The connective K-theory of finite groups
Author(s)
Bibliographic Information
The connective K-theory of finite groups
（Memoirs of the American Mathematical Society, no. 785）
American Mathematical Society, 2003
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"Volume 165, number 785 (second of 4 numbers)."
Includes bibliographical references (p. 125-127) and indexes
Description and Table of Contents
Description
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 Kunneth 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
Introduction General properties of the $ku$-cohomology of finite groups Examples of $ku$-cohomology of finite groups The $ku$-homology of finite groups The $ku$-homology and $ku$-cohomology of elementary abelian groups Appendix A. Conventions Appendix B. Indices Appendix. Bibliography.
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