Some anomalous examples of lifting spaces

TL;DR

This paper explores unusual examples of lifting spaces, which are fibrations with unique path lifting, that cannot be constructed as inverse limits of covering spaces, highlighting the diversity within the theory.

Contribution

It introduces specific anomalous lifting spaces that defy representation as inverse limits of covering spaces, expanding understanding of the theory's scope.

Findings

Some lifting spaces are not inverse limits of covering spaces.

Inverse limits of finite coverings produce fibrations with Cantor set fibers.

The paper presents examples illustrating the diversity of lifting spaces.

Abstract

An inverse limit of a sequence of covering spaces over a given space $X$ is not, in general, a covering space over $X$ but is still a lifting space, i.e. a Hurewicz fibration with unique path lifting property. Of particular interest are inverse limits of finite coverings (resp. finite regular coverings), which yield fibrations whose fiber is homeomorphic to the Cantor set (resp. profinite topological group). To illustrate the breadth of the theory, we present in this note some curious examples of lifting spaces that cannot be obtained as inverse limits of covering spaces.