Publication: Basic topological properties of Fox's branched coverings.
No Thumbnail Available
Official URL
Full text at PDC
Publication Date
2004
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Editorial Complutense
Citations
Exportar
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?