Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Representing 3-manifolds by a universal branching set

José María Montesinosa1*

a1 Universidad de Zaragoza, Spain

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 S3 branched over a subcomplex G of S3 which is independent of M, and such that S3 - 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 Σh on the set {1,2, …, h}, for some h. We show how to obtain {σ, τ} from a given Heegaard diagram of M.

(Received November 18 1981)

(Revised September 09 1982)

Footnotes

* Supported by ‘Comisión Asesora del Ministerio de Educación y Ciencia’.