorem. Let F be a formal group law and let f ( x ) E R O Q[ [ x ] ] be given by

where F2(x, y ) = aF/ay. Then f is a logarithm for F, Le., F(x, y) = f '( f ( x ) f ( y ) ) , and F is isomorphic over R O Q to the additive formal group

+

law.

Proof: Let w = f ( F ( x , y))- f ( x ) -f(y). We wish to show w =O. We have F ( F ( x , y), z ) = F(x, F(y, z ) ) . Differentiating with respect to z and setting z = O we get A2.1.7

F , ( F ( x ,Y), 0) = FAX, Y)FAY, 0).

On the other hand, we have aw/ay = f ( F ( x ,y))F2(x,y ) -f'(y), which by the definition o f f becomes

By symmetry we also have aw/dx = 0, so w is a constant. But f and F both have trivial constant terms, so w = 0. H Now we wish to consider the universal formal group law. Its construction is easy. A2.1.8 Theorem. group law

There is a ring L (called the Lazard ring) and a formal F(x, y ) = C aVx'y'

dejined over it such that for any formal group law G over any commutative

358

A2. Formal Group Laws

ring with unit R there is a unique ring homomorphism 8 : L + R such that G ( Xy, ) = z e ( a , ) x ' y i .

Pro05 Simply set L = Z [ a , , ] / I , where I is the ideal generated by the relations among the a, required by the definition A2.1.1, i.e., by ~ , , ~ - l , a,,, - 1 , a,,, and ao,, for (i), a, -a,, for (ii), and b,,, for (iii), where F(F(x,J'), z ) - F ( X , F ( y , Z ) ) = z b,kX'yiZk*

Then 8 can be defined by the equation it is supposed to satisfy. Determining the structure of L explicitly is more difficult. At this point it is convenient to introduce a grading on L by setting ~ a l , ~ = 2 ( i + j - l ) . Note that if we have 1x1 = l y l = -2 then F ( x , y ) is a homogeneous expression of degree -2.

A2.1.9 Lemma.

( a ) L O Q = Q [ m , , m , , . . . 3 with \m,\= 2 i and F ( x , y ) = f ' ( f ( x )+ f ( y ) ) w h e r e f ( x )= x + X,,, m,xl+'. ( 6 ) Let M c L O Q be Z [ m , , m 2 , . . .I. Then im L c M. Proof: (a) By A2.1.6 every formal group law G over a Q-algebra R has a logarithm g(x) so there is a unique 4: Q[ml, m,, . . . ] + R such that f ( x )= g(x).Inparticularwehave +:Q[m,, m,, . . . I - + L O Q a s w e l l a s 8: L O Q . . 3 with 64 and 46 being identity maps, so 8 and 4 are Q[ml, m2,. isomorphisms. (b) F ( x , y ) is a power series with coefficients in M, so the map from L to L O Q factors through M .

Now recall that if R is a graded connected ring (e.g., L O Q ) the group of indecomposables Q R is 1 / 1 2 where Ic R is the ideal of elements of positive degree.

A2.1.10 Theorem (Lazard [ 11). ( a ) L = Z [ z , , x 2 , . . . ] with Ix,I = 2 i f o r i > 0. ( b ) x, can be chosen so rhat its image in QL@Q is pm, rn, (c)

if i = p k forsomeprimep otherwise.

L is a subrzng of M [ A 2 . 1 . 9 ( b ) ] .

The proof of this is not easy and we will postpone the hardest part of it (A2.1.12) to the end of this section. The difficulty is in effect showing that L is torsion-free. Without proving A2.1.12 we can determine Lltorsion with relative ease. We will not give F in terms of the x , , nor will the latter be

1. Universal Formal Group Laws and Strict isomorphisms

359

given explicitly. Such formulas can be found, however, in Hazewinkel [3] and in section 5 of Hazewinkel [l]. Before stating the hard lemma we need the following exercise in binomial coefficients. A2.1.11 Proposition. Let u, be the greatest common divisor of the numbers (7) for O > i > n. Then p

'"=( 1

if n = p k forsomeprimep otherwise.

Now we are ready for the hard lemma. Define homogeneous symmetric polynomials B,(x, y ) and C , ( x , y ) of degree n for all n > 0 by BJX, y ) = ( x + y ) " - X " -y" cn(x, y ) =

{Bg:lp

if n = p k forsomeprimep otherwise.

It follows from A2.1.11 that C , ( x , y ) is integral and that it is not divisible by any integer greater than one. A2.1.12 Comparison Lemma (Lazard [ 11). Let F and G be twoformal group laws over R such that F = G mod ( x , y)". Then F = G + aC, mod ( x , y)"+' for some a E R. The proof for general R will be given at the end of this section. For now we give a proof for torsion-free R. In this case we lose no information by passing to R O Q , where we know (A2.1.6) that both formal group laws have logarithms, say f ( x ) and g ( x ) , respectively. Computing mod ( x , y)"+' we have f ( x )= g ( x ) + bx"

for some

b E ROQ

so f ' ( x ) = g - ' ( x ) - bx"

and F - G = f ' ( f ( x ) + f ( y ) )- g - ' ( g ( x ) + g ( y N

=g-'(g(x)+g(y)+b(x"+y"))-b(x+y)"-g-'(gfx)+g(y)) g-'(g(x)+g(y))+b(x"+y")-b(x+y)" -g-'(g(x)+g(y)) -bB,(x, y).

Since this must lie in R it must have the form a C , ( x , y ) , completing the proof for torsion-free R. A2.1.13 Lemma. ( a ) In Q L O Q , aV= - ( ' f ' ) m , + j - , . ( b ) QL is torsion-free.

360

A2. Formal Group Laws

Prooj (a) Over L O Q we have Z mnp1(Za,,x'y')" = Z m n - I ( x n + y n )Using . A2.1.3 to pass to Q L O Q we get C a , x ' ~ ' +C m n - l ( y + y ) " = n>l

C

mn-l(xn+yn)i

n>O

which gives the desired formula. (b) Let Q2nLdenote the component of QL in degree 2n, and let R be the graded ring Z@Q2,,L. Let F be the formal group law over R induced by the obvious map 8:L + R, and let G be the additive formal group law over R.ThenbyA2.1.12, F ( x , y ) = x + y + a C , ( x , y ) foraEQ,,L.Itfollows that QZnLis a cyclic group generated by a. By (a) Q2,,LOQ= Q, so Q,,L = Z and QL is torsion-free. It follows from the above that L is generated by elements x , whose images in Q L O Q are u,m,, where u, is as in A2.1.11, i.e., that L is a quotient of Z[x,]. By A2.1.9 it is the quotient by the trivial ideal, so A2.1.10 is proved. Note that having A2.1.12 for torsion-free R implies that Lltorsion is as claimed. The reader familiar with Quillen's theorem (4.1.6) will recognize L as n-*( M U ) = M U , . We will now define an object which is canonically isomorphic to n-,( MU A M U ) = M U , ( M U ) . This description of the latter is due to Landweber [ 11.

A2.1.14 Definition. Let R be a commutative ring with unit. Then FGL(R) is the set of formal group laws over R (A2.1.I ) and S I ( R) is the set of triples ( F , f ; G ) whereF, G E F G L ( R ) andf:F-,Gisastrictisomorphism (A2.2.5), i.e., f(x) E R"x11 w i t h f ( 0 )= O , f ( 0 ) = 1, a n d f ( F ( x ,Y ) ) = G ( f ( x ) , f ( y ) ) . A2.1.15 Proposition. FGL(-) and S I ( - ) are covariant functors on the category of cornmutative rings with unit. FGL(-) is represented by the Lazard ring L and S I ( - ) is represented by the ring LB = LOZ[b,, b 2 , .. . 3 . In the W grading introduced above, 1 b, 1 = 2 i. Prooj All but the last statement are obvious. Note that a matched pair ( F A G ) is determined by F and f and that f can be any power series of the formf(x) = X + Z , , ~ ~ ; X ' Hence + ' . such objects are in 1-1 correspondence with ring homomorphisms 8: LB-, R with 8 ( b , )=J;. Now LB has some additional structure which we wish to describe. Note that FGL(R) and S I ( R ) are the sets of objects and morphisms, respectively, of a groupoid, i.e., a small category in which every morphism is an equivalence. Hence these functors come equipped with certain natural transformations reflecting this structure. The most complicated is the one corresponding to composition of morphisms, which gives a natural (in R )

361

1. Universal Formal Group Laws and Strict Isomorphisms

map from a certain subset of SZ(R ) x SI(R ) to SI ( R ) . This structure also endows ( L , L B ) with the structure of a Hopf algebroid (Al.l.1). Indeed that term was invented by Haynes Miller with this example in mind. We now describe this structure. A2.1.16 Theorem. I n the Hopf algebroid ( L , LB) dejined above E : L B + L is dejined by B ( b,) = 0; vL: L + LB is the standard inclusion while qR: LO Q + L B O Q is given by

c

120 7 ) R ( m z ) = z o

mz(zo

I+'

c(6~))

7

where m, = b, = 1; ZlrOA( 6,) = Z,=, (ZlrO6,)'"O 6, ; and C: LB + LB is determined by c ( m , ) = v R ( m , )and Z, c ( 6 , ) ( X J z obJ)'+l=1. These are the structure formulas for M U , ( M U ) (4.1.11). Proof: E and vL are obvious. For c, if f ( x ) = Z b,x'+' then f - ' ( x ) = Z c(b,)x'+'. Expanding f-'(j"(l)) = 1 gives the formula for c(b,). For vR, let log x = Z m,x'+' and mog x = Z vR(m,)x'+' be the logarithms for F and G, respectively. Then we have

f-'(G(x, Y ) )

=

F(f - ' ( x ) , f - Y Y U

so log(f-'(G(x, y ) ) ) = log(f -'(x)) +log(f-'(y)). We also have mog(G(x, Y)) =mog(x)+mog(y) for which we deduce mog(x) = mogf-'(x). Setting x = 1 gives the formula for v R .For A let fl(x) = b:x'+', f2(x) = Z byx'+', a n d f ( x ) =f2(fi(x)). Then expanding and setting x = 1 gives Z b, = Z 6y(Z 6:)'+'. Since f 2 follows f l this gives the formula for A. Note that ( L , LB) is split (Al.1.22) since A defines a Hopf algebra structure on B = Z[b,]. Next we will show how the theory simplifies when we localize at a prime p , and this will lead us to BP, and BP,(BP). A2.1.17 Definition. A formal group law over a torsion-jiree Z(,,-aigebra is p-typical if its logarithm has the form Z,?, A,xP' with A, = 1.

Later (A2.1.22) we will give a form of this definition which works even when the Z(,,-algebra R has torsion. Assuming this can be done, we have

362

A2. Formal Group Laws

A2.1.18 Theorem (Cartier [ 11). Every formula group law ouera Z,,,-algebra is canonically strictly isomorphic to a p-typical one. Actually A2.1.17 is adequate for proving the theorem because it suffices to show that the universal formal group law is isomorphic over L@Z(,, to a p-typical one. The following notation will be used repeatedly.

A2.1.19 Definition. Let F be a formal group law over R. If x and y are elements in an R-algebra A which also contains the power series F ( x , y ) , let x

+FY =

F(x, Y ) ,

This notation may be iterated, e.g., x + F y + F z = F ( F ( x , y ) , z ) . Similarly, x - F y = F ( x , i ( y ) ) (A2.1.2). For nonnegative integers n, [ n I F ( x ) = F ( x , [ n - l ] F ( X ) )with [ o ] F ( x )= 0. (The subscript F will be omitted whenever possible.) Z F ( ) will denote the formal sum of rhe indicated elements. A2.1.20 Proposition. If the formal group law F above is defined over a K-algebra R where K is a subring of Q, then for each r E K there is a unique power series [ r I F ( x )such that ( a ) i f r is a nonnegative integer, [ r I F ( x )is thepower series dejned above, ( b ) [ r l + r 2 1 F ( X ) = F([rllF(X),[ r Z I F ( X ) ) , (c) [rIr21F(X) =[rlIF([rZIF(X))*

Proo$ Let [ - 1 I F ( x )= i ( x ) (A2.1.2), so [ r I F ( x )is defined for all r e Z. We have [ r I F ( x = ) rx mod (x'), so if d E Z is invertible in K, the power series [ ~ ] F ( X is ) invertible a n d w e can define [ d p l ] F ( ~ ) = [ d ] F 1 ( ~H) . Now we suppose q is a natural number which is invertible in R. Let

where 5 is a primitive qth root of unity. A priori this is a power series over R [ l l , but since it is symmetric in the l i it is actually defined over R. I f R is torsion-free and log(x)= Z l z om i x i + ' , we have

1. Universal Formal Group Laws and Strict Isomorphisms

363

The expression Z:==, Si"+') vanishes unless i( j + 1) is divisible by q, in which case its value is q. Hence, we have log(f,(x))

=

C

rndplxqJ.

j>O

If F is p-typical for p # q, this expression vanishes, so we make A2.1.22 Definition. A formal group law F over a Z(,,-algebra is p-typical i f f , ( x ) = 0 f i r all primes q # p. Clearly this is equivalent to our earlier definition A2.1.17 for torsionfree R. To prove Cartier's theorem (A2.1.18) we claim that it suffices to construct a strict isomorphism f ( x ) = X j x ' E LOZ,,,[[x]] from the image of F over L O Z , , , to a p-typical formal group law F'. Then if G is a formal group law over a Z,,!-algebra R induced by a homomorphism 8: L x Z ( , , + R, g ( x ) = Z O ( j ) x ' E R[[x]] is a strict isomorphism from G to a p-typical formal group law G'. Recall that if mog(x) is the logarithm for F' then mog(x) = log(f-'(x)). We want to use the fq(x) for various primes q f p to concoct an f-'(x) such that

It would not do to set f-'(x)=x

-F

CFf4(X) qfP

because if n is a product of two or more primes f p then a negative multiple of Mn-lx" would appear in logf-'(x). What we need is the Mobius function p ( n ) defined on natural numbers n by

A n )=

{

0 (-l)r

if if

n is divisible by a square n is the product of r distinct primes.

