Open 3-manifolds, wild subsets of S3 and branched coverings Jos¶e Mar¶a MONTESINOS-AMILIBIA¤

Dedicated with respect and a®ection to Professor Laurent Siebenmann 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 re-markable simple closed curve of Fox [4] (a wild knot). Moves are de¯ned such that when applied to a branching set, the corresponding covering manifold re-mains unchanged, while the branching set changes and becomes wild. As a consequence every closed, oriented 3-manifold is represented as a 3-fold cov-ering of S3 branched over a wild knot, in plenty of di®erent ways, con¯rming the versatility of irregular branched coverings. Other collection of examples is obtained by pasting the members of an in¯nite 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