Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the construction of simply-connected $ 3$-manifolds as branched covering spaces of $ S\sp{3}$


Author: Joan S. Birman
Journal: Proc. Amer. Math. Soc. 55 (1976), 440-442
DOI: https://doi.org/10.1090/S0002-9939-1976-0394629-3
MathSciNet review: 0394629
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: Let $ K$ be a knot in $ {S^3}$, and let $ \omega :{\pi _1}({S^3} - K) \to {\Sigma _n}$ be a transitive representation into the symmetric group $ {\Sigma _n}$ on $ n$ letters. The pair $ (K,\omega )$ defines a unique closed, connected orientable $ 3$-manifold $ M(K,\omega )$, which is represented as an $ n$-sheeted covering space of $ {S^3}$, branched over $ K$. A procedure is given for representing $ M(K,\omega )$ by a Heegard splitting, and a formula is given for computing the genus of that Heegard splitting of $ M(K,\omega )$. This formula is then applied to the $ 3$-sheeted irregular covering spaces studied by Hilden (Bull. Amer. Math. Soc. 80 (1974), 1243-1244) and Montesinos (Bull. Amer. Math. Soc. 80 (1974), 845-846), and, also, Tesis (Univ. de Madrid, 1971) to show that these particular covering spaces cannot yield counterexamples to the Poincaré Conjecture if the branch set has bridge number $ < 4$.


References [Enhancements On Off] (What's this?)

  • [1] J. S. Birman and H. M. Hilden, The homeomorphism problem for $ {S^3}$, Bull. Amer. Math. Soc. 79 (1973), 1006-1010. MR 47 #7726. MR 0319180 (47:7726)
  • [2] -, Heegaard splittings of branched coverings of $ {S^3}$, Trans. Amer. Math. Soc. 213 (1975), 315-352. MR 0380765 (52:1662)
  • [3] R. H. Fox, Construction of simply connected $ 3$-manifolds, Topology of $ 3$-Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 213-216. MR 25 #3539. MR 0140116 (25:3539)
  • [4] H. M. Hilden, Every closed, orientable $ 3$-manifold is a $ 3$-fold branched covering space of $ {S^3}$, Bull. Amer. Math. Soc. 80 (1974), 1243-1244. MR 0350719 (50:3211)
  • [5] -, Three-fold branched coverings of $ {S^3}$, Amer. J. Math. (to appear).
  • [6] J. M. Montesinos, Sobre la conjetura de Poincaré y los recubridores ramificados sobre un nudo, Tesis, Universidad de Madrid, 1971.
  • [7] -, A representation of closed orientable $ 3$-manifolds as $ 3$-fold branched coverings of $ {S^3}$, Bull. Amer. Math. Soc. 80 (1974), 845-846. MR 0358784 (50:11243)
  • [8] -, Three-manifolds as $ 3$-fold branched covers of $ {S^3}$, Bull. Amer. Math. Soc. 80 (1974), 845-846. MR 0358784 (50:11243)
  • [9] G. Springer, Introduction to Riemann surfaces, Addison-Wesley, Reading, Mass., 1957. MR 19, 1169. MR 0092855 (19:1169g)
  • [10] F. Waldhausen, Über Involution der $ 3$-Sphäre, Topology 8 (1969), 81-91. MR 38 #5209. MR 0236916 (38:5209)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0394629-3
Keywords: $ 3$-manifold, homotopy sphere, branched covering space, knot, Heegard splitting
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society