Group actions and covering maps in the uniform category

In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of X is the universal covering in categorical sense. Also, we introduce the notion of semi-locally small loop space which is the necessary and sufficient condition for existence of universal cover for non-homotopically hausdorff spaces, equivalently existence of small covering spaces. Also, we prove that for semi-locally small loop spaces, X is a small loop space if and only if every cover of X is trivial if and only if π top 1 (X) is an indiscrete topological group.2010 Mathematics Subject Classification. 57M10, 57M05, 55Q05, 57M12.

Section: Small Loop Groupsmentioning

confidence: 98%

Small loop spaces and covering theory of non-homotopically Hausdorff spaces

Pakdaman

Torabi

Mashayekhy

2011

“…Finally, when X is compact geodesic, Brodskiy, Dydak, Labuz, and Mitra showed (Corollary 6.5, [8]) that the uniform fundamental group is isomorphic to the first shape group of X,π 1 (X). We will suppress the formal and rather technical definition ofπ 1 (X) (cf.…”

Section: Background: Discrete Homotopy Theorymentioning

confidence: 99%

The revised and uniform fundamental groups and universal covers of geodesic spaces

Wilkins

2013

Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of the revised and uniform fundamental groups. We show that a compact geodesic space X has a universal cover if and only if the following hold: 1) its revised and uniform fundamental groups are finitely presented, or, more generally, countable; 2) its revised fundamental group is discrete as a quotient of the topological fundamental group π top 1 (X). In the process, we classify the topological singularities in X, and we show that the above conditions imply closed liftings of all sufficiently small path loops to all covers of X, generalizing the traditional semilocally simply connected property. A geodesic space X with this new property is called semilocally r-simply connected, and X has a universal cover if and only if it satisfies this condition. We then introduce a topology on π 1 (X) called the covering topology, which always makes π 1 (X) a topological group. We establish several connections between properties of the covering topology, the existence of simply connected and universal covers, and geometries on the fundamental group.

“…Characterization 1 uses ideas of Berestovskii-Plaut [1] later expanded in [2]. In case of surjective maps p : X → Y between connected metrizable spaces we characterize overlays as local isometries: p : X → Y is an overlay if and only if one can metrize X and Y in such a way that p|B(x, 1) : B(x, 1) → B(p(x), 1) is an isometry for each x ∈ X.…”

Section: Introductionmentioning

confidence: 99%

Overlays and group actions

Dydak

2016

Overlays were introduced by R.H.Fox [6] as a subclass of covering maps. We offer a different view of overlays: it resembles the definition of paracompact spaces via star refinements of open covers. One introduces covering structures for covering maps and p : X → Y is an overlay if it has a covering structure that has a star refinement.We prove two characterizations of overlays: one using existence and uniqueness of lifts of discrete chains, the second as maps inducing simplicial coverings of nerves of certain covers. We use those characterizations to improve results of Eda-Matijević concerning topological group structures on domains of overlays whose range is a compact topological group.In case of surjective maps p : X → Y between connected metrizable spaces we characterize overlays as local isometries: p : X → Y is an overlay if and only if one can metrize X and Y in such a way that p|B(x, 1) : B(x, 1) → B(p(x), 1) is an isometry for each x ∈ X.