Generalized universal covering spaces and the shape group

Abstract

If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism φ : π1 (X) -π̌1h (X) from the fundamental group to its first shape homotopy group is an isomorphism. We present a partial converse to this result: a path-connected topological space X admits a generalized universal covering space if φ : π1 (X) →̌1 (X) is injective. This generalized notion of universal covering p : X̃ → X enjoys most of the usual properties, with the possible exception of evenly covered neighborhoods: the space X is path-connected, locally path-connected and simply-connected and the continuous surjection p : X̃ → X is universally characterized by the usual general lifting properties. (If X is first countable, then p : X̃ → X is already characterized by the unique lifting of paths and their homotopies.) In particular, the group of covering transformations G = Aut(X̃ p→X) is isomorphic to π1(X) and it acts freely and transitively on every fiber. If X is locally path-connected, then the quotient X̃/G is homeomorphic to X. If X is Hausdorff or metrizable, then so is X, and in the latter case G can be made to act by isometry. If X is path-connected, locally path-connected and semilocally simply-connected, then p : X̃ → X agrees with the classical universal covering. A necessary condition for the standard construction to yield a generalized universal covering is that X be homotopically Hausdorff, which is also sufficient if π1 (X) is countable. Spaces X for which φ: π1 (X) → π̌1 (X) is known to be injective include all subsets of closed surfaces, all 1-dimensional separable metric spaces (which we prove to be covered by topological ℝ-trees), as well as so-called trees of manifolds which arise, for example, as boundaries of certain Coxeter groups. We also obtain generalized regular coverings, relative to some special normal subgroups of π1(X), and provide the appropriate relative version of being homotopically Hausdorff, along with its corresponding properties. © Instytut Matematyczny PAN, 2007.