Representing 3-manifolds by a universal branching set

Abstract

In this paper all 3-manifolds will be supposed to be compact, connected, oriented and without 2-spheres in the boundary . Given a 3-manifold M we obtain a closed pseudomanifold M^ by capping off each boundary component of M with a cone. We prove that such an M^ is a covering of S 3 branched over a subcomplex G of S 3 which is independent of M , and such that S 3 - G has free fundamental group on two generators. Hence M^ (and also M ) can be represented by a transitive pair {σ, τ} of permutations in the symmetric group Σ h on the set {1,2, …, h }, for some h . We show how to obtain {σ, τ} from a given Heegaard diagram of M .