The Powell conjecture offers a finite generating set for the genus $g$ Goeritz group, the group of automorphisms of $S^3$ that preserve a genus $g$ Heegaard surface ${\mathrm{\Sigma }}_g$, generalizing a classical result of Goeritz in the case $g=2$. We study the relationship between the Powell conjecture and the reducing sphere complex $\mathcal{R}({\mathrm{\Sigma }}_g)$, the subcomplex of the curve complex $\mathcal{C}({\mathrm{\Sigma }}_g)$ spanned by the reducing curves for the Heegaard splitting. We prove that the Powell conjecture is true if and only if $\mathcal{R}({\mathrm{\Sigma }}_g)$ is connected. Additionally, we show that reducing curves that meet in at most six points are connected by a path in $\mathcal{R}({\mathrm{\Sigma }}_g)$; however, we also demonstrate that even among reducing curves meeting in four points, the distance in $\mathcal{R}({\mathrm{\Sigma }}_g)$ between such curves can be arbitrarily large. We conclude with a discussion of the geometry of $\mathcal{R}({\mathrm{\Sigma }}_g)$.

中文翻译：

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鲍威尔猜想和约化球体络合物

鲍威尔猜想为该类提供了一个有限的生成集 $G$ Goeritz群，自同构群 ${\mathrm{小号}}^3$ 保留一个属 $G$ Heegaard表面 ${\mathrm{\Sigma }}_G$，在该案例中推广了Goeritz的经典结果 $G=2$。我们研究了鲍威尔猜想和还原球体之间的关系$\mathcal{[R}（{\mathrm{\Sigma }}_G）$，曲线复数的子复数 $\mathcal{C}（{\mathrm{\Sigma }}_G）$Heegaard分裂的归约曲线所覆盖。我们证明Powell猜想是当且仅当$\mathcal{[R}（{\mathrm{\Sigma }}_G）$已连接。此外，我们证明了在最多六个点处相交的归约曲线是通过一条路径连接的$\mathcal{[R}（{\mathrm{\Sigma }}_G）$; 但是，我们也证明了即使在四个点相交的简化曲线中，$\mathcal{[R}（{\mathrm{\Sigma }}_G）$这样的曲线之间的距离可以任意大。最后，我们讨论了$\mathcal{[R}（{\mathrm{\Sigma }}_G）$。