The Powell Conjecture and reducing sphere complexes

Abstract

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 $Σ_g$, generalizing a classical result of Goeritz in the case $g=2$. We study the relationship between the Powell Conjecture and the reducing sphere complex $\mathcal{R}(Σ_g)$, the subcomplex of the curve complex $\mathcal{C}(Σ_g)$ spanned by the reducing curves for the Heegaard splitting. We prove that the Powell Conjecture is true if and only if $\mathcal{R}(Σ_g)$ is connected. Additionally, we show that reducing curves that meet in at most six points are connected by a path in $\mathcal{R}(Σ_g)$; however, we also demonstrate that even among reducing curves meeting in four points, the distance in $\mathcal{R}(Σ_g)$ between such curves can be arbitrarily large. We conclude with a discussion of the geometry of $\mathcal{R}(Σ_g)$.