Full paper in PDF:
$% J. M. Montesinos-Amilibia, Open 3-manifolds, wild subsets of S3 and branched coverings , Rev. Mat. Complut. 16
(2003), 577–600. %$

Open 3-Manifolds, Wild Subsets of S3 and Branched Coverings

José María MONTESINOS-AMILIBIA
Facultad de Matemáticas
Universidad Complutense
28040 Madrid Spain

Received: January 31, 2002
Accepted: December 9, 2002
Dedicated with respect and affection
to Professor Laurent Siebenmann
ABSTRACT

In this paper, a representation of closed 3-manifolds as branched coverings of the 3-sphere, proved in Montesinos-Amilibia (2002), 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 (1949) (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.

Key words: Wild knots, open manifolds, branched coverings.
2000 Mathematics Subject Classification:
57M12, 57M30, 57N10.