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

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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 σ:M/Gto L such that σ∘τ is PL. If M and the G-action are of analytic class, we can choose subanalytic τ and then unique P and L.