Covering Maps with respect to Topologies on the Fundamental Group

Abstract: In this paper, using the classical covering theory, we introduce a generalization of covering maps of a space $X$ with respect to a topology $τ$ on the fundamental group of $X$. We show that the famous notions, covering, semicovering, generalized covering and fibration maps are of special cases of this new notion $π_1<sup>τ$-covering map. Moreover, among presenting some properties for this new notion, we compare $π_1<sup>τ$-covering maps of a space $X$ for several famous topologies on the fundamental group of $X$.

• The paper introduces π₁^τ-covering maps, defined by path lifting and openness of the induced fundamental-group image in an arbitrary topology τ.
• Theorem 3.7 shows that classical coverings, semicoverings, generalized coverings, and fibrations arise from suitable fundamental-group topologies, including Spanier, path Spanier, and fibration subgroup topologies.
• The paper establishes comparison and product results while identifying unresolved strictness questions, with wild spaces such as the Hawaiian earring providing key test cases.

Overview

The paper under review introduces a unified framework for generalized covering space theory by defining, for any topology $\tau$ on the fundamental group $\pi_1(X,x_0)$, the notion of a $\pi_1^{\tau}$-covering map. A continuous map $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$ with path connected domain is a $\pi_1^{\tau}$-covering map if $p$ has the path lifting property and $\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$ is open in $\pi_1^{\tau}(X,x_0)$. The central contribution is Theorem 3.7, which shows that four classical notions — covering maps, semicovering maps, generalized covering maps, and fibrations — are all instances of this single definition for suitably chosen topologies: the Spanier topology for coverings, the path Spanier topology for semicoverings, the generalized covering topology for generalized coverings, and the subgroup topology determined by $\pi^{fib}(X,x_0)$ (the intersection of images of induced homomorphisms of all fibrations over $X$) for fibrations.

The motivation comes from the classical Spanier–Rotman construction of the endpoint projection $\pi_1(X,x_0)$0 with the whisker topology, together with Spanier's criterion that a covering with image $\pi_1(X,x_0)$1 exists precisely when some Spanier subgroup $\pi_1(X,x_0)$2 is contained in $\pi_1(X,x_0)$3. Translating this into topological language, for connected locally path connected $\pi_1(X,x_0)$4, coverings with image $\pi_1(X,x_0)$5 exist exactly when $\pi_1(X,x_0)$6 is open in $\pi_1(X,x_0)$7; the paper's definition replaces "Spanier" with an arbitrary topology $\pi_1(X,x_0)$8.

Basic properties

Several structural results are established. First, every $\pi_1(X,x_0)$9-covering map is automatically surjective, and for any open subgroup $\pi_1^{\tau}$0 of $\pi_1^{\tau}$1, the endpoint projection $\pi_1^{\tau}$2 is a $\pi_1^{\tau}$3-covering map provided $\pi_1^{\tau}$4 is a semitopological group. Brazas's characterization that $\pi_1^{\tau}$5 has unique path lifting if and only if $\pi_1^{\tau}$6 connects openness to uniqueness of lifts.

The paper also proves two technical lemmas used throughout: a map $\pi_1^{\tau}$7 over $\pi_1^{\tau}$8 inherits the path lifting property whenever $\pi_1^{\tau}$9 has it and $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$0 has unique path lifting (Theorem 3.2), and $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$1 is injective if and only if $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$2 has the unique path homotopy lifting property (Theorem 3.3). For subgroup topologies, continuity of $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$3 holds whenever $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$4 (Theorem 3.4), extending known functoriality results for the Tau, quotient, whisker, and lasso topologies.

Two closure-type results follow. If $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$5 and $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$6 are $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$7-coverings, $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$8 has uphl, and $p:(\widetilde{X},\widetilde{x}_0)\to(X,x_0)$9 is continuous, then any morphism $\pi_1^{\tau}$0 between them is itself a $\pi_1^{\tau}$1-covering (Theorem 3.5). Products behave well: $\pi_1^{\tau}$2 is a $\pi_1^{\tau}$3-covering whenever the canonical map $\pi_1^{\tau}$4 is a homeomorphism (Theorem 3.6). This hypothesis is satisfied for the quotient topology under conditions, for the Tau topology, for the lasso topology, and for certain subgroup topologies, but it is a genuine restriction for arbitrary $\pi_1^{\tau}$5.

