Heegaard surfaces and measured laminations, II: Non-Haken 3–manifolds

Li, Tao

doi:10.1090/S0894-0347-06-00520-0

Heegaard surfaces and measured laminations, II: Non-Haken 3–manifolds
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by Tao Li;

J. Amer. Math. Soc. 19 (2006), 625-657

DOI: https://doi.org/10.1090/S0894-0347-06-00520-0

Published electronically: February 3, 2006

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Abstract:

A famous example of Casson and Gordon shows that a Haken 3–manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3–manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the generalized Waldhausen conjecture. Another immediate corollary is that for any irreducible non-Haken 3–manifold $M$, there is a number $N$ such that any two Heegaard splittings of $M$ are equivalent after at most $N$ stabilizations.

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