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Representing 3-manifolds by a universal branching set

  • José María Montesinos (a1)

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.

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(1) J. W. Alexander Note on Riemann spaces. Bull. Amer. Math. Soc. 26 (1920), 370372.

(2) R. H. Fox Covering spaces with singularities. In Algebraic Geometry and Topology: a Symposium in Honor of S. Lefschetz (Princeton, 1957).

(6) H. Poincaré Cinquième complément à l'analysis situs. Rend. Circ. Mat. Palermo18 (1904), 45110.

(8) H. Seifert & W. Threlfall A textbook of topology (Academic Press, 1980).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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