Heegaard structure respects complicated JSJ decompositions

Abstract

Let M be a 3-manifold with torus boundary components $T_{1}$ and $T_2$. Let $\phi :T_{1} \rightarrow T_{2}$ be a homeomorphism, $M_\phi$ the manifold obtained from M by gluing $T_{1}$ to $T_{2}$ via the map $\phi$, and T the image of $T_{1}$ in $M_\phi$. We show that if $\phi$ is “sufficiently complicated” then any incompressible or strongly irreducible surface in $M_\phi$ can be isotoped to be disjoint from T. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.