Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups

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 $p_H:{\tilde{X}}_H\to X$ 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 ${\pi }_1^{qtop}(X,x_0)$.

17 pages

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