Heegaard structure respects complicated JSJ decompositions

TL;DR

The paper demonstrates that in 3-manifolds with complex JSJ decompositions, Heegaard splittings can be understood as amalgamations of splittings of simpler components, especially after complicated gluings.

Contribution

It proves that sufficiently complicated gluings in 3-manifolds ensure Heegaard surfaces are disjoint from the glued boundary, simplifying their analysis.

Findings

Incompressible surfaces can be isotoped away from the boundary in complex gluings.

Heegaard splittings decompose into amalgamations of component splittings under complex JSJ conditions.

Sufficiently complicated gluings restrict the topology of embedded surfaces.

Abstract

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $\phi \colon T_1 \to T_2$ 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.