Fundamental groups and finite sheeted coverings

Abstract

It is well known that for a connected locally path-connected semi-locally 1-connected space X , there exists a bi-unique correspondence between the pointed d -fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σ d of degree d . The classification problem becomes more difficult if X is a more general space, particularly if X is not locally connected. In attempt to solve the problem for general spaces, several notions of coverings have been introduced, for example, those given by Lubkin or by Fox. On the other hand, different notions of ‘fundamental group’ have appeared in the mathematical literature, for instance, the Brown–Grossman–Quigley fundamental group, the Čech–Borsuk fundamental group, the Steenrod–Quigley fundamental group, the fundamental profinite group or the fundamental localic group. The main result of this paper determines different ‘fundamental groups’ that can be used to classify pointed finite sheeted connected coverings of a given space X depending on topological properties of X .