Heegaard structure respects complicated JSJ decompositions

Let M be a 3-manifold with torus boundary components T 1 and T 2 . Let ϕ : T 1 → T 2 be a homeomorphism, M ϕ the manifold obtained from M by gluing T 1 to T 2 via the map ϕ , and T the image of T 1 in M ϕ . We show that if ϕ is “sufficiently complicated” then any incompressible or strongly irreducib...

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Bibliographic Details
Published inMathematische annalen Vol. 365; no. 3-4; pp. 1137 - 1154
Main Authors Bachman, David, Derby-Talbot, Ryan, Sedgwick, Eric
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2016
Subjects
Mathematics
Mathematics and Statistics
57M99
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Summary:Let M be a 3-manifold with torus boundary components T 1 and T 2 . Let ϕ : T 1 → T 2 be a homeomorphism, M ϕ the manifold obtained from M by gluing T 1 to T 2 via the map ϕ , and T the image of T 1 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.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-015-1314-9