Deformation of homeomorphisms on stratified sets

Abstract

T H E O E M 0. The topological group H(X) of homeomorphisms of a finite simplicial complex X onto itself is locally contractible. This result does not extend to ENR's (euclidean neighborhood retracts). To see this let a space X be obtained from S 3 = 3 u o o by crushing to a point each of a sequence of mutually disjoint wild non-cellular arcs in 3 :,41, a 2 , A 3 . . . . such that each A,, n~> 1, is a copy of the same wild arc A in the unit ball in 3 translated by the vector (4n, 0, 0). This X is an ENR; indeed X x is homeomorphic to S a x = 4 0 by a result of Andrews and Curtis [4]. Clearly this compactum admits self-homeomorphisms h : X ~ X arbitrarily near the identity which nontrivially permute the images o fA 1, A2, A3, .... But no such h is isotopic to the identity because these are isolated points at which Y fails to be a manifold. (See also the fish skeleton of w The treatment of non-manifolds rests roughly speaking on a method for deforming homeomorphisms on R x cX, cY being the open cone on X, once one is given such a method on +1 xX-. Then the proof proceeds by induction on the depth of X. Here Xis regarded as a stratified set, and depth is the greatest difference of dimensions of nonempty strata of X. Stratified sets are vital to the proof because their open subsets are themselves stratified sets, and often of a lesser depth. Thus it will only clarify matters to deal from the outset with suitable stratified sets. I take this opportunity to introduce classes of pleasant stratified sets that may come to be the topological analogues of polyhedra in the piecewise-linear realm or of Thom's stratified sets in the differentiable realm. This technique of proof almost automatically provides strong relative and respectful deformation theorems (w 4.3, w 5.10), which a counterexample (w 2.3.1) suggests are