The Connective K-Theory of Finite Groups

142 Pages·2003·3.24 MB·English

by  R. R. Bruner

MEMOIRS
of the
American Mathematical Society
Number 785
The Connective K-Theory
of Finite Groups
R. R. Bruner
J. P. C. Greenlees
September 2003 Volume 165 Number 785 (second of 4 numbers) ISSN 0065-9266
American Mathematical Society
The Connective K-Theory
of Finite Groups
This page intentionally left blank
MEMOIRS
of the
American Mathematical Society
Number 785
The Connective K-Theory
of Finite Groups
R. R. Bruner
J. P. C. Greenlees
September 2003 Volume 165 Number 785 (second of 4 numbers) ISSN 0065-9266
American Mathematical Society
Providence, Rhode Island
2000 Mathematics Subject Classification.
Primary 19L41, 19L47, 19L64, 55N15; Secondary 20J06, 55N22, 55N91, 55T15, 55U20, 55U30.
Library of Congress Cataloging-in-Publication Data
Bruner, R. R. (Robert Ray), 1950-
The connective K-theory of finite groups / R. R. Bruner, J. P. C. Greenlees.
p. cm. (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 785)
"Volume 165, number 785 (second of 4 numbers)."
Includes bibliographical references and indexes.
ISBN 0-8218-3366-9 (alk. paper)
1. K-theory. 2. Finite groups. I. Greenlees, J. P. C. (John Patrick Campbell), 1959-
II. Title. III. Series.
QA3.A57 no. 785
[QA612.33]
510 sdc21
[514'.23] 2003051902
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Contents
Chapter 0. Introduction.
0.1. Motivation.
0.2. Highlights of Chapter 1.
0.3. Highlights of Chapter 2.
0.4. Highlights of Chapter 3.
0.5. Highlights of Chapter 4. 5
0.6. Reading guide. 6
0.7. Acknowledgements. 6
Chapter 1. General properties of the ku-cohomology of finite groups. 7
1.1. Varieties for connective K-theory. 7
1.2. Implications for minimal primes. 11
1.3. Euler classes and Chern classes. 14
1.4. Bockstein spectral sequences. 18
1.5. The Kunneth theorem. 21
Chapter 2. Examples of ku-cohomology of finite groups. 27
2.1. The technique. 28
2.2. Cyclic groups. 32
2.3. Nonabelian groups of order pq. 36
2.4. Quaternion groups. 43
2.5. Dihedral groups. 51
2.6. The alternating group of degree 4. 59
Chapter 3. The ku-homology of finite groups. 63
3.1. General behaviour of ku* (BG). 63
3.2. The universal coefficient theorem. 66
3.3. Local cohomology and duality. 68
3.4. The ku-homology of cyclic and quaternion groups. 69
3.5. The ku-homology of BDg. 71
3.6. Tate cohomology. 76
Chapter 4. The ku-homology and k;u-cohomology of elementary abelian
groups. 79
4.1. Description of results. 79
4.2. The ku-cohomology of elementary abelian groups. 81
4.3. What local cohomology ought to look like. 87
4.4. The local cohomology of Q. 88
4.5. The 2-adic filtration of the local cohomology of Q. 93
4.6. A free resolution of T. 94
V
vi CONTENTS
4.7. The local cohomology of T. 99
4.8. Hilbert series. 102
4.9. The quotient P/T2. 103
4.10. The local cohomology of R. 104
4.11. The ku-homology of By. 105
4.12. Duality for the cohomology of elementary abelian groups. 109
4.13. Tate cohomology of elementary abelian groups. 111
Appendix A. Conventions. 115
Al. General conventions. 115
A.2. Adams spectral sequence conventions. 115
Appendix B. Indices. 117
B.1. Index of calculations. 117
B.2. Index of symbols. 117
B.3. Index of notation. 118
B.4. Index of terminology. 122
Appendix. Bibliography 125
Abstract
This paper is devoted to the connective K homology and cohomology of finite
groups G. We attempt to give a systematic account from several points of view.
In Chapter 1, following Quillen [50, 51], we use the methods of algebraic
geometry to study the ring ku* (BG) where ku denotes connective complex K-
theory. We describe the variety in terms of the category of abelian p-subgroups
of G for primes p dividing the group order. As may be expected, the variety is
obtained by splicing that of periodic complex K-theory and that of integral ordinary
homology, however the way these parts fit together is of interest in itself. The main
technical obstacle is that the Kiinneth spectral sequence does not collapse, so we
