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

TL;DR

This paper explores conditions under which transitive topological groupoids can be continuously isomorphic to Lie groupoids, extending Hilbert's fifth problem to this broader mathematical context.

Contribution

It provides new results and generalizations addressing when transitive topological groupoids are equivalent to Lie groupoids, including proper and compact source fiber cases.

Findings

Proper transitive groupoids are continuously isomorphic to Lie groupoids.

Transitive groupoids with compact source fibers are also isomorphic to Lie groupoids.

The paper extends classical Hilbert's fifth problem to the setting of groupoids.

Abstract

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.