#### Volume 9, issue 4 (2009)

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Amalgamations of Heegaard splittings in $3$–manifolds without some essential surfaces

### Guoqiu Yang and Fengchun Lei

Algebraic & Geometric Topology 9 (2009) 2041–2054
##### Abstract

Let $M$ be a compact, orientable, $\partial$–irreducible $3$–manifold and $F$ be a connected closed essential surface in $M$ with $g\left(F\right)\ge 1$ which cuts $M$ into ${M}_{1}$ and ${M}_{2}$. In the present paper, we show the following theorem: Suppose that there is no essential surface with boundary $\left({Q}_{i},\partial {Q}_{i}\right)$ in $\left({M}_{i},F\right)$ satisfying $\chi \left({Q}_{i}\right)\ge 2+g\left(F\right)-2g\left({M}_{i}\right)+1$, $i=1,2$. Then $g\left(M\right)=g\left({M}_{1}\right)+g\left({M}_{2}\right)-g\left(F\right)$. As a consequence, we further show that if ${M}_{i}$ has a Heegaard splitting ${V}_{i}{\cup }_{{S}_{i}}{W}_{i}$ with distance $D\left({S}_{i}\right)\ge 2g\left({M}_{i}\right)-g\left(F\right)$, $i=1,2$, then $g\left(M\right)=g\left({M}_{1}\right)+g\left({M}_{2}\right)-g\left(F\right)$.

The main results follow from a new technique which is a stronger version of Schultens’ Lemma.

##### Keywords
essential surface, Heegaard genus
##### Mathematical Subject Classification 2000
Primary: 57M99, 57N10
Secondary: 57M27