Test map characterizations of local properties of fundamental groups

Abstract

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any pathconnected metric space X, and a subgroup H ≤ π 1 (X, x 0 ), we characterize the unique path lifting property relative to H in terms of a new closure operator on the π 1 -subgroup lattice that is induced by maps from a fixed "test" domain into X. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

Dense products in fundamental groupoids

“…This completes the induction. Now, for any n ∈ N, and using any basic factorization w odd v odd v −1 even w −1 even of (r 2n ) # (p τ ), we see that As noted in [5,Remark 7.2], in the previous definition, one only needs to consider closed nowhere dense sets A.…”

“…Most of the notation in this paper agrees with that in [5]. Throughout this paper, X will denote a path-connected topological space and x 0 ∈ X will be a basepoint.…”

“…We refer to Section 2 of [5] for proofs of the following basic properties of these closure operators.…”

“…is an example of a non-normal subgroup of π 1 (H + , b + 0 ), which is (C, c τ )-closed but not (P, p τ )-closed (see the proof of [5,Theorem 3.25]). There is no known normal counterexample.…”

“…According to [3], this approach to generalized covering spaces is, for metric spaces, categorically, the most robust possible generalization of covering theory based on unique lifting, which gives a classification by a lattice of π 1 -subgroups. Conditions sufficient for the existence of a generalized universal covering include π 1 -shape injectivity [20] and the homotpically path-Hausdorff property [18]; we utilize the necessary and sufficient characterization for metric spaces in [5].…”