Skip to main content
×
Home

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.

Copyright
References
Hide All
(1)Alexander J. W.Note on Riemann spaces. Bull. Amer. Math. Soc. 26 (1920), 370372.
(2)Fox R. H. Covering spaces with singularities. In Algebraic Geometry and Topology: a Symposium in Honor of S. Lefschetz (Princeton, 1957).
(3)Lyndon R. C. & Schupp P. E.Combinatorial group theory (Springer-Verlag 1977).
(4)Neuwirth L.Knot groups. Ann. Math. Studies 56 (1965).
(5)Neuwirth L.An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Cambridge Philos. Soc. 64 (1968), 603613.
(6)Poincaré H.Cinquième complément à l'analysis situs. Rend. Circ. Mat. Palermo 18 (1904), 45110.
(7)Ramírez A.Sobre un teorema de Alexander. Anales del Instituto de Matemáticas UNAM 15 (1975), 7781.
(8)Seifert H. & Threlfall W.A textbook of topology (Academic Press, 1980).
(9)Waldhausen F.Some problems on 3-manifolds. Proceedings of Symposia in Pure Mathematics 32 (1978), 313322.
(10)Whitehead J. H. C.On certain sets of elements in a free group. Proc. London Math. Soc. 41 (1936), 4856.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 102 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th November 2017. This data will be updated every 24 hours.