Retracted: The topological fundamental group and generalized covering spaces
type: Item
instance of: scholarly article
title: Retracted: The topological fundamental group and generalized covering spaces
zbMATH Open document ID: 1016.57002
DOI: 10.1016/S0166-8641(01)00247-4
author: Daniel K. Biss
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: 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.
reviewed by: Osman Mucuk
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 55U40
zbMATH DE Number: 1879098
zbMATH Keywords: topological fundamental group
zbMATH Keywords: covering spaces
zbMATH Keywords: semilocally simply connected spaces
MaRDI profile type: Publication
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full work available at URL: s0166-8641(01)00247-4
OpenAlex ID: W2567678006
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description: scientific article; zbMATH DE number 1879098