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 necessary and sufficient condition for a $ 3$-manifold to have Heegaard genus one


Authors: Joel Hass and Abigail Thompson
Journal: Proc. Amer. Math. Soc. 107 (1989), 1107-1110
MSC: Primary 57N10
DOI: https://doi.org/10.1090/S0002-9939-1989-0984792-4
MathSciNet review: 984792
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a closed $ 3$-manifold. R. H. Bing showed that $ M$ is homeomorphic to $ {S^3}$ if and only if every simple closed curve in $ M$ can be isotoped to lie inside a $ 3$-ball. We generalize this to show that there is a solid torus $ T$ imbedded in $ M$ such that every simple closed curve in $ M$ can be isotoped to lie in $ T$ if and only if $ M$ has a genus one Heegaard splitting.


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

  • [1] R. H. Bing, Necessary and sufficient conditions that a $ 3$-manifold be $ {S^3}$, Ann. of Math. 68 (1958), 17-37. MR 0095471 (20:1973)
  • [2] C. McA. Gordon and J. M. Montesinos, Fibred knots and disks with clasps, M.S.R.I. preprint 13912-85, 1985. MR 858286 (88e:57007)
  • [3] W. Haken, Some results on surfaces in $ 3$-manifolds, Studies in Modern Topology, Math. Assoc. Amer., distributed by Prentice Hall, 1968, 34-98. MR 0224071 (36:7118)
  • [4] W. Jaco, Lectures on three-manifold topology, C.B.M.S., Vol. 43, 1980. MR 565450 (81k:57009)
  • [5] D. R. McMillan, On homologically trivial $ 3$-manifolds, Trans. Amer. Math. Soc. 98 (1961), 350-367. MR 0120639 (22:11389)
  • [6] R. Myers, Open book decompositions of $ 3$-manifolds, Proc. Amer. Math. Soc. 72 (1978), 397-402. MR 507346 (80a:57004)
  • [7] -, Simple knots in compact, orientable $ 3$-manifolds, Trans. Amer. Math. Soc. 273 (1982), 75-91. MR 664030 (83h:57018)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N10

Retrieve articles in all journals with MSC: 57N10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0984792-4
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society