Note that p ( 1 ) = 1 and p ( q ) = -1 if q is prime. Then we define f ( x ) by A2.1.23

f'(x)

=

C"p(s)lF(fq(x)). PX¶

[Note also thatf,(x) = x.] The sum is over all natural numbers q not divisible by p. This infinite formal sum is well defined because &(x) = 0 mod (x".

364

A2. Formal Group Laws

Now

It is elementary to verify that if

1

n=pk

otherwise.

P k 4. .

41"

It follows that F' has logarithm

A2.1.24

mog(x)=

1 m,~-,xP', iZO

so F' is p-typical. This completes the proof of A2.1.18. Now we will construct the universal p-typical formal group law. A2.1.25 Theorem. Let V = Z , , , [ v , , v 2 ,... ] w i t h \ u , \ = 2 ( p n - l ) . Thenthere is a universal p-typical formal group law F dejined over V ; i.e., for anyp-typical formal group law G over a commutative Z(,,-algebra R, there is a unique ring homomorphism 0 : V + R such that G ( x ,y ) = 0 ( F ( x ,y ) ) . Moreover the homomorphism from LO Z , to Vcorresponding ( A 21.8) to thisformal group law is surjective, i.e., V is isomorphic to a direct summand of L O Z,,, . rn

,

We will give an explicit formula for the u,'s in terms of the log coefficients mP"t1below (A2.2.2). In 4.1.12 it is shown that V is canonically isomorphic to T * ( B P ) .

ProoJ: Recall that the canonical isomorphism f above corresponds to an endomorphism C#J of L O Z ( , , given by

{

mi 0

if i = p k -1 otherwise.

This 4 is idempotent, i.e., C#J2 = 4 and its image is a subring V c L O Z ( , , over which the universal p-typical formal group law is defined. An argument similar to the proof of Lazard's theorem A2.1.9 shows that V has the H indicated structure.

Now we will construct a ring VT canonically isomorphic to B P , ( B P ) and representing the set of p-typical matched pairs ( C f ;G ) (A2.1.13), i.e., matched pairs with F and Gp-typical. The power series f must be chosen carefully to ensure that G is p-typical, and this choice depends on E There is no such thing as a "p-typical power series," i.e., one that sends any p-typical F to a p-typical G. To characterize the appropriate f we have

365

1. Universal Formal Group Laws and Strict Isomorphisms

A2.1.26 Lemma. Let F be a p-typical formal group law over a Z,,,-algebra R. Let f ( x ) be an isomorphism (A2.1.5)from F to a formal group law G. Then G is p-typical if f - I ( x )=

C FtiXP' iz-0

for ti E R with to a unit in R.

Proof: For a prime number q # p let

where [ is a primitive pth root of unity. By A2.1.22 we need to show that h , ( x ) = 0 for all q # p iff f is as specified. From the relation G ( x ,Y )= f ( F ( f - ' ( x ) , f - ' ( Y ) ) ) we deduce

Now for isomorphism f ( x ) there are unique ci E R such that

f-’(x ) =

c"

CiXi

I>O

with c1 a unit in R. Hence we have

= ~ ' f , ( c , x '+)F qc 1

~ ' C q , X q 1= 1

1Cqlx". 1>0

This expression vanishes for all q # p iff cq, = 0 for all i > 0 and q # p , i.e., iff f is as specified. 1 It follows immediately that VT= VOZ(,)[t,, fZ, . . . ] as a ring since for a strict isomorphism to= 1. The rings V and VT represent the sets of objects and morphisms in the groupoid of strict isomorphisms of p-typical formal group laws over a Z(,,-algebra. Hence ( V , VT), like ( L , L B ) , is a Hopf algebroid (Al.l.1) and it is isomorphic to ( B P , , B P J B P ) ) . Its structure is as follows.

366

A2. Formal Group Laws

A2.1.27 Theorem. In the Hopfalgebroid (V, VT)(see Al.l.1) ( a ) V = Z ~ ~ ) [ U ~ , U ~with , . . l. u] , , ( = 2 ( p n - 1 ) , ] ( t n ) = 2 ( p " - l ) ,and ( b ) VT= V O Z ( p ) [ t l , t 2 , . . . with ( c ) q L : V + VT is the standard inclusion and E: VT+ V is dejined by & ( t , ) = O ,&(U,)=U,. Let A, E V O Q denote the image of mpg-, E L O Q (see A2.22). Then , ~ n whereAo= to= 1, ( d ) q R : V - , V T i s d e t e n n i n e d b y 7 7 R ( A n ) = ~ o ch,t:', ( e ) A is determined by A, A( t])"' = A,$'' 0 ti'+',and (f) c is determined by El,,,krOAlt,P'c( tk)' = XrrOA,. ( g ) The forgetful functor from p-typical formal group laws to formal group lawsinducesasurjection ofHopfalgebroids (A1.1.19)(LOZ(,,, L B O Z ( , , ) +

z+J,k20

(V, VT). Note that (e) and (f) are equivalent to

x F A ( t,) = I

ro

’C20 t, O f,"'

1

and

t,c( r,)"' = 1,

I,JZo

I, J

respectively. It can be shown that unlike ( L , LB) (A2.1.16), ( V , VT) is not split (Al.1.22).

ProoJ Part (a) was proved in A2.1.23, (b) follows from A2.1.23, and (c) is obvious, as is (8). For (d) letf be a strict isomorphism between p-typical formal group laws F and G with logarithms log(x) and mog(x), respectively. If f ( x ) satisfies

1"tlXP'

f-'(x) =

120

and log(x)=

2

A,xP'

’20

then by definition of q R mog(x) =

1 qR(hi)xP'. irO

We have (see the proof of A2.1.16) mog(x) = log(f -'(x)) = log =

1 log( fixP')= 120

and (d) follows.

Ait$"xP'+' iJr0

367

1. Universal Formal Group Laws and Strict Isomorphisms

A G A H be strict isomorphisms of p-typical formal

For (e) let F group laws with

f;'(x)

=

and

CFt:xP'

f;’(x)

If we set f = f i a f l

CG tyxp'.

=

120

JZO

with f-'(x)=

1"t , x p L 120

then a formula for t, in terms of t : and t: will translate to a formula for A( l,). We have

This gives

zI F A ( t i ) = C" ti@ tj'j i j

as claimed. For (f) let f :F + G be as above. Then f ( x ) = 1"c( r j ) x p ' so x =f-'(f(x))

=

1"f-y

=f-’

c" c( tj)x

c(tj)x@)=

;

c'

9

ti( c( t j ) x @ ) P i

ij

setting x = 1 gives (f). Our only remaining task is to prove Lazard's comparison lemma A2.1.12. The proof below is due to Frohlich [l]. The lemma states that if F and G are formal group laws with F = G mod (x, y ) " then F=G+aC,,(x,y)mod(x,y)"",

where (x+y)"-x"-y"

if

n = p k for some prime p

Cfl(X, Y ) =

((x+y)" -xn

-

y"

otherwise.

368

AZ. Formal Group Laws

Let T(x, y) be the degree n component of F - G. A2.1.28 Lemma. r ( x , y ) above is a homogeneous polynomial satisfying (i) U x , y 1 = T ( Y , x ) , (ii) T ( x , 0 ) = r(0,x ) = 0, (iii) T ( x , y ) + T ( x + y , z ) = T ( x , y

+z ) +T(y, z).

ProoJ: Parts (i) and (ii) follow immediately A2.l.l(ii) and (i), respectively. For (iii) let G ( x , y) = x + y + G'(x, y ) . Then mod (x, y, z)"+’ we have F ( F ( x , y ) , z ) = G ( F ( x , y ) ,z ) + r ( F ( x , y ) , z ) _= _=

+ G'(~ ( x ,), z ) + r ( x + y , z ) G ( x, y ) + T ( x , y ) + z + G'(G ( x, y ), z ) + r ( x + y , z )

F ( ~y, ) +

= G ( G ( X , ~ ) z, ) + r ( x , y ) + r ( x + y ,

z).

Similarly, F ( x , F ( y , z ) ) = G ( x , G ( y ,z ) ) + r ( x , Y + =) + r ( y , z )

from which (iii) follows. It suffices to show that any such

r must be a multiple of C,,.

A2.1.29 Lemma. Let R be a field of characteristic p > 0. Then any T ( x , y ) ouer R as aboue is a multiple of C , ( x , y ) .

ProoJ: It is easy to verify that C,, satisfies the conditions of A2.1.28, so it suffices to show that the set of all such r is one-dimensional vector space. Let T ( x , y ) = Z aix'yn-'. Then from A2.1.28 we have a, = a,, = 0,

a, = a,,-, ,

and A2.1.30

ui

(ny

= a,+j( r : j )

for 0 < i, i +j < n.

The case n = 1 is trivial so we write n = spk with either s = p or s > 1 and s 0 mod p. We will prove the lemma by showing a, = 0 if i 0 mod ( p k ) and that acPkis a fixed multiple of a P k . If i f 0 mod ( p k ) we can assume by symmetry that i < (s - 1 ) p k and write i = c p k - j with O < c < s and O < j < p k . Then A2.1.30 gives

+

i.e. ai = 0.

+

369

2. Classification and Endomorphism Rings

To show acpkis determined by Then A2.1.30 gives

apk

for c < s let i = p k and j

= ( c - l)pk.

i.e.,

(:I :)

aph

= acpkc.

This determines ucpk provided c f 0 mod ( p ) . Since c < s we are done for the case s = p. Otherwise ucph= by symmetry and since s f 0 mod ( p ) either c or s - c is f O m o d ( p ) . Note that A2.1.29 is also true for fields of characteristic 0; this can be deduced immediately from A2.1.30. Alternatively, we have already proved A2.1.11, which is equivalent to A2.1.29, for torsion-free rings. The proof of A2.1.29 is the last hard computation we have to do. Now we will prove the analogous statement for R = Z / ( p " ) by induction on m. We have

r(x,~)=aC,(x,~)+p~-~~'(x,~), where

r' satisfies A2.1.28 mod ( p ) .Hence by A2.1.29 r'(x, y ) = bC,(x, y ) so T(x, y ) = ( a + bp"-')C,(x, y )

as claimed. To prove A2.1.29 (and hence A2.1.12) for general R note that the key ingredient A2.1.30 involves only the additive structure of R ; i.e., we only have to compute in a finitely generated abelian group A containing the coefficient of r. We have to show that symmetry and A2.1.30 imply that the coefficients ai are fixed in relation to each other as are the coefficients of C,. We have shown that this is true for A = Z (from the case R = Q) and A = Z/( p " ) . It is clear that if it is true for groups A , and A2 then it is true for A 1 0 A 2 ,so it is true for all finitely generated abelian groups A. This completes the proof of A2.1.12.

2. Classification and Endomorphism Rings In order to proceed further we need an explicit choice of the generators first such choice was given by Hazewinkel [ 2 ] , which was circulating in preprint form six years before it was published. The same generators for 0,. The

370

AZ.

Formal Group Laws

p = 2 were defined earlier still by Liulevicius [3]. A second choice, which we will use, was given by Araki [l]. Hazewinkel's generators are defined by

A2.2.1 which gives, for example, A,=-,

01

v2

A2=-+-

v:+p

P P2' v 3 vlu;+ v2uf2 v : + p + p 2 A3=-+ +-----. P P2 P3 P

Of course, it is nontrivial to prove that these v, are contained in and generate V. Araki's formula is nearly identical,

A2.2.2 where uo= p . These v, can be shown to agree with Hazewinkel's mod ( p ) . They give messier formulas for A,, e.g.,

but a nicer formula (A2.2.5) for v R .

