Groups of uniform homeomorphisms of covering spaces

TL;DR

This paper proves the local contractibility of the group of uniform homeomorphisms on certain covering spaces and establishes a global homotopy property, including the contractibility of the Euclidean space's homeomorphism group.

Contribution

It introduces a local deformation lemma for uniform embeddings in covering spaces and derives global homotopy properties for groups of uniform homeomorphisms.

Findings

The group of uniform homeomorphisms of a metric covering space is locally contractible.

The identity component of the uniform homeomorphism group of Euclidean space is contractible.

Global homotopy properties are established for spaces with Euclidean ends.

Abstract

In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under the uniform topology. Further more, combining with similarity transformation, this enables us to induce a global homotopy property of groups of uniform homeomorphisms of metric spaces with Euclidean ends. In particular, we show that the identity component of the group of uniform homeomorphisms of the standard Euclidean n-space is contractible.