On subgroup topologies on fundamental groups

Abstract

It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.

On topological homotopy groups and relation to Hawaiian groups

“…However, some nice properties occur if it is open. The following proposition generalizes Proposition 2.4 in [1], by a similar argument, for the whisker topology on the nth homotopy group (n ≥ 1). Proposition 2.5.…”

A natural pseudometric on homotopy groups of metric spaces

“…) is a map with image in B d (x, r/2) for some x ∈ X. Let g : S n → X be the constant map at α (1). Since e = [α * g] and…”

“…There are many ways to enrich the n-th homotopy group π n (X, x 0 ) of a based topological space (X, x 0 ) with a geometric or topological structure that remembers local features of the space X, which are "unseen" by the usual group-theoretic structure, for example, the natural quotient topology [11], the τ -topology [9], the Spanier topology [1,3], the whisker topology [2], and variations on these (e.g. coreflections in a convenient category).…”

“…To prove this result, we show that for a compact metric space (X, d) the pseudometric topology on π n (X, x 0 ) lies between the shape topology and a third topology called the Spanier topology (Definition 5.5). For Case (1), we apply a result from [3] to conclude that, in the LC n−1 case, the Spanier and shape topologies (and thus all three topologies) agree. We prove Case (2) separately using the fact that we may alter the metric on X without affecting the topologies of π met n (X, x 0 ) or π sh n (X, x 0 ).…”