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 [Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana10 (2004) 503–514] uses this action to prove that $G_2$ 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 $S^3$. Using this we give an alternate proof of the finite generation of $G_2$ assuming the finite generation of the stabilizer of the standard reducing sphere.

标准格列的 Goeritz 群 -三球体的g Heegaard 分裂，$G_G$，作用于此 Heegaard 分裂的还原球体同位素类空间。Scharlemann [保留属二 Heegaard 分裂的 3 球体自同构，Bol。苏克。垫。Mexicana 10 (2004) 503–514] 使用此操作来证明$G_2$是有限生成的。在本文中，我们给出了一种算法，用于从标准约化球构造任意约化球，以实现 genus-2 Heegaard 分裂$S^3$。使用这个我们给出了有限生成的替代证明$G_2$假设标准减径球的稳定器是有限代的。