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

Sangbum Cho; Yuya Koda; Jung Hoon Lee; Nozomu Sekino

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arXiv · 2406.13309

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

Sangbum Cho·Yuya Koda·Jung Hoon Lee·Nozomu Sekino

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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 $g \geq 4$.

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BibTeX RIS

Sangbum Cho, Yuya Koda, Jung Hoon Lee, Nozomu Sekino. 2024-06-19. On the Invalidity of Lemma 2.5 in our previous work on the Powell Conjecture. https://arxiv.org/abs/2406.13309

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