Basic topological properties of Fox's branched coverings.

Abstract

Motivated by applications to open manifolds and wild knots, the author in this article revisits R. H. Fox's theory [in A symposium in honor of S. Lefschetz, 243–257, Princeton Univ. Press, Princeton, N.J., 1957; MR0123298 (23 #A626)] of singular covering spaces. Central to this theory is the notion of spread: a continuous map between T1-spaces such that the connected components of inverse images of open subsets of the target space form a basis for the topology of the source space. The fiber of a spread is shown to embed into an inverse limit of discrete spaces. If this embedding is actually surjective for all fibers, then the spread is called complete. Every spread admits a unique completion up to homeomorphism. This understood, a ramified covering f:Y→Z is a complete spread between connected spaces whose set of ordinary points, and its preimage, are dense and locally connected in Z, respectively Y. Moreover, f is the completion of its associated unramified covering, which in fact determines it uniquely. The interpretation of spreads via inverse limits is used by the author to show that a ramified covering f:Y→Z is surjective and open if Z satisfies the first countability axiom, and that it is discrete if all the ramification indices are finite. An interesting example is constructed of a ramified covering of infinite degree of the 3-sphere branched over a wild knot and having a compact but non-discrete fiber. A few intriguing open problems end the article: Are there non-surjective or non-open ramified coverings? Is there a ramified covering with a fiber homeomorphic to the Cantor set?