Representing 3-manifolds by a universal branching set

Abstract

The author shows that every compact connected oriented 3-manifold, after capping off boundary components by cones, is a covering of S3 branched over the 1-complex G which is "a pair of eyeglasses''. The author gives algorithms for passing between a Heegaard decomposition of a 3-manifold and this covering description. He also determines necessary and sufficient conditions for such a covering to have cone singularities. In a paper by W. Thurston ["Universal links'', Preprint], a link with similar properties (for closed 3-manifolds) to G is constructed.