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Proceedings of the American Mathematical Society
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
MathSciNet review: 984792
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0984792-4
PII: S 0002-9939(1989)0984792-4
Article copyright: © Copyright 1989 American Mathematical Society