### Equivariant-bivariant Chern character for profinite groups, by Paul Baum and Peter Schneider

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 Y "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. Under certain basic finiteness conditions on
the first space Y we conjecture the existence of a Chern character from the
equivariant Kasparov KK-theory of Y and X into our cohomology theory made
two-periodic which becomes an isomorphism upon tensoring the KK-theory with
the complex numbers. This conjecture is proved in this paper for profinite
groups G. An essential role in our construction is played by a bivariant
version of Segal localization which we establish for KK-theory.

Paul Baum <baum@math.psu.edu>

Peter Schneider <pschnei@math.uni-muenster.de>