Arc complexes, sphere complexes and Goeritz groups

2014/03/30 by Sangbum Cho, Cho, Sangbum, Yuya Koda +3
Mathematics · #57M60 #57N10 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1403.7832

openalex publication_date 2014/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of S² × S¹, then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact orientable surface to be contractible, which generalizes the result by Hatcher that the arc complexes are contractible. We then construct infinitely many Heegaard splittings, including the above-mentioned Heegaard splitting, for which suitably defined complexes of Haken spheres are contractible.

Citations

• Mapping class groups of once-stabilized Heegaard splittings
• Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting
• The genus two Goeritz group of S² × S¹

Related

• Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds
• Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting
• Reducing spheres of genus-2 Heegaard splitting of S³
• The Goeritz group of a Heegaard splitting of genus two of a Seifert manifold whose base orbifold is sphere with three exceptional points of sufficiently complex coefficients
• Heegaard splittings and virtually special square complexes