The Connective K-theory of Finite Groups

Robert Bruner and John Greenlees

MSC2000:   Primary 19L41, 19L47, 19L64, 55N15.
Secondary 20J06, 55N22, 55N91, 55T15, 55U20, 55U25, 55U30.

Department of Mathematics,      School of Mathematics and Statistics, 
Wayne State University,         Hicks Building, 
Detroit MI 48202-3489,          Sheffield S3 7RH, 
USA.                            UK.


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Abstract:

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, 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.  The variety is
obtained by splicing that of periodic complex K-theory and that of
integral ordinary homology, the interest lying in the way these parts
fit together.  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).

In Chapter 3 we consider the associated homology ku_*(BG), 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).