Rips complexes and covers in the uniform category (Q2839891)

The paper is devoted to developing a covering theory for uniform spaces. The authors define a uniform covering map between uniform spaces using Rips complexes and then characterize uniform covering maps via the chain lift property and the uniqueness of the chain lift property. Their definition of uniform covering map is equivalent to the one introduced by I. M. James. Firstly, generalizing the classical construction of universal covering spaces, to each uniform space \(X\) can be associated a uniform space \(\widetilde{X}\) and a uniformly continuous map \(\pi_{X}:\widetilde{X}\rightarrow X\). However, the projection \(\pi_{X}\) is a uniform covering map if and only if the base space \(X\) is a uniform Poincaré space i.e. a space that is path-connected and uniformly locally nice. To adjust the construction of \(\widetilde{X}\) to uniform spaces \(X\) with bad local properties, applying Rips complexes, the authors define a uniform space \(GP(X,x_{0})\) of generalized uniform paths such that the end-point map \(\pi_{X}:GP(X,x_{0})\rightarrow X\) is uniformly continuous. The notion of a uniform generalized path is a generalization of the notion of a generalized path introduced by Krasinkiewicz and Minc. Then a generalized uniform covering map between uniform spaces is defined by requiring that it has liftings and approximate uniqueness lift properties for both chains and generalized uniform paths. The class of uniform spaces characterized by the requirement of \(\pi_{X}:GP(X,x_{0})\rightarrow X\) being a generalized uniform covering map is the class of locally uniform joinable spaces. In the case of metric continua that class coincides with the class of pointed \(1\)-movable spaces. Finally, the authors relate their construction to that of Berestovskii and Plaut who defined a (generalized) uniform covering map in terms of group actions. They show that Berestovskii and Plaut's inverse limit construction produces a space that is uniformly equivalent to their space of generalized uniform paths.