A2.2.3 Theorem (Hazewinkel [2], Araki.111). The sets of elements defined by A2.2.1 and A2.2.2 are contained in and generate V as a ring, and they are congruent mod ( p ) .

Roo$ yields

We first show that Araki's elements generate V. Equation A2.2.2

1 pAixP'= 1 iz0

A,uT'xP'+'.

ijz0

Applying exp (the inverse of log) to both sides gives

A2.2.4 which proves the integrality of the u,, i.e., that v,

E

V. To show that they

371

2. Classification and Endomorphism Rings

generate V it suffices by A2.1.10 to show u, =pu,h. in Q V B Q , where u, is a unit in Z(p ) . Reducing A2.2.2 modulo decomposables gives p h , = u,

+ Anpp"

so the result follows. We now denote Hazewinkel's generators of A2.2.1 by wi.Then A2.2.1 gives

1 log WiXP'

p log x - px =

i>O

or px = p log x -

1 log w,xp'. 1>0

Exponentiating both sides gives exppx=[pl(x)

C'wiX"'

-F

1>0

=PX

C F ~ , ~- FP 11 ’

+F

by A2.2.4.

W,X"

1>0

l>O

If we can show that (exp px)/p is integral then the above equation will give

1"u,xp' C" _=

r>O

w,xp'

mod ( p )

l>O

and hence u, = w, mod ( p ) as desired. To show that (exp px)/p is integral simply note that its formal inverse is W (log px)/p = Z hlpPL-'xP',which is integral since plh, is.

From now on u, will denote the Araki generator defined by A2.2.2 or equivalently by A2.2.4. The following formula for vR(v,)first appeared in Ravenel [l], where it was stated mod ( p ) in terms of the Hazewinkel generators; see also Moreira [2].

A2.2.5 Theorem. The behavior o f q R on u, is dejined by

cF

f , 7 ) R ( UJ)"'

1,32O

Pro05 Applying

r]R

=

2" sf!'. *,,20

to A2.2.2 and reindexing we get by A2.1.26(d) pA,f!' = Z h , f f ' 7 ) R ( ~ k ) ~ ' + ' .

Substituting A2.2.2 on the left-hand side and reindexing gives

1 h,u,P'tP,'+'= 1 A r t f ' 7 ) R ( ~ k ) P ' + ' . Applying the inverse of log to this gives the desired formula. This formula will be used to prove the classification theorem A2.2.11 below. Computational corollaries of it are given in Section 4.3.

372

A2. Formal Group Laws

We now turn to the classification in characteristic p. We will see that formal group laws over a field are characterized up to isomorphism over the separable algebraic closure by an invariant called the height (A2.2.7). In order to define it we need

A2.2.6 Lemma. Let F be a formal group law over Q commutative F,-algebra R and let f ( x ) be a nontrivial endomorphism of F (A2.2.5).Then for some n, f ( x ) = g ( x " ) with g ' ( 0 ) # 0. In particular f has leading term axp". For our immediate purpose we only need the statement about the leading term, which is easier to prove. The additional strength of the lemma will be needed below (A2.2.19). The argument we use can be adapted to prove a similar statement about a homomorphism to another formal group law G.

ProoJ Suppose inductively we have shown that f ( x ) = f ; ( x p ' )this , being trivial for i = O , and suppose f i ( 0 ) =0, as otherwise we are done. Define F"'(x, Y ) by F ( x , y)" = F"'(xP',y"'). It is straightforward to show that F"' is also a formal group law. Then we have f;(F"'(x"*,Y " ' ) ) = f ; ( F ( x ,Y ) " ' ) = f ( F ( x , y ) ) =F(f(X),f(Y)= ) F(f;(XP'),f;(YP'))

so f ; ( F " ' ( x ,Y ) )= F ( f ; ( x ) J ( y ) ) . Differentiating with respect to y and setting y

=0

we get

f : ( F " ' ( x , O))F:"(x, 0)= F * ( f ; ( x ) , f ; ( o ) ) f : ( o ) . Since f : ( O )

= 0,

F:"(x, 0 ) f 0, and F'"(x, 0) = x, this gives us f X x ) =0

so

L ( X ) 'J;+l(XP).

We repeat this process until we get an f n ( x ) with fn(0)+O and set g =

f". A2.2.7 Definition. A formal group law F over a commutative F,-algebra R has height n if [ p I F ( x )has leading term axp".I f [ p I F ( x )= 0 then F has height 00.

A2.2.8 Lemma. Proof

The height of a formal group law is an isomorphism invariant.

Let f be an isomorphism from F to G. Then f([PIF(X))

=[ P I G ( f ( X ) )

since f ( x ) has leading term ux for u a unit in R, and the result follows.

1

373

2. Classification and Endomorphism Rings

A2.2.9 Examples. Just for fun we will compute the heights of the mod ( p ) reductions of the formal group laws in A2.1.4. (a) (b) (c) group

[ p I F ( x ) = 0 for all p so F has height 00. [pIF(x) = upxp so F has height 1. As remarked earlier, F is isomorphic over 2(2) to the additive formal law, so its height at p = 2 is a.Its logarithm is

c-2X2'+l i+ 1

t20

so for each odd prime p we have A , = mp-, = l/p, so u1f 0 mod p by A2.2.2, so the height is 1 by A2.2.4 and A2.2.7. ( d ) Since F is not defined over Z(2,(as can be seen by expanding it through degree 5) it does not have a m o d 2 reduction. To compute its logarithm we have F2(x,0)=J1-x4

so by A2.1.6 dt

1 .3 . 5

. . . . . (2i- 1)x4i+1

=c 2'i!(4i+ 1) (2i)! = c 22'(i!)2(4i+ 1)' 820

X41+i

rzO

Now if p = 1 mod (4), we find that A , = mP-]is a unit (in Z,,,) multiple of l/p, so as in (c) the height is 1. However, if p = -1 mod (4), u1 = A l = 0 so the height is at least 2. We have (2i)! 4 (i!)'p2

A 2 = mp2-, = 7 where

Since

p2- 1 i=4 .

314

A2. Formal Group Laws

( 2 i ) ! is a unit multiple of P ( ~ - ' ) / since ~; P2-1 -=p(

P-3

4

)

I 3p-1 4

( i ! )is a unit multiple of p'p-3'/4.It follows that A2 is a unit multiple of l/p, so u2 0 mod p and the height is 2. It is known that the formal group law attached to a nonsingular elliptic curve always has height 1 or 2. (See Hazewinkel [ l ] .It is not stated there explicitly, but can be deduced from 33.1.11 and 31.4.3.) Now we will define a formal group law of height n for each n.

+

A2.2.10 Definition. F,(x, y ) = x + y. For a natural number h let F,, be /he p-typicalformal group law (of height n ) induced by the homomorphism 8: V + R (A2.1.24) defined by 8( v,) = 1 and e( u , ) = 0 for i # n. A2.2.11 Theorem (Lazard [ 2 ] ) . Let I< be a separably closedjield of characteristic p > 0. A formal group law G over K of height n is isomorphic to F,,.

Pro05 By Cartier's theorem (A2.1.18) we can assume G is p-typical (A2.1.22) and hence induced by a homomorphism 8: V + K (A2.1.24). I f n = 00 then by A2.2.4 8( v,) = 0 for all n and G = Fa. For n finite we have e ( v i ) = 0 for i 2 , and to W ( F , ) O F , " O Z / ( 2 )for p = 2 , the extra summand being generated by - 1. , unramified degree n extension of ( 8 ) W ( F , ) O Q = Q , [ x ] / ( h ( x ) ) the Q p , the field of p-adic numbers.

5

A proof can be found in Mumford [1, lecture 261 and in Serre [ l , section II.5,6]. We will sketch the proof of (f). For p > 2 there is a short exact sequence 1 -+ w(F,) G

w ( F , )4~F; + 1

where j is mod ( p ) reduction and i( w ) = exp p w = X,20( p w ) ' / i ![this power series converges in W ( F , ) ] . To get a splitting F,"+ W(F,)" we need to produce ( q - 1)th roots of unity in W(F,), i.e., roots of the equation x 4 - x = 0. [This construction is also relevant to (c).] These roots can be produced by a device known as the Teichmuller construction. Choose a lifting u of a given efement in F, and consider the sequence {u, u4, uq2,.. . }. It can be shown that it converges to a root of x 4 - x = 0 which is independent of the choice of u.

377

2. Classification and Endomorphism Rings

For p = 2 the power series exp 2w need not converge, so we consider instead the short exact sequence 1 + W(F,)A W(F,)"

W(F,)/(4)" + 1,

where j is reduction mod ( 4 ) and i( w ) = exp 4w, which always converges. This sequence does not split. We have W(F9)/(4)" s F 9 0 F , " . Since W(F,)@Q is a field, W(F,)" can have no elements of order 2 other than * l , so the other elements of order 2 in W(F,)/(4)" lift to elements in W(F,)" with nontrivial squares. Next we describe the noncommutative Z,-algebra En, which we will show to be isomorphic to the endomorphism ring of Fn for finite n. A2.2.16 Lemma. Let En be the algebra obtained from W(F,) by adjoining an indeterminate S and setting S" = p and Sw = w"S for w E W(F,). Then

( a ) En is a free Z, module of rank n2. ( b ) Each element e E En can be expressed uniquely as X i z 0 eiSi with ey - ei = 0. ( c ) En is generated as a Z,-algebra by S and a primitive ( q - 1)th root of unity w with relations S" - p = 0, Sw - w p S = 0 , and h ( w )= 0, where h( x) is an irreducible degree n factor of x q - x over Z,. ( d ) En is the maximal order in D,, = En OQ which is a division algebra with center Q, and invariant l/n. The proofs of (a), (b), and (c) are elementary. To see that D,, is a division algebra, note that any element in D, can be multiplied by some power of S to give an element in En which is nonzero mod ( S ) . It is elementary to show that such an element is invertible. The invariant referred to in (d) is an element in Q / Z which classifies division algebras over Q,. Accounts of this theory are given in Serre [ l , chapters XI1 and XIII], Cassels and Frohlich [l, pp. 137-1391, Hazewinkel [ l , sections 20.2.16 and 23.1.41. We remark that for O < i < n and i prime to n a division algebra with invariant i / n has a description similar to that of 0, except S" = p i instead of p . Our main results on endomorphism rings are as follows. A2.2.17 Theorem (DieudonnC [ l ] and Lubin [l]). Let K be a field of characteristic p containing F, with q = p". Then the endomorphism ring of the formal group law F,, (A2.2.10) over K is isomorphic to En. The generators w and S [ A2.2.16( c)] correspond to endomorphisms Gx and x p , respectively. A2.2.18 Theorem. Let R be a commutative F,-algebra. Then the endomorphism ring of the additive formal group law F, over R is the noncommutative power series ring R((S)) in which Sa = aPSfor a E R. f i e elements a and S correspond to the endomorphisms a x and x p , respectively.

378

A2. Formal Group Laws

Proof of A2.2.18. An endomorphism f ( x ) of F, must satisfy f ( x + y ) = f(x)+f(y). This is equivalent to f ( x ) = z~?, a,xP' for a, E R. The relation Sa = aPS corresponds to ( a x ) "= aPxP. W There is an amrsing connection between this endomorphism ring and the Steenrod algebra. Theorem A2.2.18 implies that the functor which assigns to each commutative F,-algebra R the strict automorphism group of the additive formal group law is represented by the ring

P = FP[a,, ~

2

. ., . 3

since a, = 1 in this case. The group operation is, represented by a coproduct A: P+ POP. To compute Aa,, letf,(x) = Z a,lxp ,fi(x) = Z a;xpk,andf(x) = fi( fi(x)) = Z a,xp' with ah= a,N = a, = 1. Then we have f ( x ) = C a;(C a;xpJ)ph= a;(a,) I I P Jx P J + k . It follows that ha,=

1

aP,~,Oa,

with

a,=1,

OStCn

i.e., P is isomorphic to the dual of the algebra of Steenrod reduced powers. Before proving A2.2.17 we need an improvement of A2.2.6.

A2.2.19 Lemma. Let F be a formal group law over afield K of characteristic p > 0, and let f(x) be an endomorphism of F. Then f(x) =

CFU,X"' 120

for some a, E K .

