On quasi-small loop groups

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.

14 pages, 4 figures, journal paper, under review by Mathematica Slovaca

References in corpus (3)

• On small homotopies of loops
• Homotopical Smallness and Closeness
• A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions