Heegaard surfaces and the distance of amalgamation

Let $M_1$ and $M_2$ be orientable irreducible 3–manifolds with connected boundary and suppose $\mathrm{\partial }{M}_1\cong \mathrm{\partial }{M}_2$. 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, $g(M)=g(M_1)+g(M_2)-g(\partial M_i)$,where$g(M)$denotestheHeegaardgenusof$M$. This theorem is also true for certain manifolds with multiple boundary components.