The Chern character in equivariant K-homology
Peter Schneider <pschnei@escher.uni-muenster.de>
This is a report on joint work with P. Baum. For the action of a locally
compact and totally disconnected group G (the most important examples of such
being all discrete groups as well as all p-adic reductive groups) on a pair
of locally compact spaces X and Y we construct, by sheaf theoretic means, a
new equivariant and bivariant cohomology theory. In contrast to the
classical Borel construction of equivariant cohomology our construction is
not localized in the unit element of the group. If we take for the first
space "the" universal proper G-action then we obtain for the second space its
delocalized equivariant homology. All this is in exact formal analogy to the
definition of equivariant K-homology by Baum, Connes, Higson starting from
the bivariant equivariant Kasparov KK-theory. Using a homotopy theoretic
description of the latter developed by Baum and Lueck independently and based
upon the classical approach to topological K-theory through Fredholm
operators we set up a Chern character isomorphism between equivariant
K-homology and our new theory.