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

Abstract

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 : S 3 S 3 branched over the re markable simple closed curve of Fox [4] (a wild knot). Moves are defined 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 S 3 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 pro jections in R3 . 2000 Mathematics Sub ject Classification: 57M12, 57M30, 57N10.