Roo$ Suppose inductively we have f ( x ) = ZZi' alxp'+ F f m ( ~ P mthis ), being trivial for rn = 0. Then set a, =f1,(0) and consider the power series g(x"") = f , ( x " " ) -Famx"m. By A2.2.13 this is an endomorphism and we have g L ( 0 ) = 0, so by A2.2.6 g ( x p " )=fmtl(xPmi'),completing the inductive step and the proof. A2.2.17 will follow easily from the following.

A2.2.20 Lemma. Let E ( F,,) be the endomorphism ring of F,, (A2.2.10)over afieid K containing F, where q = p n . Then ( a ) i f f ( x ) = E F ~ a , xisp 'in E ( F , , ) , then each u,EF,; ( b ) for ~ E F , a, x e E ( F , ) ;

( c ) X"E E ( F , ) ; and ( d ) E ( F , ) / ( p ) = E , / ( p ) = F , ( S ) / ( S " ) with S a = aPS.

2. Classification and Endomorphism Rings

379

Proof: (a) By the definition of F,, A2.2.10 and A2.214 we have

A2.2.21

[ p](x) = XP".

Any endomorphism f commutes with [ p ] so by A2.2.19 we have

This must equal

Hence = ai for all i and a, E F,. (b) It suffices to prove this for K = F,. F,, can be lifted to a formal group law @,, over w(F,) (A2.2.15) by the obvious lifting of 8: V+F, to W(F,). It suffices to show that wx is an endomorphism of F,,if w 4 - w = 0. By A2.2.2 F,, has a logarithm of the form log(x ) = C a,xq'

so log(wx) = w log(x) and wx is an endomorphism. (c) This follows from the fact that F,, is defined over F, so Fn(xp,y p )= Fn(x, Y I P . ( d ) By A2.2.21, ( b ) and ( c ) , f ( x ) ~ p E ( F iff , , )ai= O for i < n . It follows that for f ( x ) , g(x) E E(F,,), f- g mod ( p ) E ( F , ) iff f ( x ) = g ( x ) mod (x'). Now our lifting F,,of F, above has l o g x = x m o d ( x 4 ) , so F , , ( x , y ) = x + y mod ( x , y),. It follows that E ( F,,)/(p ) is isomorphic to the corresponding quotient of E(F,) over F,, which is as claimed by A2.2.17. ProofofA2.2.17. By A2.2.16(c) En is generated by w and S. The corresponding elements are in E ( F,,) by A2.2.20(b) and (c). The relation Sw = w p S corresponds as before to the fact that (Wx)" = Wpxp,where W is mod ( p ) reduction of w. Hence we have a homomorphism A : En-,E ( F , , ) which is onto by A2.2.19. We know [A2.2.13(c)] that E(F,) is a free Z,-module. It has rank n z by A2.2.20(d), so A is 1-1 by A2.2.16(a). H

APPENDIX 3

TABLES OF HOMOTOPY GROUPS OF SPHERES

In this appendix we collect most of the known values of the stable homotopy groups of spheres for the primes 2, 3, and 5 . The results of Toda [ 5 ] on unstable homotopy groups are shown in Table A3.6. In Fig. A3.1 we display the classical Adams &-term for p = 2, Ext >‘( z/ ( 2 ) , z/ (2)1 for t - s 5 61, along the differentials and group extensions. The main reference for the calculation of Ext is Tangora [ I], which includes a table showing the answer for t - s 5 70. We use his notation for the many generators shown in Ext. His table is preceded by a dictionary (not included here) relating this notation to that of the May SS, which is his main computational tool. In our table each basis element is indicated by a small circle. Multiplication by the elements ho, h , , and h, is indicated, respectively, by vertical lines and lines with slope 1 and f . Most multiplicative generators are labeled, but there are a few unlabeled generators due to limitations of space. In each case the unlabeled generator is in the image of the periodicity operator P (denoted by n in Section 3.4), which sends an element x E Ext”’ to the Massey product (Section A1.4) (x,

h i , h , ) ~EX^^+^.'+'^.

Differentials are indicated by lines with negative slope. For t - s s 2 0 these can be derived by combining the calculation of Ext in this range due to May [l] with the calculation of the corresponding homotopy groups by 380

I

I

I

,

\

m 0

w N W

3

M 8

“3

‘0

9

c!

0

00

W

P

0

N

l l

382

22

20

18

16

14

t 10

a

6

4

2

0

A3. Tables of Homotopy Groups of Spheres

383

A3. Tables of Homotopy Groups of Spheres

30 P’h 1 A p 7 h 2 28

26

24

22

20

1 ’

16

14

12

10

8

6

d

_

44

46

48

50

t Figure A3.lc

54

52 I

56

58

_

60

~

~

62

d

The Adams spectral sequence for p

= 2 , 4 4 5 r - s 5 61.

(Differentials tentative.)

I

t-

I

W

I

v)

I

v

I

0

I

N

I

b -

m

*

0

r-l. Do

0 \o

0 9

m N

m O

00 N

N W

N 0

, -

% $

A3. Tables of Homotopy Groups of Spheres

385

Toda [ 5 ] . For 21 5 t - s I45 the results can be found in various papers by Barratt, Mahowald, Milgram, and Tangora and most recently in Bruner [2], where precise references to the earlier work can be found. Differentials in the range 46 5 t - s 5 61 have been computed (tentatively in some cases) by Mahowald (unpublished) and are included here with his kind permission. Exotic group extensions and some exotic multiplications by h, and h2 are indicated by broken lines with nonnegative slope. In Fig. A3.2 we display the Adams-Novikov E,-term for p = 2 in the range t - s 5 39. The method used is that of Section 4.4, where the calculation is described in detail through dimension 25. The small circles in the chart indicate summands of order 2 . Larger cyclic summands are indicated by squares. All such summands in this range have order 4 except the one in which has order 8. The solid and broken lines in this figure means the same thing as in Fig. 3.1 as described above. This figure does not include the v,-periodic elements described in 5.3.7, i.e., the elements in the image of the J-homomorphism and the elements constructed in Adams [l]. In Table A3.3 we list the values of the 2-component of the stable stems 7 ~ ;for k 5 45. Again we omit the v,-periodic elements described in 5.3.7. These omitted summands are as follows. Z Z/(2) z/(4) Z/(2"'") Z/(2) (Z/(2))2 Z/(8)

for for for for for for for

k = 0, k = l or 2, k = 3, k = 8 t -1, where t is an odd multiple of 2", k = O or 2 m o d ( 8 ) and k > 7 , k = 1 mod (8) and k > 7 , and k = 3 mod(8) and k > 7 .

In Tables A3.4 and A3.5 we do the same for the primes 3 and 5, recapitulating the results obtained in Sections 7.4 and 7.5, respectively. Again we omit the v,-periodic elements described in 5.3.7, which in these cases are (in positive dimensions) precisely im J, i.e., Z for k = O and Z / ( p m + ' ) for k = (2p - 2) t - 1, where t = sp" and s is prime to p. In Fig. A3.6 we reproduce the table of unstable homotopy groups of spheres through the 19-stem, given in Toda [5].

A3. Tables of Homotopy Groups of Spheres

386

Table A 3 3

ns at p = 2 " Stem 6 8 9 14

Toda's name

h:

82/2

= ( Y 2 , 2, 9 ) Y3 U2

7)K

1)*

=(a,2u, 7 )

9v* YK

18

ANSS name

Y2 E

K

15 16 17

Tangora's name

Y* = (u,2u, Y)

C O

82

h:h3 h: d0 h1do hlh4 h34 h2do

a,8 2

h2h4r

h0h2h4

hih2h4= h:h4

84/4

83

"I83

a184/4

8413

a18413 a2/283= a2/284/4 84/2,2

48412.2 = a:84/3

19

d =(u2+

C1

92

20

K

g

84

22 4K

2p4 = x20= (2, a?,B4/d z x 2 0 = .&283

q2K

hog h k h34 h, g h2c1 PdO

Stem

Group

Tangora's name

ANSS name

23

Z /2 @ Z / 8

K,

q, P)

21 1);

22

24 26 28

30 31 32

33

34

35

YC?

a2/284/2.2

a,P4

a2/21)2

484

A3. Tables of Homotopy Croups of Spheres

387

Table A3.3 (continued)

36 31

38 39

40

41

42

44

45

a

All elements have order 2 unless otherwise indicated. (im I and p 8 k + l ,p8k+2omitted.)

388

A3. Tables of Homotopy Group of Spheres Table A3.4

$Primary Stable Homotopy Excluding im J

Srem EIemenr

-~ Stem Element

(See 7.4.8 and subsequent discussion.) All elements have order 3 unless otherwise indicated.

389

A3. Tables of Homotopy Groups of Spheres Table A3.5. 5-Primary Stable Homotopy Excluding im J . Stem Element 38 45 76 83 86 93 114 121 124 131 134 141 152 159 162 169 172 179 182 189 190 200 205 206 207 210 213 214 217 220 22 1 222 227 228 230 237 238 243 245 248 255 258 265 266 268

Stem Element

390

A3. Tabla d H o a n ~ g yC m p of Spheres

Table A35 ( q Sfem Element

Stem Ekmenr 529

530 53 I 532 536 541 546 551 552 555 556

558 56 I 565 567 570 57 1 572 573 5 74 579 583 589 590 594 5%

599 601 602

604 605 608

609 610 614 617

620 62 1 622 627

39 1

A3. Tables of Homotopy Groups of Spheres

Table A3.5 (continued) Stem Element

628 632 635 636 637 636 642 643 644 646 65 1 652

Stem Element

714 715 716 717 718 72 1 723 724

655 659

727 728 730 73 1

660 662 665

738 739 741

666 667 670 675 677

742 748

678 680 685 686 689 690 692 693 694 700 701 702 703 704 710 713

75 1 753 756 758 761 762 764 765 766 768 769 77 1 776 777 778 779

392

A3. Tables of Homotopy Groups of Spheres

Table A3.5 (continued) Stem Element 780 786 789 794 796 799

800 803 806 807 809 810 81 1 812 813 814 815 816 817

818 824 825 826 827 833 834 837 838 810 841 842 844 847 849 850 853

Stem Element

393

A3. Tables of Homotopy Groups of Spheres

Table A3.5 (continued)

Stem Element 914 916 917 918 920 923 925

926 928 930

93 1 932 933 934 931 940 941

942 950 95 1

952 953 954 955

Stem Eiemenf

k

O

1

l

a

2

m

2

3

0

0

0

w

2

2

2

2

12

2

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

n

3 4

5 6 7

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

2

3

15

2

2’

12.2

84.2’

2’

6

30

30

6.2

12.2’

12.2’

2

2

3

15

2

2‘

12.2

84.2’

2’

6

30

30

62

122’

122’

m 12

2’

2’

24.3

15

2

23

30

6’2

24

2

2

2

30

2

23

72.2

504.2’

23

6.2

6.2

30.2

2’

4.2’

2422’

2642

0

m

2

60

242

23

72.2

504.4

240

6

12.2

60.6

504.2’

20

2462

10568

o

2

120

2.

24.2

504.2

0

6

24.4

2‘

242

2642

0 1

2

120.12.2 ~ 4 . 2 ~20

2 4 6 2 252062

120.2~ 2‘

240.24.4 120.2~ 2’

0

1322

241242’ 1201225 13225

6.2‘

25

24’2

504.2

0

6.2

23

2.

24.2

504.2

0

6

16.4

240.2~

2’

w.23

12.2

504

12

6

16.2

11

6.2

504

2

6.2

16.2

12

6

m504

2

6.2

48.4.2

240.2

2

2‘

504

2

6

16.2

480.2

2

24

8’2

21x2~

0

m.3

8.2

480.2

24.2

2.

8’2

26442

3

4.2

480.2

23

25

8’2

2642’

26

248’2

2642’

8 9

10

13 14 15 16 17 18 19 20 21

2

~ 1 2 0 2‘ 240

2=

504242 2642

24

242

240.2’ 240.2

23

242’

2646

240.2

23

842

21342~

2‘

2

~0.480.2 2‘ 480.2

2642

480 4’ 2 264 25

23

25

8’2

2642’

2’

00.2.

