Replacement of Fixed Sets for Compact Group Actions: The 2ρ Theorem

The replacement problem for stratified spaces asks whether any manifold (simple) homotopy equivalent to the bottom stratum of a stratified space X is the bottom stratum of a stratified space stratified (simple) homotopy equivalent to X. This is impossible in general, as one sees by considering X = M × [0, 1] and working rel M ×0. On the other hand, the classic theorem of Browder, Casson, Haefliger, Sullivan and Wall (see [20] Corollary 11.3.1) asserts that if X = (W,M) is a pair consisting of a manifold with a codimension c submanifold, and c > 2, then replacement is always possible in the topological and PL locally flat settings. The key technical reason for such replacement is the stability theorem (and its topological analogue) that says that the map of classifying spaces, Gc/PLc → G/PL,