On the space of Lipschitz homeomorphisms of a compact polyhedron

Katsuro Sakai and Raymond Y. T. Wong

Vol. 139 (1989), No. 1, 195–207

DOI: 10.2140/pjm.1989.139.195

Abstract

Let X be a positive dimensional compact Euclidean polyhedron. Let H(X), H_LIP(X) and H_PL(X) be respectively the space of homeomorphisms, the space of Lipschitz homeomorphisms and the space of piecewise-linear homeomorphisms of X onto itself. In this paper, we establish a homeomorphism taking the triple (H(X),H_LIP(X),H_PL(X)) onto the triple (H(X) × s,H_LIP(X) × Σ,H_PL(X) × σ), where s = (−1,1)^ω, Σ = {(x₁) ∈ s|sup|x_i| < 1} and σ = {(x_i) ∈ s|x_i = 0 except for finitely many i}. As a consequence we prove that when X is a PL manifold with dimx≠4 and ∂X = ∅, in case dimX = 5, (H(X),H_LIP(X)) is an (s,Σ)-manifold pair if H(X) is an s-manifold. We also prove that if dimX = 1 or 2, then (H(X),H_PL(X)) is an (s,σ)-manifold pair and (H(X),H_LIP(X)) is an (s,Σ)-manifold.

Mathematical Subject Classification 2000

Primary: 57Q99

Secondary: 58D15

Milestones

Received: 30 November 1987

Revised: 8 June 1988

Published: 1 September 1989