The fundamental group as a topological group

Abstract

This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as the quotient of the loop space often fails to result in a topological group; we use free topological groups to construct a topology which promotes the fundamental group of any space to topological group structure. The resulting invariant, denoted π τ 1 , takes values in the category of topological groups, can distinguish spaces with isomorphic fundamental groups, and agrees with the quotient fundamental group precisely when the quotient topology yields a topological group. Most importantly, this choice of topology allows us to naturally realize free topological groups and pushouts of topological groups as fundamental groups via topological analogues of classical results in algebraic topology. qtop 1: hTop * → qTopGrp from the homotopy category of based spaces to the category of quasitopological groups 1 and continuous homomorphisms [3,7], however, recent results indicate that very often π qtop 1 (X, x 0 ) fails to be a topological group. In 1 A quasitopological group is a group with topology such that inversion is continuous and multiplication is continuous in each variable. See [1] for basic theory.

On topological shape homotopy groups

“…Endowed with the quotient topology induced by the natural surjective map q : Ω k (X, x) → π k (X, x), where Ω k (X, x) is the kth loop space of (X, x) with the compact-open topology, the familiar homotopy group π k (X, x) becomes a quasitopological group which is called the quasitopological kth homotopy group of the pointed space (X, x), denoted by π qtop k (X, x) (see [2,3,4,13]). Biss [2] proved that π qtop 1 (X, x) is a topological group.…”

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

“…Calcut and McCarthy [7] proved that the quotient topology induced by the compact-open topology of the fundamental group of a locally path connected and semilocally simply connected space is discrete which implies that π qtop 1 (X, x 0 ) is a topological group. Brazas [4] introduced a new topology on fundamental groups made them topological groups. Torabi et al [15] showed that π qtop 1 (X, x 0 ) is a topological group when X is a locally path connected, semilocally small generated space.…”

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

“…For more details, see [1,2,4]. Also, π τ 1 (X, x) is the fundamental group endowed with another topology introduced by Brazas [3]. In fact, the functor π τ 1 removes the smallest number of open sets from the topology of π qtop 1 (X, x) so that make it a topological group.…”

“…In [3], it is proved that topological fundamental groups of shape injective spaces are Hausdorff. Since the spaces with trivial Spanier group are not necessarily shape injective (see [6, Step 18, Section 6]), triviality of Spanier groups can not certify the Hausdorffness of topological fundamental groups in general.…”