have to show that it collapses up to isomorphism of varieties.
In Chapter 2 we give several families of new complete and explicit calculations
of the ring ku* (BG). This illustrates the general results of Chapter 1 and their
limitations.
In Chapter 3 we consider the associated homology ku*(BG). We identify this as
a module over ku* (BC) by using the local cohomology spectral sequence. This gives
new specific calculations, but also illuminating structural information, including
remarkable duality properties.
Finally, in Chapter 4 we make a particular study of elementary abelian groups
V. Despite the group-theoretic simplicity of V, the detailed calculation of ku* (BV)
and ku* (BV) exposes a very intricate structure, and gives a striking illustration
of our methods. Unlike earlier work, our description is natural for the action of
GL(V).
Received by the editor January 1, 2002.
1991 Mathematics Subject Classification. Primary: 19L41, 19L47, 19L64, 55N15; Sec-
ondary: 20J06, 55N22, 55N91, 55T15, 55U20, 55U25, 55U30.
Key words and phrases. connective K-theory, finite group, cohomology, representation the-
ory, Chern classes, Euler classes, cohomological varieties, Gorenstein, local cohomology, dihedral,
quaternion, elementary abelian.
The first author is grateful to the EPSRC, the Centre de Recerca Matematica, and the Japan-
US Mathematics Institute, and the second author is grateful to the Nuffield Foundation and the
NSF for support during work on this paper.
vii
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CHAPTER 0
Introduction.
0.1. Motivation.
This paper is about the connective complex K theory ku* (BC) and ku (BC)
of finite groups G. The first author and others [3, 4, 5, 6, 8, 35, 48] have made
many additive calculations of ku* (BC) and ku* (BC) for particular finite groups
G. The purpose of the present paper is to give a more systematic account.
More precisely, ku* (X) is the cohomology represented by the connective cover
of the spectrum K representing Atiyah-Hirzebruch periodic complex K theory.
Their values on a point are
K* = Z[v, v-1] and ku* = 7L[v],
where v is the Bott periodicity element in degree 2.
Connective K theory is relatively easy to calculate, and it has been used to
great effect as a powerful and practical invariant by homotopy theorists. However,
it is not well understood from a theoretical point of view. Although it can be
constructed by infinite loop space theory, and there are ad hoc interpretations of
its values in terms of vector bundles trivial over certain skeleta [54], these fall short
of a satisfactory answer (for instance because they fail to suggest a well behaved
equivariant analogue). Similarly, ku is also a complex oriented theory, and is the
geometric realization of the representing ring for multiplicative formal groups if we
allow a non-invertible parameter. This does appear to generalize to the equivariant
case [27] but, since ku* is not Landweber exact, it does not give a definition.
Our response to this state of affairs is to exploit the calculability. In practical
terms this extends the applicability of the theory, and in the process we are guided
in our search for geometric understanding.
Our results fall into four types, corresponding to the four chapters.
(1) General results about the cohomology rings ku* (BC), describing its vari-
ety after Quillen. In the course of this we exploit a number of interesting
general properties, such as the behaviour of the Bockstein spectral se-
quence and the fact that the Ki nneth theorem holds up to nilpotents for
products of cyclic groups. The use of Euler classes of representations is
fundamental.
(2) Explicit calculations of cohomology rings for low rank groups. The input
for this is the known group cohomology ring (processed via the Adams
spectral sequence) and the complex character ring.
(3) General results about the homology modules ku* (BC), and the curious
duality phenomena which appear. The connections with the cohomology
1