Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting

Abstract

An updated proof of a 1933 theorem of Goeritz, exhibiting a finite set of generators for the group of automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. The group is analyzed via its action on a certain connected 2-complex. (The analogous problem for higher genus Heegaard splittings appears to remain unresolved.)

arXiv comment13 pages, 10 figures

Cited by in corpus (25)

• Powell moves and the Goeritz group
• Mapping class groups of Heegaard splittings
• Primitive disk complexes for lens spaces
• Mapping class groups of once-stabilized Heegaard splittings
• Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots
• The Powell Conjecture and reducing sphere complexes
• Heegaard splittings of 3--manifolds (Haifa 2005): Problems
• The genus two Goeritz group of $S^2\times S^1$
• Tunnel leveling, depth, and bridge numbers
• The depth of a knot tunnel
• Connected primitive disk complexes and genus two Goeritz groups of lens spaces
• Arc complexes, sphere complexes and Goeritz groups
• Knot homotopy in subspaces of the 3-sphere
• Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds
• Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two
• Convex-cocompact subgroups of the Goeritz group
• Additional cases of positive twisted torus knots
• Essential surfaces of non-negative Euler characteristic in genus two handlebody exteriors
• The mapping class groups of reducible Heegaard splittings of genus two
• A Complex of Incompressible Surfaces for handlebodies and the Mapping Class Group
• Thick isotopy property and the mapping class groups of Heegaard splittings
• Twisted book decompositions and the Goeritz groups
• Haken spheres for genus two Heegaard splittings
• Constructing knot tunnels using giant steps
• Describing elements of the genus-2 Goeritz group of $S^3$