Orbit projections of proper Lie groupoids as fibrations

TL;DR

This paper characterizes when the orbit projection of a proper Lie groupoid is a fibration, showing it occurs precisely when the groupoid is regular, thus linking geometric structure to topological properties.

Contribution

It provides a necessary and sufficient condition for the orbit projection of a proper Lie groupoid to be a fibration, connecting regularity of the groupoid with fibration properties.

Findings

Orbit projection is a fibration if and only if the groupoid is regular.

Proper Lie groupoids with finite type orbits have a well-defined fibration structure.

The result characterizes the geometric conditions for fibration in the context of Lie groupoids.

Abstract

Let $\mathcal{G} \rightrightarrows M$ be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection $M \to M/\mathcal{G}$ is a fibration if and only if $\mathcal{G} \rightrightarrows M$ is regular.