Branched coverings after Fox

Abstract

General branched coverings, folded coverings, and branched folded coverings are all special cases of the spreads introduced by R. H. Fox in the 1950's [in A symposium in honor of S. Lefschetz, 243–257, Princeton Univ. Press, Princeton, N.J., 1957;] to put the point set topology of these objects on a solid footing. This largely expository article revisits the subject, giving it a full treatment. Certain definitions are extended and several essentially new sufficient conditions for a map to be a spread are given. The concept of a singular covering is introduced, allowing a common treatment of branched coverings and branched folded coverings. One motivation for re-examining and extending the theory is for applications to possibly wild knots. In addition, the theory contains, for example, the end compactification of Freudenthal as a special case, which is given a complete treatment here. The paper opens with a useful, detailed historical survey putting all the work in the area in a common framework.