Inverse Limits of Uniform Covering Maps

TL;DR

This paper explores the relationship between generalized uniform covering maps and inverse limits of uniform covering maps, providing characterizations and conditions for their equivalence and approximation.

Contribution

It establishes that the universal generalized uniform covering map is equivalent to an inverse limit of uniform covering maps and characterizes when such maps are approximated by or equivalent to inverse limits.

Findings

Universal generalized uniform covering map is an inverse limit of uniform covering maps.

Characterization of generalized uniform covering maps approximated by uniform covering maps.

Conditions for generalized uniform covering maps to be equivalent to inverse limits.

Abstract

In ``Rips complexes and covers in the uniform category'' the authors define, following James, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper notes that the universal generalized uniform covering map is uniformly equivalent to the inverse limit of uniform covering maps and is therefore approximated by uniform covering maps. A characterization of generalized uniform covering maps that are approximated by uniform covering maps is provided as well as a characterization of generalized uniform covering maps that are uniformly equivalent to the inverse limit of uniform covering maps. Inverse limits of group actions that induce generalized uniform covering maps are also treated.