# Baum–Connes conjecture

The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for a discrete torsion-free group, and the injectivity is closely related to the Novikov conjecture.

The conjecture is also closely related to index theory, as the assembly map {\displaystyle \mu } is a sort of index, and it plays a major role in Alain Connesnoncommutative 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.