Title of Paper:
The Burnside Ring and Equivariant Cohomotopy for Infinite Groups
Author:
Wolfgang Lueck
AMS Classification numbers:
55P91, 19A22.
If already submitted to the xxx LANL archive, include the id. no:
math.AT/0504051
Addresses of Authors
Fachbereich Mathematik
Universitaet Muenster
Einsteinstr. 62
48149 Muenster
Germany
Email address of Author:
lueck@math.uni-muenster.de
Abstract:
After we have given a survey on the Burnside ring of a finite
group, we discuss and analyze various extensions of this notion to
infinite (discrete) groups. The first three are the
finite-$G$-set-version, the inverse-limit-version and the
covariant Burnside group. The most sophisticated one is the fourth
definition as the equivariant zero-th cohomotopy of the
classifying space for proper actions. In order to make sense of
this definition we define equivariant cohomotopy groups of finite
proper equivariant CW-complexes in terms of maps between the
sphere bundles associated to equivariant vector bundles. We show
that this yields an equivariant cohomology theory with a
multiplicative structure. We formulate a version of the Segal
Conjecture for infinite groups. All this is analogous and related
to the question what are the possible extensions of the notion of the
representation ring of a finite group to an infinite group. Here
possible candidates are projective class groups, Swan groups and
the equivariant topological K-theory of the classifying space
for proper actions.