Reducing spheres of genus-2 Heegaard splitting of S3

Abstract

The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, [Formula: see text], acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann [Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana 10 (2004) 503–514] uses this action to prove that [Formula: see text] is finitely generated. In this paper, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the [Formula: see text]. Using this we give an alternate proof of the finite generation of [Formula: see text] assuming the finite generation of the stabilizer of the standard reducing sphere.