On the Generalization of Hilbert's Fifth Problem to Transitive Groupoids

In the following paper we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter which may be considered generalizations of the Hilbert's fifth problem to this context. Most notably we present a "solution" to the problem for proper transitive groupoids and transitive groupoids with compact source fibers

The Solution of Hilbert's Fifth Problem for Transitive Groupoids

In the following paper we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter which may be considered generalizations of the Hilbert's fifth problem to this context. Most notably we present a "solution" to the problem for proper transitive groupoids and transitive groupoids with compact source fibers.Comment: The preliminary sections partially coincide with the ones of our previous article arXiv:1710.1144

On a notion of maps between orbifolds, I. function spaces

This is the first of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we define the maps in the more general context of orbispaces, and establish several basic results concerning the topological structure of the space of such maps. In particular, we show that the space of such maps of C^r-class between smooth orbifolds has a natural Banach orbifold structure if the domain of the map is compact, generalizing the corresponding result in the manifold case. Motivations and applications of the theory come from string theory and the theory of pseudoholomorphic curves in symplectic orbifolds.Comment: Final version, 46 pages. Accepted for publication in Communications in Contemporary Mathematics. A preliminary version of this work is under a different title "A homotopy theory of orbispaces", arXiv: math. AT/010202

Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney B stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid