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Representing 3-manifolds by a universal branching set

  • José María Montesinos (a1)
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 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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

(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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