**Memoirs of the American Mathematical Society**

2003;
127 pp;
Softcover

MSC: Primary 19; 55;
Secondary 20

Print ISBN: 978-0-8218-3366-7

Product Code: MEMO/165/785

List Price: $62.00

Individual Member Price: $37.20

**Electronic ISBN: 978-1-4704-0383-6
Product Code: MEMO/165/785.E**

List Price: $62.00

Individual Member Price: $37.20

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# 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 [

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

#### Readership

Graduate students and research mathematicians interested in algebra, algebraic geometry, geometry, and topology.