On Strong Small Loop Transfer Spaces Relative to Subgroups of Fundamental Groups

TL;DR

This paper introduces the concept of strong H-SLT spaces relative to subgroups of fundamental groups, exploring their properties, covering maps, and topological relationships, with examples illustrating the new definitions.

Contribution

It extends the strong SLT space concept to a relative version with respect to subgroups, analyzing covering maps and topological equivalences in this context.

Findings

Existence of covering maps for strong H-SLT spaces

Conditions for whisker and lasso topology agreement

Examples illustrating strong H-SLT space properties

Abstract

Let $H$ be a subgroup of the fundamental group $\pi_{1}(X,x_{0})$. By extending the concept of strong SLT space to a relative version with respect to $H$, strong $H$-SLT space, first, we investigate the existence of a covering map for strong $H$-SLT spaces. Moreover, we show that a semicovering map is a covering map in the presence of strong $H$-SLT property. Second, we present conditions under which the whisker topology agrees with the lasso topology on $\widetilde{X}_{H}$. Also, we study the relationship between open subsets of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{l}(X,x_{0})$. Finally, we give some examples to justify the definition and study of strong $H$-SLT spaces.