Twisted longitudinal index theorem for foliations

I report on the joint work with Paulo Carrillo-Rouse. For a foliation with a twisting on its leaf space, we establish the equivalence between the twisted topological index and the twisted analytic index, both taking values in the K-theory of the twisted C*-algebra of the honolomy groupoid. We also develop a notion of geometric cycles and the geometric K-homology for a foliation with a twisting. As an application of our twisted longitudinal index theorem, we

show that a twisted version of Baum-Connes assembly map is well-defined, and is expected to be an isomorphism whenever the untwisted assembly map is an isomorphism.