Generalized universal covering spaces and the shape group

Hanspeter Fischer ; Andreas Zastrow

Fundamenta Mathematicae, Tome 193 (2007), p. 167-196 / Harvested from The Polish Digital Mathematics Library

Résumé

If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism φ: π₁(X) → π̌₁(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) → π̌₁(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 X̃ 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 $G = Aut(X̃ \stackrel{p}{→} X)$ 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 X̃, 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.

Détail

Publié le : 2007-01-01

Zbl 1137.55006

EUDML-ID : urn:eudml:doc:282916