Triangulation of the map of a \(G\)-manifold to its orbit space (Q2873896)

Let \(G\) be a Lie group with a smooth proper action on a manifold \(M\). It is proved that the orbit projection \(M \to M/G\) is topologically equivalent to a PL map from a PL manifold to a polyhedron. The first ingredient of the proof is the proof of the Thom conjecture [\textit{R. Thom}, Enseign. Math. (2) 8, 24--33 (1962; Zbl 0109.40002)] by the second author [Topology 39, No. 2, 383--399 (2000; Zbl 0934.57028)]. This asserts that a locally triangulable map between subsets of Euclidean spaces is triangulable if it is proper and satisfies a condition which roughly says that its fibers do not explode. The orbit projection is shown to possess the latter property; however, it is not proper in general, hence an extra ingredient is needed. This is the second author's theory of subanalytic sets and maps [\textit{M. Shiota}, Geometry of subanalytic and semialgebraic sets. Progress in Mathematics (Boston, Mass.). 150. Boston, MA: Birkhäuser (1997; Zbl 0889.32006)]. A subset of \(\mathbb R^n\) is subanalytic if it can be obtained from finitely many analytic images of real analytic manifolds by taking set differences and unions, and a map is subanalytic if its graph is. To be able to apply this theory, the authors endow \(M\), \(M/G\) and the orbit map with subanalytic structures.