Heegaard splittings with (disk, essential surface) pairs that intersect in one point
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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 $|D\cap F|=1$.
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
- Email: jhlee@kias.re.kr
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1877-1888
- MSC (2000): Primary 57M50, 57M25
- DOI: https://doi.org/10.1090/S0002-9939-09-10147-8
- MathSciNet review: 2587472