TRIANGULATION OF THE MAP OF A G-MANIFOLD TO ITS ORBIT SPACE

Let G be a Lie group, and let M be a smooth proper G-manifold. Let M/G denote the orbit space, and let π : M → 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 τ : P → M and σ : M/G → L such that σ o π o τ 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 τ and σ subanalytic and P and L unique up to PL homeomorphisms.