Heegaard structure respects complicated JSJ decompositions

الأصل · EN

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