# The Powell Conjecture and reducing sphere complexes

## TL;DR

This paper explores the Powell Conjecture's equivalence to the connectivity of the reducing sphere complex in genus g Heegaard splittings, providing new insights into the structure and geometry of these complexes.

## Contribution

It establishes that the Powell Conjecture holds if and only if the reducing sphere complex is connected, and analyzes the complexity of paths between reducing curves.

## Key findings

- Powell Conjecture is equivalent to the connectivity of the reducing sphere complex.
- Reducing curves meeting in at most six points are connected within the complex.
- Distances between certain reducing curves can be arbitrarily large.

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