Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation

2009/04/02 by David Bachman, Bachman, David
Mathematics · #57M99 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M99

paper · pdf · doi:10.48550/arxiv.0904.0485

16 pages, 4 figures

arxiv created 2009/04/02 · openalex publication_date 2009/04/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M₁ and M₂ be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of \bdy M₁ to some component of \bdy M₂ by a homeomorphism ϕ. We show that when ϕis "sufficiently complicated" then (1) the amalgamation of low genus, unstabilized, boundary-unstabilized Heegaard splittings of M_i is an unstabilized splitting of M, (2) every low genus, unstabilized Heegaard splitting of M can be expressed as an amalgamation of unstabilized, boundary-unstabilized splittings of M_i, and possibly a Type II splitting of F × I, and (3) if there is no Type II splitting in such an expression then it is unique.

Related

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• Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)
• Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization
• The Heegaard genus of amalgamated 3-manifolds
• 3-Manifolds with irreducible Heegaard splittings of high genus