Triangulation of the map of a $G$-manifold to its orbit space

TL;DR

This paper demonstrates that for a smooth proper G-manifold, the orbit space can be triangulated via PL manifolds and polyhedra, with analytic cases allowing subanalytic homeomorphisms, facilitating topological and geometric analysis.

Contribution

It establishes a method to triangulate the orbit space of a G-manifold using PL manifolds and polyhedra, including the analytic case with subanalytic homeomorphisms.

Findings

Existence of a PL manifold and polyhedron representing the orbit space

Homeomorphisms making the orbit map PL

Subanalytic homeomorphisms in the analytic case

Abstract

Let $G$ be a Lie group and $M$ a smooth proper $G$-manifold. Let $pi:Mto M/G$ denote the natural map to the orbit space. Then there exist a PL manifold $P$, a polyhedron $L$ and homeomorphisms $tau:Pto M$ and $\sigma:M/Gto L$ such that $\sigma\circpi\circ\tau$ is PL. If $M$ and the $G$-action are of analytic class, we can choose subanalytic $\tau$ and then unique $P$ and $L$.