On amalgamations of Heegaard splittings with high distance

Yang, Guoqiu; Lei, Fengchun

doi:10.1090/S0002-9939-08-09642-1

On amalgamations of Heegaard splittings with high distance
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by Guoqiu Yang and Fengchun Lei

Proc. Amer. Math. Soc. 137 (2009), 723-731

DOI: https://doi.org/10.1090/S0002-9939-08-09642-1

Published electronically: September 9, 2008

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Abstract:

Let $M$ be a compact, orientable 3-manifold and $F$ an essential closed surface which cuts $M$ into $M_{1}$ and $M_{2}$. Suppose that $M_{i}$ has a Heegaard splitting $V_{i}\cup _{S_{i}}W_{i}$ with distance $D{(S_{i})}\geqslant {2g(M_{i})+1}$, $i=1, 2$. Then $g(M)=g(M_1)+g(M_2)-g(F)$, and the amalgamation of $V_{1}\cup _{S_{1}}W_{1}$ and $V_{2}\cup _{S_{2}}W_{2}$ is the unique minimal Heegaard splitting of $M$ up to isotopy.

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