Stability and invariants of Hilsum-Skandalis maps

TL;DR

This paper explores the properties of Hilsum-Skandalis maps, focusing on their stability, invariants, and the induced foliations, providing new insights into the structure of topological groupoids and their Morita equivalences.

Contribution

It introduces a stability theorem for foliations induced by generalized morphisms and studies Morita invariants like homotopy groups and the Connes algebra.

Findings

Fibers induce singular foliations of the topological groupoid.

A Reeb-Thurston stability theorem for these foliations is established.

Morita invariants such as homotopy groups and Connes algebra are analyzed.

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.