In this paper, a representation of closed 3-manifolds as branched coverings of the 3-sphere, proved in [13], and showing a relationship between open 3-manifolds and wild knots and arcs will be illustrated by examples. It will be shown that there exist a 3-fold simple covering p : S3 --> S3 branched over the remarkable simple closed curve of Fox [4] (a wild knot). Moves are defined such that when applied to a branching set, the corresponding covering manifold remains unchanged, while the branching set changes and becomes wild. As a consequence every closed, oriented 3-manifold is represented as a 3-fold covering of S3 branched over a wild knot, in plenty of different ways, confirming the versatility of irregular branched coverings. Other collection of examples is obtained by pasting the members of an infinite sequence of two-component strongly-invertible link exteriors. These open 3-manifolds are shown to be 2-fold branched coverings of wild knots in the 3-sphere. Two concrete examples, are studied: the solenoidal manifold, and the Whitehead manifold. Both are 2-fold covering of the euclidean space R3 branched over an uncountable collection of string projections in R3.
@article{urn:eudml:doc:44439,
title = {Open 3-manifolds, wild subsets of S3 and branched coverings.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {16},
year = {2003},
pages = {577-600},
zbl = {1054.57005},
mrnumber = {MR2032934},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44439}
}
Montesinos-Amilibia, José María. Open 3-manifolds, wild subsets of S3 and branched coverings.. Revista Matemática de la Universidad Complutense de Madrid, Tome 16 (2003) pp. 577-600. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44439/