Heegaard structure respects complicated JSJ decompositions

Abstract

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $\varphi :T_1\to T_2$ be a homeomorphism, $M_{\varphi }$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $\varphi$, and $T$ the image of $T_1$ in $M_{\varphi }$. We show that if $\varphi$ is "sufficiently complicated" then any incompressible or strongly irreducible surface in $M_{\varphi }$ 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.

Incorporates updated references and improved exposition

References in corpus (3)

• On the Heegaard splittings of amalgamated 3-manifolds
• The structure of a solvmanifold's Heegaard splittings
• Algorithmically detecting the bridge number of hyperbolic knots