Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups

2017/04/24 by Pashaei, S. Z., Mashayekhy, B., Torabi, H. +1 · 1 citation
#55Q05 #57M05 #57M10 #57M12 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.07408

Abstract

Let H be a subgroup of π₁(X,x₀). In this paper, we extend the concept of X being SLT space to H-SLT space at x₀. First, we show that the fibers of the endpoint projection p_H:X_H→ X are topological group when X is H-SLT space at x₀ 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 π₁^qtop(X,x₀) when X is SLT at x₀. Second, we present conditions under which the whisker topology is agree with the quotient of compact-open topology on X_H. Also, we study the relationship between open subsets of π₁^wh(X,x₀) and π₁^qtop(X,x₀).

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• Comparison of Topologies on Fundamental Groups with Subgroup Topology Viewpoint

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