Rouse, Paule CarilloWang, Bai-Ling2015-12-101631-073Xhttp://hdl.handle.net/1885/58673For any Lie groupoid with a twisting, we define an analytic index morphism using the Connes tangent groupoid. This morphism agrees with the one of the Lie groupoid when the twisting is trivial. We discuss a longitudinal index theorem, geometric cycles, push-forward maps and Baum-Connes assembly maps for foliations with a twisting on the space of leaves.201010.1016/j.crma.2010.10.0372015-12-10