842

2642

2‘

82’

2642

82

m2642 264 2

Toda scalculationsofunstablehomotopygroups~ , , + ~ ( S ” ) f o=s rkn+2and k L 19. For n > k + 1 the group is isomorphic to the one for n = k + 1. The notation a.b.c. . . . denotes the direct sum of cyclic groups of orders a, b, c, etc. T h e notation a’ denotes j cyclic groups, each having order j. (After Toda [ 5 ] . ) FigureA3.6

REFERENCES

ADAMS,J. F. [I] On the groups J ( X ) , IV, Topology 5 (1966), 21-71. [2] On the groups J ( X ) , 11, Topology 3 (1965), 137-171. [3] On the structure and applications of the Steenrod algebra, Comm. Math. Helu. 32 (1958), 180-214. [4] Part I11 and [5] Part I1 of “Stable Homotopy and Generalised Homology.” Univ. of Chicago Press, Chicago, Illinois and London, 1974. [6] Lectures on generalised cohomology, Lecture Notes in Math. 99 (1969). [7] Stable homotopy theory, Lecture Notes in Math. 3 (1966). [8] On Chern characters and the structure of the unitary group, Roc. Cambridge Philos. SOC.57 (1961), 189-199. [9] Infinite loop spaces, Ann. of Math. Srudies No. 90 (1978). ADAMS,J. F., A N D PRIDDY,S. B. [lo] Uniqueness of BSO, Math. hoe. Cambridge Philos. Soc. 80 (1978), 475-509. ADAMSJ. F., A N D MARGOLIS,H.R. [ l l ] Modules over the Steenrod algebra, Topology 10 (1971), 271-282. ADAMS,J. F. [12] On the non-existence of elements of Hopf invariant one, Ann. of Math. 7 2 (1960), 20-104. ADAMS,J. F., A N D ATIYAH,M. F. [13] K-theory and the Hopf invariant, Quart. J. Math. Oxford Ser. (2) 17 (1966), 31-38. ADAMS,J. F. [ 141 Operations of the nth kind in K-theory and what we don’t know about RPm, New developments in topology, London Marh. SOC.Lecture Note Ser. 11 (1974). [15] The Kahn-Priddy theorem, Roc. Cambridge Philos. SOC.73 (1973), 45-55. [ 161 Vector fields on spheres, Ann. of Marh. 75 (1962), 603-632. [17] A periodicity theorem in homological algebra, Proc. Cambridge Philos. SOC.62 (1966), 365-377. J. H. C., A N D MILLER,H. R. ADAMS,J. F., GUNWARDENA, [I81 The Segal conjecture for elementary abelian p-groups, to appear in Topo/ogy. 395

396

References

AIKAWA,T. [ l ] 3-dimensional cohomology of the mod p Steenrod algebra, Math. Scand. 47 (1980), 91-115. ANDERSON,D. W., A N D DAVIS,D. W. [ l ] A vanishing theorem in homological algebra, Comm. Math. Helu. 48 (1973), 318-327. ANDERSON,D. W., A N D H O D G K I NL. , [2] The K-theory of Eilenberg-MacLane complexes, Topology 7 (19681, 317-329. ARAKI,S. [ l ] “Typical Formal Groups in Complex Cobordism and K-Theory.’’ Lecture Notes Math., Kyoto Univ. 6, Kinokuniya Book Store, 1973. AUBRY,M. [l] Calculs d e groupes d’homotopie stables de la sphere, par la suite spectral d’AdamsNovikov, Math. Z. 185 (1984), 45-91. BAAS, N. A. [ l ] On bordism theory of manifolds with singularity, Math. Scand. 33 (1973). 279-302. BAHRI,A. P. [ l ] A direct summand in H*( M0(8), Z2),Proc. Amer. Math. SOC.78 (1980). 295-298. BARRATI-,M. G., MAHOWALD,M. E., A N D TANGORA,M. C. [ l ] Some differentials in the A d a m spectral sequence-11, Topology 9 (1970), 309-316. M. E. BARRATT,M. G., JONES, J. D. s., A N D MAHOWALD, [2] Relations amongst Toda brackets and the Kervaire invariant in dimension 62,J. London Mafh.SOC.30 (1985), 533-550. BENDERSKY,M., CURTIS,E. B., A N D MILLER,H. R. [ l ] The unstable Adams spectral sequence for generalized homology, Topology 17 ( 197R), 229-248. BENDERSKY,M. [2] Some calculations in the unstable Adams-Novikov spectra sequence, Publ. RIMS, Kyoto Uniu. 16 (l980), 739-760. BENDERSKY,M., CURTIS,E. B., A N D RAVENEL,D. C. [3] EHP sequences in BP theory, Topology 21 (1982), 373-391. BOTT, R. [ l ] The stable homotopy of the classical groups, Ann. of Math. ( 2 ) 70 (1959). 313-337. [2] The space of loops on a Lie group, Michigan Math. J. 5 (1958). 35-61. BOUSFIELD,A. K., A N D KAN, D. M. [ l ] Homotopy limits, completions and localizations, Lecture Notes in Math. 304 (1972). , G . , RECTOR, D. L., A N D BOUSFIELD, A. K., CURTIS, E. B., KAN, D. M.,Q U I L L E N D. SCHLESINGER, J. W. [2] The mod-p lower central series and the A d a m spectral sequence, Topology 5 (19661, 331-342. BOUSFIELD,A. K., A N D KAN, D. M. [3] The homotopy spectral sequence of a space with coefficients in a ring, Topology I 1 (1972), 79-100. BOUSFIELD,A. K., A N D CURTIS,E. B. [4] A spectral sequence for the homotopy of nice spaces, Trans. Amer. Math. SOC. 151 (1970), 457-479. BROWDER,W. [l] The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. 90 (1969), 157-186. BROWN,E. H., A N D PETERSON,F. P. [l] A spectrum whose 2, cohomology is the algebra of reduced p-th powers, Topology 5 (1966), 149-154.

References

397

BROWN,E. H. [2] Cohomology theories, Ann. ofMath. 75 (1962), 467-484; 78 (1963), 201. BROWN,E. H., A N D GITLER,S. [3] A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283-295. M. BRUNER,R., LEWIS,G., MAY,J. P., MCCLURE,J., A N D STEINBURGER, [ l ] “H, ring spectra and their applications.” To appear in Lecture Notes in Math. [2] A new differential in the A d a m spectral sequence, Topology 23 (1984), 271-226. CARLSSON,ti. [ l ] Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. oJ’Math. 120 (1984), 189-224. CARTAN,H., A N D EILENBERC,S. [ 11 “Homological Algebra.” Princeton Univ. Press, Princeton, New Jersey, 1956. CARTIER,P. [ l ] Modules associCs a un groupe formel commutatif, Courbes typiques, C. R. Acad. Sci. Paris Sir. A 265 (1967), 129-132. CASSELS,J. W. S., A N D FROHLICH,A. [ l ] “Algebraic Number Theory.” Academic Press, New York, 1967. COHEN, F., MOORE,J. c . , A N D NEISENDORFER, J. [l] The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. 110 (1979), 549-565. COHEN,R. L., A N D GOERSS,P. [ 11 Secondary cohomology operations that detect bomotopy classes, Topology 23 (1984). 177-194. COHEN,R. L. [2] “Odd Primary Infinite Families in Stable Hornotopy Theory.” Amer. Math. SOC.242 (1981). CONNER,P. E., A N D FLOYD,E. E. [ 11 The relation of cobordism to K-theories, Lecture Notes in Moth. 28 (1966). [2] “Differentiable Periodic Maps.” Ergeb. Math. Grenzgeb. (N.F.), Band 33. SpringerVerlag, Berlin and New York, 1964. CONNER,P. E., A N D SMITH,L. [3] On the complex bordism of finite complexes, Publ. Math. 1.H.E.S. 37 (1969), 117-221. CONNER,P. E. [4] Differentiable periodic maps, 2nd edition, Lecfure Notes in Math. 738 (1979). CURTIS,E. B. [ I ] Simplicia1 homotopy theory, Ada in Math. 6 (1971), 107-209. DAVIS,D. M., A N D MAHOWALD, M. E. [ l ] v,- and v,-periodicity in stable homotopy theory, Amer. J. Math. 103 (1981), 615-659. DAVIS,D. M. [2] An infinite family in the cohomology of the Steenrod algebra, J. Pure Appl. Algebra 21 (1981), 145-150. [3] On the cohomology of M0(8), Adem Symposium, Contemp. Math. 12 (1982), 91-104. DAVIS,D. M.,A N D MAHOWALD,M. E. [4] The nonrealizability of the quotient A//A, of the Steenrod algebra, Amer. J. Math. 104 (1982), 1211-1216. [5] Ext over the subalgebra A, of the Steenrod algebra for stunted projective spaces, Canad. Math. Soc. Con$ Roc. 2 Part 1 (1982), 297-342. DAVIS,D. M. [6] The splitting of BO(8) A bo and MO(8) A bo, Trans. Amer. Math. SOC.276 (1983),671-683.

398

References

DIEUDONNE,J. [ l ] Groupes de Lie et hyperalgebris de Lie sur un corps de caractiristique p > 0 (VII), Math. Ann. 134 (1957), 114-133. DOLD, A., A N D THOM, R. [I] Quasifaserungen und unendliche symmetriche Produkte, Ann. of Math. (2) 67 (19S8), 239-281. ECKMANN,B. [I] Gruppentheoretischer Beweis des Satzes von Hurwitz-Radon iiber die Komposition quadratischer Formen, Comment. Math. Helu. 15 (1942), 358-366. EILENBERG,S., A N D MOORE, J. C. [I] Foundations of relative homological algebra, Mem. Amer. Math. SOC.55 (1965). EILENBERG,S., AND MACLANE,S. I23 Relations between homology and homotopy groups of spaces, Ann. of Math. (2) 46 (194S), 480-509. FLOYD, E. E. [I] Actions of (2,)’ without stationary points, Topology 10 (1971). 327-336. FRojHLIcn, A. [I] Formal groups, Lecture Notes in Math. 74 (1968). GIAMBALVO, V., A N D PENGELLEY, D. J. [ I ] The homology of M spin, Math. Proc. Cambridge Philos. SOC.95 (1984), 427-436. GIAMBALVO, V. [2] On (S)-cobordism, Illinois J. Math. 15 (1971). 533-541. GRAY, B. W. [I] On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8 (1969), 219-232. GUNAWARDENA, J. H. C. [I] Segal’s conjecture for cyclic groups of (odd) prime order, J. T. Knight Prize Essay, Cambridge, 1979. HARPER, J. R., A N D MILLER,H. R. [ l ] On the double suspension homomorphism at odd primes, Trans. Amer. Math. SOC.273 (1982), 319-331. HAZEWINKEL,M. [I] “Formal Groups and Applications.” Academic Press, New York, 1978. [2] Constructing formal groups I: the local one dimensional case, J. Pure Appl. Algebra ( 9 ) (1977), 131-150. [ 3 ] Constructing formal groups 11: the global one dimensional case, J. Pure Appl. Algebra (9) (1977), 151-162. HIGGINS,P. J. [l] “Categories and Groupoids.” Van Nostrand Reinhold, London, 1971. HILTON,P., A N D STAMMBACH,U. [ l ] “A Course in Homological Algebra.” Springer-Verlag, Berlin and New York, 1970. HIRZEBRUCH,F. [ 11 “Topological Methods in Algebraic Geometry.” Springer-Verlag, Berlin and New York. 1956. HOPF, H. [ l ] h e r die Abbildungen von Spharen auf Spharen neidrigerer Dimension, Fund. Math. 25 (1935), 427-440. HUREWICZ,W. [ l ] Beitrage zur Topologie der Deformationen, Nederl. Akad. Werensch. Proc. Ser. A 38 (1935), 112-119, 521-528; 39 (1936). 117-126, 215-224.

