Around Palais' Covering Homotopy Theorem

The classification by Palais of G-spaces, topological spaces acted on by homeomorphisms by a compact Lie group G, is refined. Under mild topological hypotheses, it is shown that when a sequence of orbit spaces is 'close' to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequence are strongly equivalent to the limit G-space. Three applications of the theory to Alexandrov and Riemannian geometry are then given. The Covering Homotopy Theorem, which is key to the classification theory, is used to prove a version of the Slice Theorem for Alexandrov spaces, showing that the local action of a group of isometries is topologically determined by its infinitesimal action. The refinement of the classification theory is used to prove an equivariant version of Perelman's Stability Theorem for equicontinous sequences of isometric actions by a fixed compact Lie group. The class of Riemannian orbifolds of a given dimension defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter is shown to be finite up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is also shown to be finite up to orbifold homeomorphism.