Assembly map

Summary

An assembly map compares a homology theory of the classifying space of a group with K-theory or L-theory of its group ring. In the Baum–Connes setting, the target is K-theory of the group C*-algebra. The source is computable by standard methods of algebraic topology, while the target is hard to compute. The isomorphism conjectures of Farrell–Jones and Baum–Connes predict that suitable assembly maps are isomorphisms. Assembly maps are central in the topology of high-dimensional manifolds.