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Heegaard splittings with (disk, essential surface) pairs that intersect in one point

Author: Jung Hoon Lee
Journal: Proc. Amer. Math. Soc. 138 (2010), 1877-1888
MSC (2000): Primary 57M50, 57M25
Published electronically: December 18, 2009
MathSciNet review: 2587472
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Abstract: We consider a Heegaard splitting $ M=H_1\cup_S H_2$ of a $ 3$-manifold $ M$ having an essential disk $ D$ in $ H_1$ and an essential surface $ F$ in $ H_2$ with $ \vert D\cap F\vert=1$.

From $ H_1\cup_S H_2$, we obtain another Heegaard splitting $ H'_1\cup_{S'} H'_2$ by removing a neighborhood of $ F$ from $ H_2$ and attaching it to $ H_1$. As an application, by using a theorem due to Casson and Gordon,

we give examples of $ 3$-manifolds admitting two Heegaard splittings of distinct genera, where one of them is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction.

We also show that all Heegaard splittings of a Seifert fibered space are related via the above construction.

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Additional Information

Jung Hoon Lee
Affiliation: School of Mathematics, Korea Institute for Advanced Study, 207-43, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, Korea

Keywords: Heegaard splitting, essential surface, Seifert fibered space, strongly irreducible
Received by editor(s): January 4, 2009
Received by editor(s) in revised form: May 25, 2009, and August 4, 2009
Published electronically: December 18, 2009
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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