Stability and invariants of Hilsum-Skandalis maps

Abstract

We consider principal bundles as generalized morphisms between topological groupoids. In the category of these generalized morphisms two topological groupoids are isomorphic if and only if they are Morita equivalent. We show that the fibers of a generalized morphism from H to G induce a singular foliation of the topological groupoid H, and we prove a Reeb-Thurston stability theorem for such foliations. Next, we use generalized morphisms to study some Morita invariants of topological groupoids, in particular the homotopy groups of a topological groupoid and the Connes convolution algebra of an etale groupoid.