References

399

JAMES, I. M. [l] Reduced product spaces, Ann. of Math. 62 (1953), 170-197. [2] Whitehead products and vector fields on spheres, Proc. Cambridge Philos. SOC.53 (1957), 817-820. [3] Spaces associated with Stiefel manifolds, Proc. London Math. SOC.(3) 9 (1959), 115-140. JOHNSON,D. C., MILLER,H. R., WILSON,W. S., A N D ZAHLER,R. S. [ l ] Boundary homomorphisms in the generalised Adams spectral sequence and the nontriviality of infinitely many yt in stable homotopy. Reunion sobre teorie de homotopia, Northwestern Univ. 1974, SOC.Mat. Mexicana (1975), 47-59. JOHNSON, D. C., A N D WILSON, W. S. [2] Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327-353. [3] BP-operations and Morava’s extraordinary K-theories, Math. Z. 144 (1975), 55-75. JOHNSON,D. C., A N D YOSIMURA,Z. [4] Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), 117-136. JOHNSON,D. C., AND WILSON,W. S. [5] The Brown-Peterson homology of elementary p-groups, Amer. 1. Math. 107 (1985), 427-453. KAHN, D. S. [ 11 Cup4 products and the Adams spectral sequence, Topology 9 (1970), 1-9. KAHN,D. S., A N D PRIDDY, S. 8. [2] The transfer and stable homotopy, Math. Proc. Cambridge Philos. SOC.83 (1978), 103-1 11. KOCHMAN,S. 0. [ l ] The symplectic cobordism ring I, Mern. Amer. Math. SOC.228 (1980). [2] The symplectic cobordism ring 11, Mern. Amer. Math. SOC.271 (1982). [3] The symplectic cobordism ring 111. To appear. [4] A chain functor for bordism, Trans. Amer. Math. Soc. 239 (1978), 167-190. [5] Uniqueness of Massey products on the stable homotopy of spheres, Can. J. Math. 32 (19801, 576-589. KUHN, N. J. [ l ] A Kahn-Pnddy sequence and a conjecture of G. W. Whitehead, Math. Proc. Cambridge Philos. SOC.92 (1982), 467-483; 95 (1984), 189-190. LANDWEBER,P. S. [ l ] BP,(BP) and typical formal groups, Osaka J. Math. 12 (1975), 357-363. [2] Cobordism operations and Hopf algebras, Trans. Amer. Math. Soc. 129 (1967), 94-110. [3] Homological properties of comodules over MU,(MU) and BP,(BP), Amer. J. Math. 98 (1976), 591-610. [4] Annihilator ideals and primitive elements in complex cobordism, Illinois J. Math. 17 (1973), 273-284. [5] Associated prime ideals and Hopf algebras, J. Pure Appl. Algebra 3 (1973), 43-58. [6] Complex bordism of classifying spaces, Roc. Amer. Math. SOC.27 (1971), 175-179. [7] Invariant regular ideals in Brown-Peterson homology, Duke Math. J. 42 (1975), 499-505. LANG, S. [ 11 “Algebra.” Addison-Wesley, Reading, Massachusetts, 1965. LAZARD, M. [ l ] Lois d e groupes et analyseurs, Ann. kcoles Norm. Sup. 72 (1955), 299-400. [2] Sur les groupes de Lie formels a un paramitre, Bull. SOC.Math. France 83 (1959,251-274. [3] La non-existence des groupes de Lie formels non abtliens 6 un paramitre, C. R. Acad. Sci. Paris 239 (1954), 942-945. [4] Groupes analytiques p-adic, Publ. Math. IHES 26 (1965).

400

References

LI, H. H., A N D SINGER,W. M. [ l ] Resolutions of modules over the Steenrod algebra and the classical theory of invariants, Math. 2. 181 (1982). 269-286. LIN, W. H. [I] On conjectures of Mahowald, Segal, and Sullivan, Math. Roc. Cambridge Philos. SOC. 87 (1980), 449-458. LIN, W. H., DAVIS,D. M., MAHGWALD,M. E., A N D ADAMS,J. F. [2] Calculation of Lin’s Ext groups, Math. Proc. Cambridge Philos. SOC. 87 (1980), 459-469. LIN, W. H. [3] Algebraic Kahn-Priddy theorem, Pacific J. Math. 96 (1981). 435-455. LIULEVICIUS,A. L. [ l ] A proof of Thorn’s theorem, Comm. Math. Helu. 37 (1962-1963). 121-131. [2] The factorization of cyclic reduced power by secondary cohomology operations, Mum. Amer. Math. SOC. No. 42 (1962). [3] On the algebra BP,(BP), Lectures Notes in Math. 249 (1972), 47-53. LUBIN, [ l ] J. One parameter formal Lie group over p-adic integer rings. Ann. of Math. 80 (1964). 464-484; Correction, 84 (1966). 372. LUBIN,J., A N D TATE, J. [2] Formal moduli for one parameter formal Lie groups, Bull. SOC.Math. France 94 (1966). 49-60. MAC LANE, S. [ l ] “Homology.” Springer-Verlag. Berlin and New York, 1963. [2] “Categories for the Working Mathematician.” Springer-Verlag, Berlin and New York, 1971. MAHOWALD,M. E. [ l ] The metastable homotopy of S “ , Mem. Amer. Math. SOC. No. 72 (1967). [2] Descriptive homotopy of the image of J, Proc. of Internat. Con$ on Manifolds (1975), 255-253. Univ. of Tokyo Press, Tokyo. [3] The double suspension homorphisrn, Trans. Amer. Math. SOC.214 (1975). [4] The image of J in the EHP sequence, Ann. of Math. 116 (1982), 65-1 12. MAHOWALD,M. E., A N D MILGRAM,R. J. [5] Operations which detect Sq4 in connective K-theory and their applications, Quarr. J. Math. Oxford Ser. (2) 27 (19761, 415-432. MAHOWALD,M. E. [6] A new infinite family in 2ns, Topology 16 (1977). 249-256. [7] Some remarks on the Kervaire invariant problem from the homotopy point of view, Proc. Sympos. Pure Murh. 22 (1971), 165-169. MAHOWALD,M. E., A N D TANGORA,M. c . [8] On secondary operations which detect homotopy classes, Bol. Sot. Mar. Mexitana 12 (1967), 71-75. [9] Some differentials in the Adams spectral sequence, Topology 6 (1967), 349-369. MAHOWALD,M. E. [lo] Some homotopy classes generated by T,, Algebraic Topology Aarhus 1978, Lecture Notes in Math. 763 (1979), 23-37. [ l l ] The construction of small ring spectra, Lecture Notes in Math. 658 (1978), 234-239. [I21 The primary v,-periodic family, Math. 2. 177 (1981), 381-393. [13] The order of the image of the J-homomorphism, Bull. Amer. Math. SOC.76 (1970). 13 10- 13 13.

References

40 1

MAHOWALD,M. E., A N D TANGORA,M. C. [14] An infinite subalgebra of Ext,(Z,,Z,), Trans. Amer. Math. SOC. 132 (1968), 263-274. MAHOWALD,M. E. [15] bo resolutions, Pacijc J. Math. 192 (1981), 365-383. MARGOLIS,H. R., PRIDDY,S. B., A N D TANGORA, M. C. [ I ] Another systematic phenomenon in the cohomology of the Steenrod algebra, Topology 10 (1971), 43-46. MASSEY,W. S. [ 13 “Algebraic Topology: An Introduction.” Harcourt, New York, 1967. [2] Exact couples in algebraic topology, Ann. of Math. 56 (1952), 363-396. [3] Some higher order cohomology operations, Sympos. Internat. Alg. Top., Mexico City (1958), 145-154. MAUNDER,C. R. F. [ l ] On the differentials in the Adams spectral sequence for the stable homotopy groups of spheres, Proc. Cambridge Philos. SOC.61 (1965), 53-60; 61 (1965), 855-868. MAY, J. P. [ l ] The cohomology of restricted Lie algebras and of Hopf algebras; application to the Steenrod algebra. Thesis, Princeton Univ., 1964. [2] The cohomology ofrestricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123- 146. [3] Matric Massey products, J. Algebra 12 (1969), 533-568. [4] The Steenrod algebra and its associated graded algebra, University of Chicago preprint, 1974. [5] A general algebraic approach to Steenrod operations, Lecture Notes in Mafh. 168 (1970), 153-23 1. [6] (1982) Private communication. MILGRAM,R. J. [ l ] The Steenrod algebra and its dual for connective K-theory, Reunion sobre teoria de homotopia, Northwestern Uniu. 1974, SOC.Mat. Mexicana (1975), 127-158. [2] Group representations and the Adams spectral sequence, Pacific J. Math. 41 (1972), 157- 182. MILLER,H. R., RAVENEL,D. C., A N D WILSON,W. S. [ 11 Periodic phenomena in the Adams-Novikov spectral sequence, Ann. ofMarh. 106 (1977), 469-516. MILLER,H. R. [2] Some algebraic aspects of the Adams-Novikov spectral sequence. Thesis, Princeton Univ., 1974. MILLER,H. R., A N D WILSON,W. S. [3] On Novikov’s Ext’ modulo on invariant prime ideal, Topology 5 (1976), 131-141. MILLER, H. R. [4] On relations between A d a m spectral sequences, with an application to the stable homotopy of a Moore space, J. Atre Appl. Algebra 20 (1981), 287-312. MILLER,H. R.,A N D RAVENEL,D. C. [5] Morava stabilizer algebras and the localization of Novkov’s E,-term, Duke Math. J. 44 (1977), 433-447. MILLER,H. R. [6] An algebraic analogue of a conjecture of G. W. Whitehead, Proc. Amer. Math. SOC.84 (1982), 131-137. [7] A localization theorem in homological algebra, Mafh. Proc. Cambridge Philos. SOC.84 (1978), 73-84.

402

References

MILLER,H. R., A N D WILKERSON,c. [8] Vanishing lines for modules over the Steenrod algebra, J. Pure Appl. Algebra 22 (1981 1, 293-307. [9] The Segal conjecture for periodic groups, Northwestern Homotopy Theory Conf., Contemp. Math. 19 (1983). 233-246. MILNOR,J. W. [l] Morse theory, Ann. of Math. Studies No. 51 (1956). [2] The Steenrod algebra and its dual, Ann. ofMath. 67 1958), 150-171. MILNOR,J. W., A N D MOORE, J. c. [3] On the structure of Hopf algebras, Ann. of Math. 81 1965), 211-264. MILNOR, J. W. [4] On the cobordism ring R* and a complex analogue, Amer. J. Math. 82 (1960). 505-521. MILNOR,J . W., A N D STASHEFF,J . D. [S] Characteristic classes, Ann. ofhfarh. Studies No. 76 (1974). MIRONOV,0. K. [ l ] Existence of multiplicative structures in the theory of cobordism with singularities, Izu. Aknd. Nauk SSSR Ser. Mat. 39 (1975). No. 5, 1065-1092, 1219. MISCHENKO [ l ] Appendix 1 in Novikov [ l ] . MOORE, J. c . , A N D PETERSON, F. P. [ l ] Nearly Frobenius algebras, PoincarC algebras and their modules, J. Pure Appl. Aigebra 3 (1973), 83-93. MORAVA,J. [ l ] A product for the odd primary bordism of manifolds with singularities, Topology 18 (1979), 177-186. [2] Completions of complex cobordism, Lecture Notes in Math. 658 (1978). 349-361. [3] Noetherian localisations of categories of cobordism comodules, Ann. of Math. 121 (1985), 1-39. MOREIRA,M. R. F. [ l ] Primitives of BP,BP modulo an invariant prime ideal, Amer. J. Math. 100 (1975). 1247-1273. [2] Formal groups and the BP spectrum, J. Pure Appl. Algebra 18 (1960). 79-89. [3] The last distinguished primitive of BP,BP modulo an invariant prime ideal. To appear. Moss, R. M. F. [l] On the composition pairing of Adams spectral sequences. Roc. London Math. Soc. ( 3 ) 18 (1968), 179-182. MUMFORD,D. [ l ] “Lectures on Curves on an Algebraic Surface.” Princeton Univ. Press, Princeton, New Jersey, 1966. NAKAMURA,0. [ 11 On the squaring operations in the May spectral sequence, Mem. Fac. Sci. K-vushu Unia Ser. A 26 (1972), 293-308. [2] On the cohomology of the mod p Steenrod algebra, BUM. Sci. Engrg. Diu. Uniu. R-vukyus Math. Natur. Sci. 18 (1975), 9-58. NEISENDORFER,J. [l] 3-primary exponents, Math. Proc. Cambridge Philos. SOC.90 (1981), 63-83. NEW YORK TIMES [ l ] Editorial page, June 2, 1976. NISHIDA,G. [ I ] The nilpotence of elements of the stable homotopy groups of spheres. J. Math. Soc. Japan 25 (1973), 707-732.

