Heegaard structure respects complicated JSJ decompositions

Details Heegaard structure respects complicated JSJ decompositions Heegaard structure respects complicated JSJ decompositions

2009-11-27

arxiv.org/abs/0911.5078v1

Mathematics

Geometric Topology

22 pages, 7 figures

Scientific paper

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.

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