Recovery of classical theories

Theorem 3.7 is the substantive core. Its converses deserve emphasis:

• Coverings: a $\pi_1^{\tau}$6-covering map with unique lifting property is equivalent to $\pi_1^{\tau}$7 for $\pi_1^{\tau}$8, hence is a covering map.
• Semicoverings: analogously, a $\pi_1^{\tau}$9-covering map with unique lifting property is a semicovering map, using Torabi–Pakdaman–Mashayekhy's result that $p$0 is a semicovering when $p$1 for some path open cover $p$2.
• Generalized coverings: every generalized covering map is a $p$3-covering map, since its image is by definition a generalized covering subgroup, hence open in $p$4.
• Fibrations: every fibration is a $p$5-covering map trivially, since fibrations have the path lifting property.

A further bridge (Theorem 3.8): if $p$6 is a generalized covering map with finite index image $p$7, then $p$8 is a $p$9-covering map, because generalized covering subgroups are closed in the whisker topology and closed finite-index subgroups of the homogeneous space $\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$0 are open.

Comparison across topologies

The paper assembles the known chain of comparisons among topologies on $\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$1,

$\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$2

and translates it into a corresponding hierarchy of covering notions (Diagram 2), where finer topologies yield smaller classes of $\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$3-covering maps. Notable instances include:

Statement	Status
$\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$4-covering = $\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$5-covering	Always (same open subgroups)
$\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$6-covering = $\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$7-covering	Always (topologies coincide)
$\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$8-covering = $\pi_1(p)\pi_1(\widetilde{X},\widetilde{x}_0)$9-covering	Always
Covering ⟹ $\pi_1^{\tau}(X,x_0)$0-covering; converse with ul	Theorem 3.7(i)
Semicovering ⟹ $\pi_1^{\tau}(X,x_0)$1-covering; converse with ul	Theorem 3.7(ii)
$\pi_1^{\tau}(X,x_0)$2 vs. $\pi_1^{\tau}(X,x_0)$3-covering	Incomparable in general

The Hawaiian earring $\pi_1^{\tau}(X,x_0)$4 provides strictness witnesses: $\pi_1^{\tau}(X,x_0)$5 and $\pi_1^{\tau}(X,x_0)$6 strictly. On $\pi_1^{\tau}(X,x_0)$7, the endpoint projection $\pi_1^{\tau}(X,x_0)$8 is a $\pi_1^{\tau}(X,x_0)$9-covering (since $\pi^{fib}(X,x_0)$0 makes the topology indiscrete) but not a $\pi^{fib}(X,x_0)$1-covering, because its image is trivial and not open in the non-discrete $\pi^{fib}(X,x_0)$2. Similarly, $\pi^{fib}(X,x_0)$3 for $\pi^{fib}(X,x_0)$4 is a $\pi^{fib}(X,x_0)$5-covering but not a $\pi^{fib}(X,x_0)$6-covering, since that would force $\pi^{fib}(X,x_0)$7 to admit a universal covering, contradicting known results. For the Harmonic Archipelago, $\pi^{fib}(X,x_0)$8 while $\pi^{fib}(X,x_0)$9 for $X$0, separating the corresponding covering classes conjecturally.

The paper also records equalities under hypotheses: for locally path connected, paracompact Hausdorff $X$1, the shape, thick Spanier, and Spanier topologies coincide, collapsing the associated covering notions; for connected locally path connected spaces, GCOV$X$2COV$X$3 exactly when $X$4 is semilocally $X$5-connected.

Limitations and open questions

The comparison diagram contains several arrows justified only by conjecture or guesswork rather than proof. In particular, the authors state they "guess" that $X$6 fails to be open in $X$7 and in $X$8, which would establish strictness of the corresponding inclusions; these claims remain unverified. Seventeen explicit questions (Q1–Q17) are posed, asking whether the various inclusions of covering classes are strict — e.g., whether there exists a $X$9-covering that is not a $\pi_1(X,x_0)$00-covering (Q3), whether every $\pi_1(X,x_0)$01-covering is a semicovering (Q9), and whether every generalized covering is a fibration with upl (Q12). Additionally, several results depend on side hypotheses: the product theorem requires $\pi_1(X,x_0)$02 to be a homeomorphism, the semitopological group condition is needed for endpoint projections to be $\pi_1(X,x_0)$03-coverings, and equality cases throughout require semilocal connectedness assumptions of various kinds. The paper does not resolve whether the framework yields new existence theorems beyond re-expressing known criteria topologically.

Conclusion

This paper systematizes generalized covering theory by parameterizing covering-like maps over topologies on the fundamental group, proving that coverings, semicoverings, generalized coverings, and fibrations are special cases of $\pi_1(X,x_0)$04-covering maps, and organizing the resulting classes into a coherent hierarchy mirroring the known lattice of fundamental group topologies. The main unresolved issue is the strictness of most inclusions in this hierarchy, for which the authors supply candidate counterexamples on wild spaces such as the Hawaiian earring and Harmonic Archipelago but only conjectural arguments.