# Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups

## TL;DR

This paper generalizes the concept of small loop transfer spaces to subgroups of fundamental groups, exploring topological properties of associated covering spaces and the relationships between different topologies on fundamental groups.

## Contribution

It introduces the notion of H-SLT spaces, analyzes the topological structure of fibers in covering spaces, and compares various topologies on fundamental groups.

## Key findings

- Fibers of the endpoint projection are topological groups when H is normal.
- Homotopically path Hausdorff relative to H coincides with homotopically Hausdorff relative to H under certain conditions.
- The endpoint projection has the unique path lifting property iff H is closed normal in the quasitopological fundamental group.

## Abstract

Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$. In this paper, we extend the concept of $X$ being SLT space to $H$-SLT space at $x_0$. First, we show that the fibers of the endpoint projection $p_{H}:\tilde{X}_{H}\rightarrow X$ are topological group when $X$ is $H$-SLT space at $x_0$ and $H$ is a normal subgroup. Also, we show that under these conditions the concepts of homotopically path Hausdorff relative to $H$ and homotopically Hausdorff relative to $H$ coincide. Moreover, among other things, we show that the endpoint projection map $p_{H}$ has the unique path lifting property if and only if $H$ is a closed normal subgroup of $\pi_{1}^{qtop}(X,x_{0})$ when $X$ is SLT at $x_{0}$. Second, we present conditions under which the whisker topology is agree with the quotient of compact-open topology on $\tilde{X}_{H}$. Also, we study the relationship between open subsets of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{qtop}(X,x_{0})$.