References

403

NOVIKOV,S. P. [ l ] The methods of algebraic topology from the viewpoint of cobordism theories (in Russian), Izu. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855-951; translation, Math. USSR-Izu. (1967), 827-913. [2] Some problems in the topology of manifolds connected with the theory of Thom spaces (in Russian), Dokl. Akad. Nauk SSSR 132 (1960), 1031-1034; translations, Soviet Math. Dokl. 1 (1960), 717-720. [3] Homotopy properties of Thom complexes (Russian), Mat. Sb. ( N . S.) 57 (99) (1962), 407 -442. OKA, S. [ l ] The stable homotopy groups of spheres I, 11,111, Hiroshima Math. J. 1 (1971), 305-337; 2 (1972), 99-161; 5 (1975), 407-438. [2] A new family in the stable homotopy groups of spheres I, 11, Hiroshima Math. J. 5 (1975), 87-114; 6 (1976), 331-342. [3] Realizing some cyclic BP,-modules and applications to stable homotopy groups of spheres, Hiroshima Math. J. 7 (1977), 427-447. [4] Ring spectra with few cells, Japan. J. Math. 5 (1979), 81-100. O m , S., AND TODA, H. [5] 3-primary P-family in stable homotopy, Hiroshima Math. J. 5 (1975), 447-460. O m ,S., AND SHIMOMURA,K. [6] On products of the &elements in the stable homotopy groups of spheres, Hiroshima Math. J. 12 (1982), 611-626. OKA, S. [7] Small ring spectra with p-rank of the stable homotopy of spheres, Roc. Northwestern Univ. Homotopy Theory Con$, 1982. QUILLEN,D. G. [3] The spectrum of an equivariant cohomology ring I, I1 Ann. of Math. 94 (1971), 549-572, 5 73-602. PENGELLEY, D. J. [ l ] The homotopy type of MSU, Amer. J. Math. 104 (1982), 1101-1123. [2] The homology of MSO and MSU as A* comodule algebras and the cobordism ring, J. London Math. SOC.25 (1982), 467-472. PRIDDY,S. [ l ] A cellular construction of BP and other irreducible spectra, Math. Z. 173 (1980), 29-34. QUILLEN,D. G. [ l ] The Adams conjecture, Topology 10 (1971), 1-10, [2] On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. SOC.75 (1969), 1293-1298. RAVENEL,D. C. [ I ] The structure of BP,BP modulo an invariant prime ideal. Topology 15 (1976), 149-153. RAVENEL,D. C., A N D WILSON,W. S. [2] The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1977), 241-280. [3] The Morava K-theories of Eilenberg-Mac Lane spaces and the Comer-Floyd conjecture, Amer. J. Math. 102 (1980), 691-748. RAVENEL, D. C. [4] The Segal conjecture for cyclic groups and its consequences, Amer. J. Math. 106 (1984), 41 5-446. [5] The structure of Morava stabilizer algebras, Inuent. Math. 37 (1976), 109-120. [6] The cohomology of the Morava stabilizer algebras, Math. Z. 152 (1977), 287-297. [7] The nonexistence of odd primary Arf invariant elements in stable homotopy, Math. R o c . Cambridge Philos. SOC.83 (1978). 429-443.

404

References

[8] Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351-414. [9] A geometric realization of the chromatic resolution, to appear in Roc. 1. C. Moore Con$, Princeton, 1983. SCIENCE [I] June 7, 1976. SERRE,J.-P. [ I] “Corps locaux.” Hermann, Pans, 1962. [2] Homologie singuliire des espace fibrts, Ann. of Math. 54 (1951), 425-505. [3] Groupes d’homotopie et classes de groupes abelien, Ann. o f M a t h . 58 (1953). 258-294. SHIMADA,N., A N D YAGITA,N. [1] Multiplication in the complex bordism theory with singularities, Pub!. Res. Inst. Math. Sci. 12 (1976-1977). No. 1, 259-293. SHIMADA,N., A N D IWAI, A. [2] On the cohomology of some Hopf algebras, Nagoya Math. J. 30 (1967), 103-111. SHIMADA,N., A N D YAMANOSHITA, T. [3] On the triviality of the mod p Hopf invariant, Japan. J. Math. 31 (1961). 1-24. K. SHIMOMURA, [l] Novikov’s Ext2 at the prime 2, Hiroshima Math. 1. I f (19811, 499-513. SIEGEL,C. L. [ l ] “Topics in Complex Function Theory,” Vol. 1. Wiley (Interscience), New York, 1969. SINGER,W. M. [11 A new chain complex for the homology of the Steenrod algebra, Math. Roc. Cambridge Philos. SOC.90 (1981), 279-292. [2] Invariant theory and the Lambda algebra, Trans. Amer. Math. SOC.280 (1983). 573-693. [3] The algebraic EHP sequence, Trans. Amer. Math. SOC.201 (1975), 367-382. [4] Squaring operations and the algebraic EHP sequence, J. &re Appl. Algebra 6 (1975), 13-29. [5] Extension theory for connected Hopf algebras, J. Algebra 21 (1972), 1-16. [6] Steenrod squares in spectral sequences, Trans. Amer. Math. SOC.175 (1973), I , 327-336; 11, 337-353. SMITH,L. [I] On realizing complex bordism modules, Amer. J. Math. 92 (1970), 793-856. [2] On realizing complex bordism modules, IV, Applications to the stable homotopy groups of spheres, Amer. J. Math. 99 (1971), 418-436. SNAITH,V. P. [I] Stable decomposition of Cl”Z”X, J. London Math. SOC.(2) 7 (1974), 577-583. SPANIER,E. H. [I] “Algebraic Topology.” McGraw-Hill, New York, 1966. STEENROD,N. E., A N D EPSTEIN, D. 8. A. [I] Cohomology operations, Ann. of Math. Studies No. 50 (1962). STONG,R. E. [I] “Notes on Cobordism Theory.” Princeton Univ. Press, Princeton, New Jersey, 1968. Z,) and H*(BU(k, . . . , a), Z2). Trans. Amer. Math. [2] Determination of H*(BO(k, . . . ,a), SOC.107 (1963). 526-544. [3] Cobordism of maps, Topology 5 (1966). 245-258. SULLIVAN, D. P. [l] Genetics of homotopy theory and the A d a m conjecture, Ann. ofkfath. I 0 0 (1974), 1-79. TANGORA,M. C. [l] On the cohomology of the Steenrod algebra, Math. Z. 116 (1970), 18-64.

References

405

[2] Some remarks on the lambda algebra, Lecture Notes in Math. 658 (1978), 476-487. [3] Generating Curtis tables, Lecture Notes in Math. 673 (1978), 243-253. [4] Computing the homology of the lambda algebra, Mem. Amer. Math. SOC.337 (1985). [S] Some extension questions in the A d a m spectral sequence, Proc. Adv. Study Inst. Algebra Top., Aarhus (1970). 578-587. SWEEDLER, M. E. [l] Hopf Algebras, Benjamin, New York, 1969. SWITZER,R. M. [ 11 Algebraic topology, homoropy and homology, Springer-Verlag, Berlin and New York, 1975. THOM, R. [l] Quelques propri6tks globales des varietks differentiables, Comm. Math. Helu. 28 (1954), 109- 18 1. THOMAS,E., A N D ZAHLER,R. S. [ l ] Nontriviality of the stable homotopy element y,, J. Pure and Applied Algebra 4 (1974), 189-203. TODA, H. [ l ] On spectra-realizing exterior parts of the Steenrod algebra, Topology 10 (1971), 53-65. [2] Extended p-th powers of complexes and applications to homotopy theory, Proc. Japan Acad. 44 (1968), 198-203. [3] An important relation in the homotopy groups of spheres, Proc. Japan Acad. 43 (1967), 893-942. [4] p-primary components of homotopy groups, I. Exact sequences in Steenrod algebra; 11. Mod p Hopf invariant; 111. Stable groups of the sphere; IV. Compositions and toric constructions, Mem. Coll. Sci., Uniu. Kyofo, Ser. A 31 (1958), 129-142, 143-160, 191-210; 32 (1959), 288-332. [5] Composition methods in homotopy groups of spheres, Ann. of Marh. Studies No. 49 (1962). [6] Algebra of stable homotopy of 2,-spaces and applications, J. Math. Kyoto Uniu. 11 (1971), 197-251. TOM DIECK, T. [ 11 Actions of finite abelian p-groups without stationary points, Topology 9 (1970), 359-366. WANG, J . S. P. [ 13 On the cohomology of the mod-2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Illinois J. Math. 11 (1967), 480-490. WELCHER,P. J . [ I ] Symmetric products and the stable Hurewicz homomorphism, Illinois J. Math. 24 (19801, 527-544. [2] Symmetric fiber spectra and K9(n)-homology acyclicity, Indian J. Math. 30 (1981), 801-812. WELLINGTON,R. J. [I] The computation of Ext groups for modules overthe Steenrod algebra. (Tobe published.) WELLINGTON,R. J. [ l ] The computation of Ext groups for modules over the Steenrod algebra, to appear. WILSON,W. S. [I] The a-spectrum for Brown-Peterson chomology, Part I, Comment. Marh. Helu. 48 (1973), 45-55. [2] The n-spectrum for Brown-Peterson cohornology, Part 11, Amer. J. Math. 97 (1975), 101-123. [3] “Brown-Peterson Homology, an Introduction and Sampler.” Regional Conference Series in Math., No. 48. AMS, Providence, Rhode Island, 1980.

406

References

WURGLER, V.

[ [] On products in a family of cohomology theories associated to the invariant prime ideals of ?r,(BP), Comment. Math. Helu. 52 (1977). 457-481. [2] A splitting for certain cohomology theories associated to BP*(-), Manuscripta Math. 29 (1979), 93-111. [3] On the relation of Morava K-theories to Brown-Peterson homology, Enreign. Math. NO.26 (1978), 269-280. [4] Cobordism theories of unitary manifolds with singularities and formal group laws, Math. 2. 150 (1976), 239-260. YACITA, N. [ I ] On the algebraic structure of cobordism operations with singularities, J. London Math. SOC.16 (1977). 131-141. [2] On the Steenrod algebra of Morava K-theory, J. London Math. SOC.22 ( 1980), 423-438. [3] The exact functor theorem for BP,/I,-theory, Proc. Japan Acad. 52 (1976). 1-3. YOSIMURA,

z.

[I] Projective dimension of Brown-Peterson homology with modulo (p, v,, . . . vn-,) coefficients, O s a h J. Math. 13 (1976), 289-309. ZACHARIOU,A. [l] A subalgebra of Ext,(Z,, Z2), Bull. Amer. Math. SOC.73 (1967), 647-648. ZAHLER,R. [ I ] The Adams-Novikov spectral sequence for the spheres, Ann. of Math. 96 (1972),480-504. [2] Fringe families in stable homotopy, Trans. Amer. Math. SOC.224 (1976). 243-253.

INDEX

A

ai, 73, 82, 152 an,,,190 A, see Steenrod algebra A( l), 76-79, 1 10- 1 1 1, see also P( 1) A(2), 88-89, 1 1 1 A(n), 103, 11 1 - I 1 3

ABC theorem, 283 Abelian category, 3 1 1 Adams conjecture, 6, 196 Adams filtration, 62, 187- 188 Adams-Novikov spectral sequence (ANSS), 13, 18-29, 151-169 algebraic, 152, 153 differentials, 151, 159, 195, 199, 206, 289, 291,384 &-term, 18, 19, 25, 151, 159, 160, 167, 195, 384 group extensions, 169, 384 sparseness, 1 5 1 unstable, 136- 137 Adams periodicity, 102, 104- 106, 108, 116- 117, 380, see also Periodicity, vIAdams resolution, 49, 55 canonical, 60 definition, 49 generalized, 57 Adams spectral sequence (ASS), 9- 12, 47-68 for bo, 76-78,83 for bu, 75 computations, 69- 1 I7

connecting homomorphism, see Connecting homomorphism, generalized convergence, 58,60 definition, 5 1 differentials, 9, 11, 12, 51 -52, 110, 117, 169,381-383 &-term, 60 &-term, 10,48, 58, 86, 99, 168, 381-383 &term, 53 generalized, 57 -68 group extensions, 55, 71, 117, 381-383 for MO, 75 modified, 117 for MU, 72-74 naturality, 6 1-62 oddprimary, 11-12, 151-152 periodicity, see Adams periodicity product, see Smash product pairing, Composition pairing resolution, see Adams resolution unstable, 9 1 vanishing line, 85,97-98, 101- 102, 116, 155 Adams vanishing line, see Adams spectral sequence Admissible monomials, see Lambda algebra “Algorithm,” 285 CY,, 19, 37, 39,40-43, 172, 213, 216 ANSS, see Adams- Novikov spectral sequence Approximation lemma, 104 Araki generators, 137, 370 407

408

Index

Arf invariant, see Kervaire invariant Artin local ring, 222 ASS, see Adams spectral sequence Associated bigraded group, 5 1 Automorphism group of formal group law, 25,219, see also Z(n) cohomology, 27,230-244 eigenspace decomposition, 228 group ring, 226 matrix representation, 229

B b, (in Ext), 11, 101, 244 b,(in N.(MU)), 18, 71, 123, 254, 361 b, 82, 143, 145, 154, 256 B(n), 133, 134 Bernoulli numbers, 198 p,,20, 157, 159, 167, 169, 177,208-215 products, 2 I6 - 17 pSB, 21, 157, 159, 167, 169, 202-203, 211-212.217 Bimodule, 59, 306 Birth, 33 Bockstein operation, see Steenrod operations Bockstein spectral sequence (BSS), 153, 175-176 bo, 59,76-78 Bordism groups, 14, 130 Bott periodicity, 5, 78, 253 BP,24, 126 BP.(BP), 128-129, 137-150 conjugation, 143- 144 coproduct, 143 filtration, 146- 150, 23 1 right unit, 137, 144- 145, 371 BP, 132 Browder's theorem, 44, 101 Brown-Gitler spectra, 114- 115, 201 Brown-Peterson spectrum, see BP Brown-Peterson theorem, 126 BZ,, see Symmetric group bsp, 39 BSS, see Bockstein spectral sequence bu, 59, 75

