Baum-Connes Conjecture

The Baum-Connes conjecture predicts that reduced assembly is an isomorphism from equivariant K-homology of a classifying space for proper group actions to the topological K-theory of the reduced group C^*-algebra. For a countable discrete group , the version with trivial complex coefficients is

where and the domain uses -compact supports. The fixed-point sets in are contractible for finite subgroups and empty for infinite subgroups. For a torsion-free group this domain identifies with , using compact supports. Allowing an arbitrary coefficient algebra gives a stronger and separate conjecture.