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

Sormani and Wei proved in 2004 that a compact geodesic space has a\ncategorical universal cover if and only if its covering/critical spectrum is\nfinite. We add to this several equivalent conditions pertaining to the geometry\nand topology of the revised and uniform fundamental groups. We show that a\ncompact geodesic space X has a universal cover if and only if the following\nhold: 1) its revised and uniform fundamental groups are finitely presented, or,\nmore generally, countable; 2) its revised fundamental group is discrete as a\nquotient of the topological fundamental group. In the process, we classify the\ntopological singularities in X, and we show that the above conditions imply\nclosed liftings of all sufficiently small path loops to all covers of X,\ngeneralizing the traditional semilocally simply connected property. A geodesic\nspace with this new property is called semilocally r-simply connected, and X\nhas a universal cover if and only if it satisfies this condition. We then\nintroduce a topology on the fundamental group called the covering topology,\nwith respect to which the fundamental group is always a topological group. We\nestablish several connections between properties of the covering topology, the\nexistence of simply connected and universal covers, and geometries on the\nfundamental group.\n