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

Abstract

Let $G$ be a Lie group, and let $M$ be a smooth proper $G$-manifold. Let $M/G$ denote the orbit space, and let $\pi :M\to M/G$ be the natural map. It is known that $M/G$ is homeomorphic to a polyhedron. In the present paper we show that there exist a piecewise linear (PL) manifold $P$, a polyhedron $L$, and homeomorphisms $\tau :P\to M$ and $\sigma :M/G\to L$ such that $\sigma \circ \pi \circ \tau$ is PL. This is an application of the theory of subanalytic sets and subanalytic maps of Shiota. If $M$ and the $G$-action are, moreover, subanalytic, then we can choose $\tau$ and $\sigma$ subanalytic and $P$ and $L$ unique up to PL homeomorphisms.