Baum–Connes conjecture

The Baum–Connes conjecture asserts that an assembly map from a group’s equivariant K-homology of its classifying space to the K-theory of its reduced group C*-algebra is an isomorphism.

Connect

Assembly map: A natural transformation that sends geometric or homological data to an analytic invariant. It is the map whose being an isomorphism is asserted by the conjecture.
Proper group action: A group action on a space for which compact sets meet only finitely many of their translates, in the discrete-group setting. Proper actions supply the geometry underlying the classifying space used in the conjecture.
Paul Baum: An American mathematician known for work in topology, K-theory, and the Baum–Connes conjecture. He helped formulate the conjecture with Alain Connes.
Novikov conjecture: A conjecture asserting homotopy invariance of higher signatures of closed manifolds. Baum–Connes implies the Novikov conjecture for groups for which its assembly map is injective.
Strong Novikov conjecture: The assertion that the Baum–Connes assembly map is rationally injective. It is a weaker injectivity statement than the full isomorphism conjecture.
Equivariant K-homology: A generalized homology theory that incorporates a group action on a space. It supplies the conjecture’s geometric input, evaluated on the classifying space.
Group cohomology: A cohomology theory that encodes algebraic information about a group through its modules. It is a classical algebraic invariant that helps situate the conjecture’s homological input.
Alain Connes: A French mathematician whose work founded noncommutative geometry. His noncommutative-geometric viewpoint shaped the conjecture’s analytic target.
Kadison–Kaplansky conjecture: A conjecture that the complex group algebra of a torsion-free group has no nontrivial idempotents. Baum–Connes implies this conjecture for torsion-free groups through K-theory.
Baum–Connes conjecture with coefficients: A version of the Baum–Connes conjecture that incorporates an auxiliary group C*-algebra coefficient. It is stronger than the plain conjecture and has distinct counterexamples.