On the Invalidity of Lemma 2.5 in our previous work on the Powell Conjecture

TL;DR

This paper identifies a critical flaw in a previous proof claiming to establish the Powell Conjecture for genus-g Heegaard splittings of the 3-sphere, leaving the conjecture unproven for g ≥ 4.

Contribution

The authors demonstrate that a key lemma in their prior work is invalid, invalidating their previous proof of the Powell Conjecture.

Findings

Lemma 2.5 does not hold in general

The previous proof of the Powell Conjecture is incomplete

The Powell Conjecture remains open for g ≥ 4

Abstract

In our previous version entitled ``The reducing sphere complexes for the 3-sphere are connected: a proof of the Powell Conjecture", we claimed to prove the Powell Conjecture, which states that the Goeritz group of the genus-$g$ Heegaard splitting of the 3-sphere is finitely generated for any non-negative integer $g$. However, we have found a critical error in the proof of Lemma 2.5 in that version. In this note, we prove that the statement of Lemma 2.5 does not hold in general. This invalidates a key step in our argument and leaves the proof of the Powell Conjecture incomplete. Consequently, the Powell Conjecture remains an open problem in the case of $g \geq 4$.