Retracted: The topological fundamental group and generalized covering spaces
A topological space \(X\) is called semilocally simply connected if each \(x\in X\) has an open neighborhood \(U\) such that the inclusion map \(\iota: (U,x)\rightarrow (X,x)\) induces the trivial map on \(\pi_1\). It is a well known result that a topological space has a universal cover if and only if it is semilocally simply connected. The space \(\Hom((S^1,1), (X,x))\) of continuous based maps can be given the compact open topology. The surjective map \( \Hom((S^1,1), (X,x))\twoheadrightarrow \pi_1(X,x)\) gives a topology on \(\pi_1(X,x)\). In this paper it is proved that, with this topology, \(\pi_1(X,x)\) is a topological group and also proved that \(X\) is semilocally simply connected if and only if \(\pi_1(X,x)\) is discrete. Furthermore fibrations with fiber \(\pi_1\) are constructed and a theory of special type of fibrations called rigid covering fibrations is developed. Remark: In [\textit{R. Brown} and \textit{G. Danesh-Naruie}, Proc. Edinb. Math. Soc. 19, 237-244 (1975; Zbl 0297.55001)] the fundamental groupoid \(\pi_1X\) of a topological space X, which is more general then the fundamental group, is given a topology such that \(\pi_1X\) is a topological groupoid. The relation between these is interesting.
scientific article; zbMATH DE number 1879098
scientific article