Memoirs of the American Mathematical Society 2003; 127 pp; softcover Volume: 165 ISBN10: 0821833669 ISBN13: 9780821833667 List Price: US$62 Individual Members: US$37.20 Institutional Members: US$49.60 Order Code: MEMO/165/785
 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 Ktheory. 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 Ktheory 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 grouptheoretic 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 \(GL(V)\). Readership Graduate students and research mathematicians interested in algebra, algebraic geometry, geometry, and topology. 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
