# One Powell generator is redundant

## TL;DR

This paper demonstrates that one of Powell's proposed generators for the Goeritz group of certain 3-manifold splittings is redundant, simplifying the understanding of the group's generating set.

## Contribution

It provides a short proof that one of Powell's five generators is unnecessary, reducing the generating set for the Goeritz group.

## Key findings

- One generator is redundant in Powell's generating set.
- The redundancy is a consequence of three other generators.
- Simplifies the structure of the Goeritz group for genus g ≥ 4.

## Abstract

In 1980 J. Powell proposed that five specific elements sufficed to generate the Goeritz group of any Heegaard splitting of $S^3$. This conjecture remains unresolved for genus $g \geq 4$. Here a short argument shows that one of his proposed generators is redundant, in fact a consequence of three of the other four.