On the space of Lipschitz homeomorphisms of a compact polyhedron

Abstract

Let X be a positive dimensional compact Euclidean polyhedron. Let H(X), HUP{X) and HPL(X) be respectively the space of homeomorphisms, the space of Lipschitz homeomorphisms and the space of piecewise-lin ear homeomorphisms of X onto itself. In this paper, we establish a homeomorphism taking the triple (H(X), HUp(X), H?L(X)) onto the triple (H(X) x s, HUp(X) x Σ, HFL(X) X σ), where 5 = (-1, l)ω, Σ = {(*,) e s\sup\Xi < 1} and σ = {(*,) e s\x t = 0 except for finitely many /}. As a consequence we prove that when X is a PL manifold with dim c Φ 4 and dX = 0, in case dimZ = 5, (H(X),HUP(X)) is an (s, Σ)-manifold pair if H(X) is an s-manifold. We also prove that if dim* = 1 or 2, then (H(X),HPL(X)) is an (5, σ)-manifold pair and (H(X),HUp(X)) is an (s, Σ)-manifold.