Small loop transfer spaces with respect to subgroups of fundamental groups

Abstract

Let H be a subgroup of π 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 :X H → 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 π qtop 1 (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 onX H . Also, we study the relationship between open subsets of π wh 1 (X, x 0 ) and π qtop 1 (X, x 0 ).

On strong small loop transfer spaces relative to subgroups of fundamental groups

“…Let p : X → X be an lpc 0 -covering map with p * π 1 ( X,x 0 ) = H. In [4, Lemma 5.10] it was shown that p is equivalent to the endpoint projection map p H : X wh H → X. Moreover, it was shown that the fiber p −1 (x 0 ) is Hausdorff and hence (p −1 H (x 0 )) wh is Hausdorff (see [13,Corollary 3.10]). On the other hand, since |p −1 (x 0 )| < ∞, we have…”

“…Also, note that the example below shows that the concepts of relative version of strong SLT and SLT spaces are not necessarily the same. (HE, x 0 ), Proposition 3.6 of [13] implies that HE is an H-SLT space at x 0 and hence, using [13,Proposition 3.7], H is an open subgroup of π wh 1 (X, x 0 ). One can see that H is not an open subgroup in π l 1 (HE, x 0 ) because it is not a covering subgroup.…”

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

“…In [6, Theorem 4.12] Brodskiy et al showed that this coincidence holds at all point x ∈ X when X is a locally path connected SLT space. Moreover, Pashaei et al [12] showed that the property of being SLT at x 0 for the space X is a necessary and sufficient condition for the coincidence π wh 1 (X, x 0 ) = π qtop 1 (X, x 0 ). In the following theorem, using path Spanier groups, we give another proof for this statement.…”

“…In this paper, we are going to investigate on the relationship between the topological group property of π wh 1 (X, x 0 ) and π qtop 1 (X, x 0 ) and SLT spaces at one point which are introduced in [12]. In Section 2, first, we address to the relationship between path Spanier groups, introduced in [14], and SLT spaces at a point.…”

“…In Section 2, first, we address to the relationship between path Spanier groups, introduced in [14], and SLT spaces at a point. Using this relationship, we give a new proof to show that π wh 1 (X, x 0 ) = π qtop 1 (X, x 0 ) if and only if X is SLT at x 0 (see also [12]). Second, we introduce the notion of small loop transfer loop at one point (SLTL for short) and show that the property of being SLTL at x 0 for X is a necessary and sufficient condition for π wh 1 (X, x 0 ) to be a topological group.…”