The Revised and Uniform Fundamental Groups and Universal Covers of Geodesic Spaces

TL;DR

This paper characterizes when compact geodesic spaces have universal covers by linking geometric and topological properties of their fundamental groups, introducing new concepts like the semilocally r-simply connected property and a topology on the fundamental group.

Contribution

It provides new equivalent conditions for the existence of universal covers in geodesic spaces, including properties of the revised and uniform fundamental groups and the introduction of the covering topology.

Findings

Universal cover existence characterized by finitely presented or countable fundamental groups.

Discreteness of the revised fundamental group as a quotient relates to universal covers.

Introduction of the semilocally r-simply connected property and the covering topology.

Abstract

Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of the revised and uniform fundamental groups. We show that a compact geodesic space X has a universal cover if and only if the following hold: 1) its revised and uniform fundamental groups are finitely presented, or, more generally, countable; 2) its revised fundamental group is discrete as a quotient of the topological fundamental group. In the process, we classify the topological singularities in X, and we show that the above conditions imply closed liftings of all sufficiently small path loops to all covers of X, generalizing the traditional semilocally simply connected property. A geodesic space with this new property is called semilocally r-simply connected, and X has a universal cover if and only if it satisfies this condition. We then introduce a topology on the fundamental group called the covering topology, with respect to which the fundamental group is always a topological group. We establish several connections between properties of the covering topology, the existence of simply connected and universal covers, and geometries on the fundamental group.