# The Connective K-Theory of Finite Groups

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*R. R. Bruner; J. P. C. Greenlees*

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
Künneth 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 \(ku^*(BG)\) 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 \(GL(V)\).

#### Table of Contents

# Table of Contents

## The Connective K-Theory of Finite Groups

- Contents v6 free
- Chapter 0. Introduction 110 free
- Chapter 1. General properties of the ku-cohomology of finite groups 716
- Chapter 2. Examples of ku-cohomology of finite groups 2736
- Chapter 3. The ku-homology of finite groups 6372
- Chapter 4. The ku-homology and ku-cohomology of elementary abelian groups 7988
- 4.1. Description of results 7988
- 4.2. The ku-cohomology of elementary abelian groups 8190
- 4.3. What local cohomology ought to look like 8796
- 4.4. The local cohomology of Q 8897
- 4.5. The 2-adic filtration of the local cohomology of Q 93102
- 4.6. A free resolution of T 94103
- 4.7. The local cohomology of T 99108
- 4.8. Hilbert series 102111
- 4.9. The quotient P/T[sub(2)] 103112
- 4.10. The local cohomology of R 104113
- 4.11. The ku-homology of BV 105114
- 4.12. Duality for the cohomology of elementary abelian groups 109118
- 4.13. Tate cohomology of elementary abelian groups 111120

- Appendix A. Conventions 115124
- Appendix B. Indices 117126
- Bibliography 125134