Heegaard structure respects complicated JSJ decompositions

2009/11/27 by Bachman, David, Derby-Talbot, Ryan, Sedgwick, Eric
#57M99 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0911.5078

Abstract

Let M be a 3-manifold with torus boundary components T₁ and T₂. Let ϕ\colon T₁ → T₂ be a homeomorphism, M_ϕ the manifold obtained from M by gluing T₁ to T₂ via the map ϕ, and T the image of T₁ in M_ϕ. We show that if ϕ is "sufficiently complicated" then any incompressible or strongly irreducible surface in M_ϕ 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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