Covering spaces

[摘要] Covering Spaces may be studied from several points of view and with respect to many kinds of spaces. The earliest form of covering spaces considered was that of the Riemann surface where the spaces involved are 2-dimensional and triangulable and the mapping function is an analytic function of a complex variable. Generalizations to n-dimensions were natural and an account of such a generalization may be found in [15] where the spaces considered are complexes. In this work we take the generalization further and assume only that the spaces dealt with are arc-connected, locally arc-connected, locally simply connected and Hausdorff. A brief account of the theory from this point of view may be found in [14] where the existence of only the universal covering space is proved. This approach has the advantage that it does not appeal to ideas of dimension and triangulability and it also includes a good portion of the topological spaces ordinarily encountered. We undertake here to prove and extend, insofar as possible the results in [15]. The theory of covering spaces can be made even more general by replacing the properties arc-connected and locally arc-connected by connectedness and local connectedness respectively [2]. However this approach has the disadvantage of giving an unorthodox definition for the fundamental group and of being rather incapable of easy visualization. Also, covering spaces may be considered as a special case of fibre bundles. Chapter 0 of this thesis consists of the prerequisites needed to develop the theory of covering spaces. Chapter 1 develops the theory of covering spaces for that type of space that we have already mentioned. Sections on the monodromy theorem, existence theorem, universal and regular coverings are included. A black bar indicates the end of a proof of a theorem. The expression ;;iffi;; takes place of the expression ;;if and only if;;. The author wishes to thank Professor G. R. MacLane for his valuable help during the writing of this thesis.

[发布机构] Rice University