Rips complexes and covers in the uniform category

摘要

James [20] introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut [3] introduced a theory of covers for uniform spaces generalizing their results for topological groups [1]-[2]. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal covering space and their theory works well for the so-called coverable spaces. As will be seen in Section 7, [3] generalizes only regular covering maps in topology and pro-discrete actions may not be preserved by compositions. In this paper we redefine the uniform covering maps and we generalize pro-discrete actions using Rips complexes and the chain lifting property. We expand the concept of generalized paths of Krasinkiewicz and Minc [21]. One way to do it is by embedding X in a space with good local properties and this is done in Section 6. Another way is by systematic use of Rips complexes. In the topological category one uses paths in X originating from a base point to construct the universal covering space eX. We use paths in Rips complexes and their homotopy classes possess a natural uniform structure, a generalization of the basic topology on eX. Applying Rips complexes leads to a natural class of uniform spaces for which our theory of covering maps works as well as the classical one, namely the class of locally uniform joinable spaces. In the case of metric continua (compact and connected metric spaces) that class is identical with pointed 1-movable spaces, a well-understood class of spaces introduced by shape theorists (see [9] or [24]). As an application of our results we present an exposition in [7] of Prajs' [30] homogeneous curve that is path-connected but not locally connected.