Generalized universal covering spaces and the shape group

Abstract

Abstract. If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism ϕ : π 1 (X) →π 1 (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 :This generalized notion of universal covering p : X → 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 : X → X is universally characterized by the usual general lifting properties. (If X is first countable, then p : X → X is already characterized by the unique lifting of paths and their homotopies.) In particular, the group of covering transformations G = Aut( X p → X) is isomorphic to π 1 (X) and it acts freely and transitively on every fiber. If X is locally path-connected, then the quotient X/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 : X → 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 π 1 (X) is countable. Spaces X for which ϕ : π 1 (X) →π 1 (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 π 1 (X), and provide the appropriate relative version of being homotopically Hausdorff, along with its corresponding properties.General Assumption. Throughout this article, we consider a pathconnected topological space X with base point x 0 ∈ X. Recall that a continuous map p : X → X is called a covering of X, and X is called a covering space of X, if for every x ∈ X there is an open subset U of X with x ∈ U and such that U is evenly covered by p, that is, p −1 (U ) is the disjoint union of open subsets of X each of which is mapped homeomorphically onto U by p.In the classical theory, one assumes that X is, in addition, locally pathconnected and wishes to classify all path-connected covering spaces of X and to find among them a universal covering space, that is, a covering p : X → X with the property that for every covering q : X → X by a path-connected space X there is a covering q : X → X such that q•q = p. If X is locally pathconnected, we have the following well-known result, which can be found, for example, in [22] and [25]:

Cotorsion-free groups from a topological viewpoint

“…Provided X is also locally path-connected, X admits a universal covering space if and only if S contains a minimal element, i.e., π s (X, x) = π(U, x) for some U, and it admits a simply-connected covering space if and only if π(U, x) = 1 for some U [32]. In general, π s (X, x) lies in the kernel of the natural homomorphism π 1 (X, x) → π 1 (X, x) to the firstČech homotopy group [20] and it has recently been shown to equal this kernel if X is locally path-connected and paracompact Hausdorff (e.g. if X is a Peano continuum) [2].…”

“…However, the automorphism group of this generalized covering projection is always naturally isomorphic to the quotient π 1 (X, x)/π s (X, x) and it acts freely and transitively on every fiber. Moreover, p is open if X is locally path-connected [20]. When π s (X, x) = 1, we speak of a generalized universal covering.…”

“…It was shown in [20], that every path-connected topological space X admits a generalized covering projection p : X → X corresponding to π s (X, x), i.e. :…”

Spanier spaces and the covering theory of non-homotopically path Hausdorff spaces

“…We recall that a space X is called shape injective, if the natural homomorphism ϕ : π 1 (X, x) −→π 1 (X, x) is injective, whereπ 1 (X, x) is the first shape group of (X, x) (see [7,Section 3] for further details). In [3], it is proved that topological fundamental groups of shape injective spaces are Hausdorff.…”

“…H. Fischer and A. Zastrow in [7] defined a generalized regular covering which enjoys most of the usual properties of classical coverings, with the possible exception of evenly covered neighborhoodness. If X is connected, locally pathconnected and semi-locally simply connected, then the generalized universal covering p : X −→ X agrees with the classical universal covering.…”