On quasi-small loop groups

TL;DR

This paper introduces the quasi-small loop group, a new subgroup of the fundamental group based on homotopical closeness, and explores its properties and relation to other fundamental group concepts.

Contribution

It defines the quasi-small loop group, proves its independence from base point, and establishes its connection to homotopically path Hausdorff spaces and quasi-topological fundamental groups.

Findings

The quasi-small loop group is a normal subgroup containing the small generated subgroup.

The triviality of the quasi-small loop group characterizes homotopically path Hausdorff spaces.

Relationships between the quasi-small loop group and the quasi-topological fundamental group are established.

Abstract

In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by $\pi_1^{qs} (X, x)$ for a pointed space $(X, x)$. Then we prove that, unlike the small loop group, the quasi-small loop group $\pi_1^{qs}(X, x)$ does not depend on the base point, and that it is a normal subgroup containing $\pi_1^{sg}(X, x)$, the small generated subgroup of the fundamental group. Also, we show that a space $X$ is homotopically path Hausdorff if and only if $\pi_1^{qs} (X, x)$ is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group.