engleski

Fundamental groups and finite sheeted coverings

It is well known that for a connected locally path-connected semilocally 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. The objective of this paper is to determine what topological conditions on a space are sufficient to establish the classification of its pointed finite-sheeted connected coverings in terms of transitive representations of a determined kind of fundamental topological group, fundamental progroup or fundamental localic group. Our main result establishes the classification of pointed finite sheeted connected coverings for important classes of spaces in terms of transitive representations in symmetric groups of different kind of fundamental groups, topological groups, localic groups and progroups.

Brown-Grossman group; Steenrod group; fundamental pro-group; overlay; covering projection

nije evidentirano