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Sormani and Wei proved in 2004 that a compact geodesic space has acategorical universal cover if and only if its covering/critical spectrum isfinite. We add to this several equivalent conditions pertaining to the geometryand topology of the revised and uniform fundamental groups. We show that acompact geodesic space X has a universal cover if and only if the followinghold: 1) its revised and uniform fundamental groups are finitely presented, or,more generally, countable; 2) its revised fundamental group is discrete as aquotient of the topological fundamental group. In the process, we classify thetopological singularities in X, and we show that the above conditions implyclosed liftings of all sufficiently small path loops to all covers of X,generalizing the traditional semilocally simply connected property. A geodesicspace with this new property is called semilocally r-simply connected, and Xhas a universal cover if and only if it satisfies this condition. We thenintroduce a topology on the fundamental group called the covering topology,with respect to which the fundamental group is always a topological group. Weestablish several connections between properties of the covering topology, theexistence of simply connected and universal covers, and geometries on thefundamental group.

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