K-homology, assembly and rigidity theorems for relative eta invariants

We connect the assembly map in C^*-algebra K-theory to rigidity properties for relative eta invariants that have been investigated by Mathai, Keswani, Weinberger and others. We give a new and conceptual proof of Keswani's theorem that whenever the C^*-algebra assembly map is an iso- morphism, the relative eta invariants associated to the signature operator are homotopy invariants, whereas the relative eta invariants associated to the Dirac operator on a manifold with positive scalar curvature vanish.