### Impacto

Montesinos Amilibia, José María
(1983)
*Representing 3-manifolds by a universal branching set.*
Mathematical Proceedings of the Cambridge Philosophical Society, 94
(1).
pp. 109-123.
ISSN 0305-0041

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Official URL: http://journals.cambridge.org/abstract_S0305004100060941

## Abstract

The author shows that every compact connected oriented 3-manifold, after capping off boundary components by cones, is a covering of S3 branched over the 1-complex G which is "a pair of eyeglasses''. The author gives algorithms for passing between a Heegaard decomposition of a 3-manifold and this covering description. He also determines necessary and sufficient conditions for such a covering to have cone singularities. In a paper by W. Thurston ["Universal links'', Preprint], a link with similar properties (for closed 3-manifolds) to G is constructed.

Item Type: | Article |
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Uncontrolled Keywords: | branched coverings of the 3-sphere; finite presentation of fundamental group; compact, connected, oriented 3-manifold without 2-spheres in the boundary; singular 3-manifold; Heegaard diagram |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 17198 |

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

Deposited On: | 26 Nov 2012 09:14 |

Last Modified: | 07 Feb 2014 09:43 |

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