Groups of uniform homeomorphisms of covering spaces

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. Furthermore, combining with similarity transformations, this enables us to induce a global deformation property of groups of uniform homeo- morphisms 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.