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January 2020 Generalized Heegaard Splittings and the Disk Complex
Jungsoo Kim
Osaka J. Math. 57(1): 103-140 (January 2020).

Abstract

Let $M$ be an orientable, irreducible $3$-manifold and $(\mathcal{V},\mathcal{W};F)$ a weakly reducible, unstabilized Heegaard splitting of $M$ of genus at least three. In this article, we define an equivalence relation $\sim$ on the set of the generalized Heegaard splittings obtained by weak reductions and find special subsets of the disk complex $\mathcal{D}(F)$ named by the ``$equivalent$ $clusters$'', where we can find a canonical function $\Phi$ from the set of equivalent clusters to the set of the equivalent classes for the relation $\sim$. These equivalent classes are more detailed than the isotopy classes of the generalized Heegaard splittings obtained by weak reductions from $F$. In the last section, we prove $\Phi$ is a bijection if the genus of $F$ is three.

Citation

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Jungsoo Kim. "Generalized Heegaard Splittings and the Disk Complex." Osaka J. Math. 57 (1) 103 - 140, January 2020.

Information

Published: January 2020
First available in Project Euclid: 15 January 2020

zbMATH: 07196618
MathSciNet: MR4052631

Subjects:
Primary: 57M50

Rights: Copyright © 2020 Osaka University and Osaka City University, Departments of Mathematics

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Vol.57 • No. 1 • January 2020
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