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 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 U p(X), H ?L (X)) onto the triple (H(X) x s, H U p(X) x Σ, H FL (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),H PL (X)) is an (5, σ)-manifold pair and (H(X),H U p(X)) is an (s, Σ)-manifold. 0. Introduction. Let X = (X 9 d) and Y = (Y, p) be metric spaces. A map /: X -> 7 is said to be Lipschitz (resp. bi-Lipschitz) if there is some A: > 0 (resp. A: > 1) such that p{f\x),f{x')) < k rf(x ? x') (resp.

Topological (\prod^ω\ell_2, \sum^ω\ell_2)-factors of diffeomorphism groups of non-compact manifolds

“…Stability properties of homeomorphism groups and their subgroups have already been studied by many authors [9,10,12,13,18,22]. K. Sakai and R.Y.…”

“…K. Sakai and R.Y. Wong [18] showed that for Euclidean polyhedra X and Y the triples of homeomorphism groups and spaces of embeddings:…”

“…Stability properties of homeomorphism groups of topological manifolds and their subgroups have already been studied by many authors (cf. [9,10,12,13,18] etc). In particular, in [22] we have treated the non-compact case in detail.…”