Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T07:11:21.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Representing 3-manifolds by a universal branching set

Published online by Cambridge University Press:  24 October 2008

José María Montesinos
José María Montesinos
Affiliation:
Universidad de Zaragoza, Spain
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 1 , July 1983 , pp. 109 - 123
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Alexander, J. W.Note on Riemann spaces. Bull. Amer. Math. Soc. 26 (1920), 370–372.CrossRefGoogle Scholar
(2)Fox, R. H. Covering spaces with singularities. In Algebraic Geometry and Topology: a Symposium in Honor of S. Lefschetz (Princeton, 1957).CrossRefGoogle Scholar
(3)Lyndon, R. C. & Schupp, P. E.Combinatorial group theory (Springer-Verlag 1977).Google Scholar
(4)Neuwirth, L.Knot groups. Ann. Math. Studies 56 (1965).Google Scholar
(5)Neuwirth, L.An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Cambridge Philos. Soc. 64 (1968), 603–613.CrossRefGoogle Scholar
(6)Poincaré, H.Cinquième complément à l'analysis situs. Rend. Circ. Mat. Palermo 18 (1904), 45–110.CrossRefGoogle Scholar
(7)Ramírez, A.Sobre un teorema de Alexander. Anales del Instituto de Matemáticas UNAM 15 (1975), 77–81.Google Scholar
(8)Seifert, H. & Threlfall, W.A textbook of topology (Academic Press, 1980).Google Scholar
(9)Waldhausen, F.Some problems on 3-manifolds. Proceedings of Symposia in Pure Mathematics 32 (1978), 313–322.CrossRefGoogle Scholar
(10)Whitehead, J. H. C.On certain sets of elements in a free group. Proc. London Math. Soc. 41 (1936), 48–56.CrossRefGoogle Scholar