On the Invalidity of Lemma 2.5 in our previous work on the Powell Conjecture
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915212604473344 |
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| author | Cho, Sangbum Koda, Yuya Lee, Jung Hoon Sekino, Nozomu |
| author_facet | Cho, Sangbum Koda, Yuya Lee, Jung Hoon Sekino, Nozomu |
| contents | 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 $g \geq 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Invalidity of Lemma 2.5 in our previous work on the Powell Conjecture Cho, Sangbum Koda, Yuya Lee, Jung Hoon Sekino, Nozomu Geometric Topology 57K30, 57K20, 20F05 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 $g \geq 4$. |
| title | On the Invalidity of Lemma 2.5 in our previous work on the Powell Conjecture |
| topic | Geometric Topology 57K30, 57K20, 20F05 |
| url | https://arxiv.org/abs/2406.13309 |