Twisted longitudinal index theorem for foliations and wrong way functoriality

Abstract: For a Lie groupoid with a twisting σ (a -principal bundle over ), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. Display Omitted For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes–Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai, Melrose, and Singer. We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map and a twisting σ on the holonomy groupoid , next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes and Skandalis when the twisting is trivial and of Carey and Wang for manifolds. [Copyright &y& Elsevier]