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

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.