A proof of Powell's conjecture on the Goeritz group of $S^3$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911703634018304 |
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| author | Iguchi, Daiki |
| author_facet | Iguchi, Daiki |
| contents | For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21905 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A proof of Powell's conjecture on the Goeritz group of $S^3$ Iguchi, Daiki Geometric Topology 57K30, 57M60 For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$. |
| title | A proof of Powell's conjecture on the Goeritz group of $S^3$ |
| topic | Geometric Topology 57K30, 57M60 |
| url | https://arxiv.org/abs/2605.21905 |