Representing open 3-manifolds as 3-fold branched coverings

Abstract

It is a celebrated result of H. Hilden and the author of the present paper that every closed, connected, oriented 3-manifold is a 3-fold irregular (dihedral) branched covering of the 3-sphere, branched over a knot. Here the author explores a generalization of this result to the case of non-compact manifolds. It is shown that a non-compact, connected, oriented 3-manifold is a 3-fold irregular branched covering of an open subspace of S3, branched over a locally finite family of proper arcs. The branched covering is constructed in such a way that it extends to a branched covering (suitably understood) of the Freudenthal end compactification over the entire 3-sphere. In particular all (uncountably many) contractible open 3-manifolds may be expressed as 3-fold branched coverings of R3, branched over a locally finite collection of proper arcs.