Open 3-manifolds, wild subsets of S3 and branched coverings

Abstract

It is proved that any closed orientable 3-manifold is a 3-fold irregular branched covering of the 3-sphere branched over a wildly embedded knot. These branched coverings are obtained by starting with such a branched covering over a tame knot and then inserting into it a particular irregular branched covering of the 3-sphere over the 3-sphere, with a wild branch set. It is also shown how to use related techniques to produce branched coverings of certain open 3-manifolds over tame, properly embedded arcs in R3. For example, the Whitehead contractible open 3-manifold is expressible as a 2-fold branched covering over such an arc, conjecturally in uncountably many different ways. These results should be considered as illustrations of the general construction given in the author's recent paper [Rev. Mat. Complut. 15 (2002), no. 2, 533–542