Three-manifolds as 3-fold branched covers of S3

Abstract

The following theorem is established: Every closed orientable 3-manifold M may be represented as a 3-fold irregular covering space of S3, with branching over a knot K, in such a way that the inverse image of each point of K consists of one point of branch index 2 and a second point of branch index 1. This result was also proved simultaneously by H. M. Hilden [Bull. Amer. Math. Soc. 80 (1974), 1243–1244] using different methods. Covering spaces of the type studied here were investigated earlier by the author [Rev. Mat. Hisp.-Amer. (4) 32 (1972), 33–51], who defined a set of moves that allow one to modify the branch set without altering the topological type of the covering space. Using these modifications, one sees that there are, in general, a very large variety of distinct knot types that can serve as the branch set. The proof of the main theorem in the present paper shows, nevertheless, that the branch set may be chosen to be a very special type of knot which looks (roughly speaking) like a ribbon knot except for certain special types of clasp singularities.