Cartier's theorem, 362 CESS, see Cartan-Eilenberg spectral sequence Change-of-ringstheorem, 26, 175,220- 226 Miller-Ravenel proof, 221 -223 Milnor-Moore, 331 Morava's proof, 223-226 Chromatic filtration, 170 Chromatic notation, 27, 173, 177 Chromatic resolution, 173 Chromatic spectral sequence (CSS),26 - 29, 117, 171-218,258-259 differentials, 181, 187, 203-204 Cobar complex, 70, 323, 347 chromatic, 174, 259 Cobar resolution, 323 Cobordant, 13- 14, 120 Cobordism ring, complex, 14, 120 Cogroup object, 24, 306 Cogroupoid object, 24, 307, 309 Cohen-Goerss theorem, 101,207 Cohen -Moore - Neisendorfer exponent theorem, 5 Coker J , 6, see also Homotopy groups of spheres Comodule, 3 10 algebra, 3 I0 extended, 321 filtered, 329 injective, 3 19 relatively injective, 32 1 tensor product, 3 10 weak injective, 259 Comodule algebra structure theorem, 3 17 Completion, 54, 58, 60, 133, 221 Composition pairing, 64 Connecting homomorphism, generalized, 63-64.64-68 Conner- Floyd conjecture, 133 Conner-Floyd isomorphism, 135 CP", 16-17, 121, 123-124 Cotensor product, 31 1 Cotor, 320 CSS, see Chromatic spectral sequence Cup product, 63, 324 Curtis algorithm, see Lambda algebra

C

c,, 85, I00 cij, 144, 145 Cartan-Eilenberg spectral sequence (CESS), 76,79,105, 112,152,259.331 -332,335

D Davis- Mahowald elements, 2 11 , 2 12-2 13 Death, 33

Index An, 143 Derived functor, 3 19 Descent, see Method of infinite descent Detection theorem, 246 Differentials, see specific spectral sequence Division algebra, 230, 243, 377 Double complex, 326, 330 E

E(n), 135, 220 Edge theorem, 181 EHP sequence, 29 - 46 algebraic, 9 I EHP spectral sequence (EHPSS), 30-46 algebraic, 9 1 differentials, 95 -96 stable, 33 stable zone, 3 I , 92 superstable, 45 vanishing line, 30.92 Eilenberg-Mac Lane space, 7, 133, 136, see also K(2, 3) Eilenberg-Mac Lane spectrum, 38, 48, 55-56, 74, 121, 130 Einhangung, see Suspension Elliptic integral, 356 Equivalence of functors, 22 1 q, see Hopf map q j , 39,43-44, 207 qR, see BP.(BP), right unit Exact couple, 50- 52 Exponent theorem, 5 Exponential series, 197 Ext over A , see Adams spectral sequence over BP.(BP), see Adams- Novikov spectral sequence definition, 320 Ext', 100, 184- 192 EXt', 100- 101,201-215 Extension of Hopf algebras, 76, 1 12, 3 17 cocentral, 105, 335 Extension of Hopf algebroids, 3 16- 3 17 F

Filtration, 5 1, 327, see also Adams filtration Finite field, 375 Finiteness theorem, 4 Formalgroup law, 16, 140-142,354-379,

see also Automorphism group, Exponential series, and Logarithm additive, 16, 17, 356 classification theorem, 25, 374 definition, 355 endomorphism ring, 226,374-375, 377-379 examples, 356, 373 height of, 25, 372-373 Hopf algebroid, 36 1, 366 inverse, 355 multiplicative, 16, 17, 356 p-typical, 361, 363 strict isomorphism, 356 universal, 17, 357-358, 364 Formal sum, 362 Freudenthal suspension theorem, 3 G g,,82-83, 101

G (power series group), 18- 19,25-26 cohomology, 18 - 2 1 r(n), 258 yt, 20, 215, 289 Generalized homology theory, 57- 58 Geometric cycle, 13 Greek letter construction, 19-23, 27 -28, 46, 172 Groupoid, 24, 307 normal sub-, 307-308

H h,, 10-11, 11-13, 188, 189,381-382 h,, 76, 81, 82, 233, 256 H , see Hopf invariant H@. q) system, 65 Hazewinkel generators, 137, 369-370 Height, see Formal group law Hom dim, 132- 133 Homotopy groups, 3 of bo, 78 of BP, 126 of bu, 75 of complex with p cells, 268-281, 283 of Eilenberg-Mac Lane space, 7 of J, 37,40 of MU, 7 1 of orthogonal group, 5

410

Index

of spheres, 1-46,86,285-305,380-397 stable, 3 unstable, 29-46, 395-397 Homotopy inverse limit, see Inverse limit of spectra Hopf algebra, 306 Hopfalgebroid, 24-25, 60, 306, 309-319 associated Hopf algebra, 3 I3 definition, 309- 3 10 equivalent, 22 1 extension, 3 16 filtered, 328 map, 313 normal map, 3 14 split, 319, 325 unicursal, 3 14 Hopf invariant, 29, 3 1 one, 10-11, 187-188 Hopf map, 4,33 Hopf ring, I36 Hopf- Whitehead J-homomorphism, see J-homomorphism Hurewicz theorem, 3, 7 Hurwitz-Radon construction, 36

I I,, 20, 133, 137, 143, 144, 146 Im J, see J-homomorphism Invariant ideal, 3 19, 325 Invariant prime ideal, see Prime ideal, invariant Invariant regular ideal, see Regular ideal, invariant Inverse limit of spectra, 52 *-isomorphism,222

K

k,, 11-13, 101, 154 K-theory,9, 17, 34, 36, 31, 108, 135 K(n), see Morava K-theory K(n)*(K(n)),134 K(Z, 3), 8, see also Eilenberg-Mac Lane space Kahn-Priddy theorem, 38,45, 118 Kervaire invariant (O,), 39, 44, 46 odd primary, 244-253 Koszul resolution, I 1 I , 233 Krull dimension, 230-23 1 Kudo transgression theorem, 105, 353 L

L, see h a r d ring Lambda algebra, 85, 90- 100 admissible monomials, 90 Curtis algorithm, 9 I -96 generalized, 1 14- 1 16 Landweber exact functor theorem, 134- 135 Landweber filtration, 134 Llandweber- Novikov theorem, 125 h a r d comparison lemma, 359,367 - 369 Lazard ring, 18, 357 Lazard theorem, 358 LB, 125,360 Lie algebra, restricted, 80, 23 1-232 Lie group, p-adic, 226 Lifts functor, 222 Lightning flash, 98 Lin's theorem, 44-45 Liulevicius theorem, 11 Localization, 60 Logarithm, 122, 356-357

M J J (spectrum), 37 -43 James periodicity, 34, 37, 96-97 J-homomorphism, 5-6, 19, 35, 37, 39, 103, 195-198 Johnson-Miller- Wilson-Zahler theorem, 63 Johnson- Wilson spectra, 132 Johnson-Yosimura theorem, 135 Juggling, see Massey products

in,, 122, 358 hf",173 Mahowald elements, 44, 101, 208 Mahowald (root) invariant, 45-46 Mahowald's theorem (on EHP sequence), 40-41,43 Mahowald- Tangora theorem, 10 1,206 Manifold, stably cornpiex, 13, I20 Massey products, 82-83, 102, 117, 154, 286-287, 335-346

Index

411

convergence, 34 1 - 342 defining system, 337 definition, 336 differentialsand extensions, 343 - 346 indeterminacy, 338 juggling, 339-340 Leibnitz formula, 342-343 matric, 337 strictly defined, 337 May spectral sequence, 10, 79-89, 380 for A( l), 83 forA(2), 88-89 differentials, 83, 85, 87, 11 1 nonassociativity, 82 -83 for Z(n),232 Method of infinite descent, 258 Miller-Ravenel- Wilson theorem, 202 Miller- Wilson theorem, 184 Milnor-Novikov theorem, 14,7 1 Mischenko’s theorem, 122 MO, 75, 130 M0, 11 1 Mobius function, 363 Moore spectrum, 21,54, 199-201,210 MoravaK-theory, 133- 134, 136, 175 Morava-Landweber theorem, 20, 134, 138, 259 Morava stabilizer algebra, see Z(n) Morava stabilizer group, see Automorphism group of formal group law Morava vanishing theorem, 27, 175 Morava’s point of view, 25-26 Moreira formula, 192, 236 MSO, 74, 111, 130 MSp, 1 1 1, 114, 130 MSpin, 111 MSU. 59, 74, 11 I , 130 P t , 195 MU, 14,71, 120- 129, 1’30 Adams spectral sequence based on, see Adams- Novikov spectral sequence Adams spectral sequence for, 72-74 R-spectrum, 18, 136 MU.(MU), 125 N

N”, 173 Newton’s formula, 197 Nishida’s theorem, 5

Novikov spectral sequence, see AdamsNovikov spectral sequence v, see Hopf map

0

0, see Orthogonal group Oka- Shimomura theorem, 2 17 Oka-Smith-Zahler elements, 21 1 Oka- Toda theorem, 2 12 Open subgroup theorem, 233 Orientation, complex, 121 degree rn, 253 Orthogonal group, 5, 34-35 Orthogonal spectral sequence, 35

P P, see Adams periodicity, Whitehead product P(I), 27 1 -274 fin), 133, 134 Pi,see Steenrod operations Periodicity, Bott, see Bott periodicity Periodicity, James, see James periodicity Periodicity, u,, 103, 170, 192-201, see also Adams periodicity Periodicity, u, , 28, 117, 170 Periodicity operators in ASS, see Adams periodicity II,, see Adams periodicity Poincare series, 238, 239, 240, 243, 263 Prime ideal, invariant, 138, see also I,, Product of spectra, 52

Q QX, 33 Quillen’s theorem, 17- 18, 1 19, 122

R Regular ideal, invariant, 2 1, 131 Resolution by relative injectives, 323 Restriction, 80 pn. 189, 192, 193,235 Right unit, see BP.(BP) Ring spectrum, finite, 209 - 2 10,2 1 1.215

Index Root invariant, see Mahowald (root) invariant RP”, 17, 33-35,40-43,44-46

S

SZm,30 S’, 98-99 Segal conjecture, 117 Serial number, 3 1,92 - 93 Serre finiteness theorem, 4 Sene’s method, 7 -9 Serre spectral sequence, 7 Shimada-Yamanoshita theorem, 1 I Shimomura’s theorem, 2 1,203 6,see Hopf map 2(n), 26, 175, 2 19,223 244, see also Automorphism group of formal group law cohomology, 23 1 -244 filtration, 23 I H’, 235 H2,236 May spectral sequence, 232 Singularity, 131 Slope +,98, 102, 104, 116 Smash product pairing, 62-68 Smith’s theorem, 20,2 1 1 Sparseness, see Adams- Novikov spectral sequence Special unitary group, 137, 253 Spectral sequence, 325-335, see also specific spectral sequence for filtered complex, 328-329. 332 for Hopf algebroid map, 329 for long exact sequence, 326 Sd, see Steenrod operations Stable zone, see EHP spectral sequence Steenrod algebra, 70, 152, 378 filtration, 82 Steenrod operations, algebraic, 346- 353, see also Steenrod algebra Adem relations, 35 1 Bockstein, 350 Cartan formula, 35 1 in Ext, 85, 105, 246 in May SS, I I I Po, 351 Sqo, 351 suspension axiom, 352

-

Stem, 3 Stong’s theorem, 76 Strict isomorphism, see Formal group law SU, see Special unitary group Sullivan- Baas construction, 13I Suspension, 3, 29,91 double, I14 Symmetricgroup, 33-34, 38, 39,347-348

T ti, 128, 143, 365, 366, see also BP.(BP)

f,,, 146

nn),253-256 r,, 70 O,, see Kervaire invariant Thom reduction (a),187- I88,20 I , 205 Thom space, 36, 120 Thom spectrum, 130 Thom’s theorem, 14, 120 Toda bracket, 286-287 Toda differential, 15 1, 159, 262 Toda’s theorem, 159 Torsion, vn-, 170, 191 Transgression, see Kudo transgression theorem U Unit coideal, 329 V u,, 19, 20, 128, 364, 366, see also BP.(BP),

Periodicity, u,- and Torsion, v,-

L, 128, 364, 366 V(n),21-23, 209, 215, 217, 289, 291 Vanishing theorem, see Morava vanishing theorem Vector field problem, 35- 36 odd primary, 36 VT, 128, 364-365. 366, see also BP.(BP) W

w,, 141 Whitehead conjecture, 1 17- I 18 Whitehead product, 29,9 I

413

Index

Y

Witt lemma, 140 Witt ring, 376

Yoneda product, 64,251

X

Z

Cn, 162, 189, 192, 235

This Page Intentionally Left Blank