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

TL;DR

This note identifies a critical flaw in the previous claim that the Powell Conjecture holds for genus-$g$ Heegaard splittings of $S^3$ by showing Lemma $2.5$ fails in general. It provides explicit counterexamples for $g=3$ and a general construction for all $g\ge 3$, exhibiting genus-$g$ Heegaard surfaces $\Sigma$ with disks $D_1,\dots,D_{g-2}\subset V$ and a non-separating disk $E\subset W$ such that no reducing sphere $P$ separates $\cup_{i=1}^{g-2} D_i$ from $E$. The counterexamples arise from compressions of $\Sigma$ to produce knotted torus components, including a component $\Sigma_0$ bounding a knotted solid torus $V_0$ with no essential disks, implying any separating reducing sphere cannot exist. Consequently, the Powell Conjecture remains open for $g\ge 4$, and this note clarifies the status by identifying the invalid step in the previous proof.

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 $g \geq 4$.

Figures (2)

• Figure 1: The genus-$3$ surface $\Sigma$.
• Figure 2: The genus-$g$ surface $\Sigma$.