Arc-smooth generalized universal covering spaces
A continuous map \(p : \widetilde{X}\to X\), where \(\widetilde{X}\) is path-connected, locally path-connected and \(1\)-connected, is called a generalized universal covering of \(X\) if for every path connected and locally path connected space \(Y\) and every continuous map \(g : (Y,y)\to (X,x)\) with \(g_{\#} \pi_1(Y,y) = 1\), and for every \(\widetilde{x}\) in \(\widetilde{X}\) with \(p(\widetilde{x}) = x\), there is a unique lifting \(h : (Y,y) \to (\widetilde{X}, \widetilde{x})\) with \(p\circ h = g\). Suppose now that \(Y\) is a path-connected subset of a CAT(0) space \(Z\) and suppose there is a continuous map \((Y,y_0) \to (X,x_0)\) onto a 1-dimensional separable metrizable space such that \({\mathcal G} = \{ f^{-1}(x) \,|\, x\in X\}\) is a null collection of convex subsets of \(Z\). Here null means that for each \(\varepsilon>0\), only finitely many members of \({\mathcal G}\) have diameter greater than \(\varepsilon\). Then the author proves that such a space \(Y\) admits a generalized universal covering space \((\widetilde{Y}, \widetilde{y_0})\). Moreover there is a continuous map \(A : (\widetilde{Y}, \widetilde{y_0}) \to (C(\widetilde{Y}), \{\widetilde{y_0}\})\) from \(\widetilde{Y}\) into the metric space of nonempty compact connected subsets of \(\widetilde{Y}\) endowed with the Hausdorff metric such that \(A(x)\) is an arc from \(x_0\) to \(x\) for \(x\neq x_0\) and \(A(y) \subset A(x)\) when \(y\in A(x)\).
scientific article; zbMATH DE number 5290936
scientific article