Small loop transfer spaces with respect to subgroups of fundamental groups (Q1676540)

Given a based path connected topological space \((X, x_0)\), the standard suspect for the universal covering space is \(\widetilde X\), which is defined as the space of homotopy classes of based paths. The construction is of particular interest in the realm of wild spaces, i.e., spaces, which are not semi-locally simply-connected. In this context, \textit{N. Brodskiy}, \textit{J. Dydak}, \textit{B. Labuz} and \textit{A. Mitra} [``Topological and uniform structures on universal covering spaces'', Preprint, \url{arXiv:1206.0071}] introduced a class of SLT (small loop transfer) spaces as follows. A space \(X\) is SLT if for each path \(\alpha: [0,1]\to X\) and each neighborhood \(U\) of \(\alpha(0)\) there exists a neighborhood \(V\) of \(\alpha(1)\) such that given a loop \(\beta : (S^1,1) \to (V, \alpha(1))\) there is a loop \(\gamma : (S^1,1)\to(U,\alpha(0))\) that is homotopic to \(\alpha*\beta*\alpha^{-1}\) rel \(\alpha(0)\). It turns out that SLT spaces have many interesting properties. In this paper the authors introduce a notion of \(H\)-SLT spaces, where \(H\) is a subset of \(\pi_1(X, x_0)\). This is a variant of the SLT property in \(\widetilde X_H\), which is obtained from \(\widetilde X\) as the appropriate quotient by \(H\). They then prove properties, analogous to the ones of SLT spaces. Amongst other results they prove that if \(H\) is normal, the following holds for \(H\)-SLT spaces: fibers of the projection \(p_H : \widetilde X_H \to X\) are topological groups; the conditions of being homotopically Hausdorff relative to \(H\) and of being homotopically path Hausdorff relative to \(H\) are equivalent; \(H\) is closed in the whisker topology iff the projection \(p_H\) has the unique path lifting property; the whisker and the compact-open induced topologies on the appropriate quotients of \(\widetilde X\) coincide.