The Connective K-Theory of Finite Groups

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 $GL(V)$.