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 $GL(V)$.

Readership

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