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On topologized fundamental groups with small loop transfer viewpoints. (English) Zbl 1431.57018

Let \(X\) be topological space and \(x_0\in X\). Then there are two topologies on the fundamental group \(\pi_1(X,x_0)\). The first one is induced by the standard topology in the construction of the universal cover of \(X\) at the point \(x_0\) and is called the whisker topology and written \(\pi_1^{wh}(X,x_0)\). The second one is the quotient topology induced by the compact-open topology written \(\pi_1^{qtop}(X,x_0)\).
In this paper, the authors study the behavior of the topologized fundamental groups \(\pi_1^{wh}(X,x_0)\) and \(\pi_1^{qtop}(X,x_0)\) with the whisker topology and the compact-open topology, respectively. In particular, they give necessary or sufficient conditions for these two topologized fundamental groups to be equal and to be topological groups, together with some examples to show that the converse of some of these implications does not hold, in general.
A path \(\alpha\) in \(X\) is called a small loop transfer path and written SLT path for abbreviation if for every open neighborhood \(U\) of \(\alpha(0)\) in \(X\), there exists an open neighborhood \(V\) of \(\alpha(1)\) in \(X\) such that for a given loop \(\gamma\) at \(\alpha(1)\) there is a loop \(\gamma'\) at \(\alpha(0)\) which is homotopic to \(\alpha\ast\gamma\ast \alpha^{-1}\). The space \(X\) is called an SLT space if all paths are SLT paths.
In the paper, the authors also investigate the relationship between the property of being a topological group for \(\pi_1^{wh}(X,x_0)\), \(\pi_1^{qtop}(X,x_0)\) and SLT spaces at one point.

MSC:

57M05 Fundamental group, presentations, free differential calculus
55P35 Loop spaces
54H11 Topological groups (topological aspects)
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References:

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