# Geometry of compact lifting spaces

## TL;DR

This paper explores the structure of profinite fibrations, generalizing inverse systems of covering spaces, and characterizes when such fibrations are limits of coverings, linking fundamental group actions with profinite topology.

## Contribution

It provides a characterization of profinite fibrations within a broad class and relates the fundamental group's action to the profinite topology on the fiber.

## Key findings

- Profinite fibrations can be distinguished from inverse limits of coverings.
- The fundamental group's action on the fiber relates to the fiber's profinite topology.
- A version of the Borel construction is developed for profinite group fibers.

## Abstract

We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in a previous paper (Conner-Herfort-Pavesic: Some anomalous examples of lifting spaces), there are many fibrations whose fibres are profinite groups, which are far from being inverse limits of coverings. We characterize profinite fibrations among a large class of fibrations and relate the profinite topology on the fundamental group of the base with the action of the fundamental group on the fibre, and develop a version of the Borel construction for fibrations whose fibres are profinite groups.