Reducing spheres of genus-2 Heegaard splitting of S³

2022/12/20 by Sreekrishna Palaparthi, Palaparthi, Sreekrishna, Swapnendu Panda +1
Mathematics · #57K30 (Secondary) #57M40 (Primary) 20F38 #Advanced Topology and Set Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2212.09997

openalex publication_date 2022/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, G_g, acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann MR2199366 (2007c:57020) uses this action to prove that G₂ is finitely generated. In this article, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the S³. Using this we give an alternate proof of the finite generation of G₂ assuming the finite generation of the stabilizer of the standard reducing sphere.

Related

• Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting
• The Powell Conjecture for the genus-three Heegaard splitting of the 3-sphere
• Describing elements of the genus-2 Goeritz group of S³
• Genus 2 Goeritz Equivalence in S³
• Generating the genus g+1 Goeritz group of a genus g handlebody