Every Continuum has a Compact Universal Cover

TL;DR

This paper introduces the concept of a compact universal cover for continua, showing it forms an inverse system with a trivial profinite fundamental group, and explores its properties and applications in covering space theory.

Contribution

It defines the compact universal cover for continua, proves its properties, and establishes a Galois correspondence, extending covering space theory to a broader class of spaces.

Findings

The inverse limit of all regular covers forms a continuum with trivial profinite fundamental group.

Every continuum has at most n! non-equivalent n-fold covers by continua.

Every non-compact manifold covering a compact manifold has a unique profinite compactification.

Abstract

We define the compact universal cover of a compact, metrizable connected space (i.e. a continuum) X to be the inverse limit of all continua that regularly cover X. We show that such covers do indeed form an inverse system with bonding maps that are regular covering maps, and the projection from the inverse limit is a generalized regular covering map in the sense of Berestovskii-Plaut. The inverse limit space is a continuum that is "compactly simply connected" in the sense that its "profinite fundamental group" (the inverse limit of the deck groups of the finite covers) is trivial. We prove a Galois Correspondence for closed normal subgroups of the compact fundamental group, uniqueness, universal and lifting properties. As an application we prove that every non-compact manifold that regularly covers a compact manifold has a unique "profinite compactification", i.e. an imbedding as a dense subset in a compactly simply connected continuum. As part of the proof of metrizability of the inverse limit we show that every continuum has at most n! non-equivalent n-fold covers by continua.