Baum–Connes conjecture

In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group.

The conjecture sets up a correspondence between different areas of mathematics, with the K-homology of the classifying space being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the group's reduced C*-algebra is a purely analytical object.

The conjecture is also closely related to index theory, as the assembly map

is a sort of index, and it plays a major role in Alain Connes' noncommutative geometry program.

The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects.

One can define a morphism called the assembly map, from the equivariant K-homology with

Paul Baum and Alain Connes introduced the following conjecture (1982) about this morphism: As the left hand side tends to be more easily accessible than the right hand side, because there are hardly any general structure theorems of the

The original formulation of the conjecture was somewhat different, as the notion of equivariant K-homology was not yet common in 1982.

is discrete and torsion-free, the left hand side reduces to the non-equivariant K-homology with compact supports of the ordinary classifying space

However, counterexamples to the conjecture with coefficients were found in 2002 by Nigel Higson, Vincent Lafforgue and Georges Skandalis.

One can then show that the assembly map is KK-theoretic Poincaré duality as defined by Gennadi Kasparov, which is an isomorphism.

The conjecture without coefficients is still open, although the field has received great attention since 1982.

This goes back to ideas of Michael Atiyah and was developed in great generality by Gennadi Kasparov in 1987.