Heegaard surfaces and the distance of amalgamation

Abstract

Let $M_1$ and $M_2$ be orientable irreducible $3$–manifolds with connected boundary and suppose $\partial M_1\cong \partial M_2$. Let $M$ be a closed $3$–manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S^3$ and all small-genus Heegaard splittings of $M$ are standard in a certain sense. In particular, $g\left(M\right)=g\left(M_1\right)+g\left(M_2\right)-g\left(\partial M_i\right)$, where $g\left(M\right)$ denotes the Heegaard genus of $M$. This theorem is also true for certain manifolds with multiple boundary components.