Dehn twists and the Nielsen realization problem for spin 4-manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866913422008909824 |
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| author | Konno, Hokuto |
| author_facet | Konno, Hokuto |
| contents | We prove that, for a closed oriented smooth spin 4-manifold $X$ with non-zero signature, the Dehn twist about a $(+2)$- or $(-2)$-sphere in $X$ is not homotopic to any finite order diffeomorphism. In particular, we negatively answer the Nielsen realization problem for each group generated by the mapping class of a Dehn twist. We also show that there is a discrepancy between the Nielsen realization problems in the topological category and smooth category for connected sums of copies of $K3$ and $S^{2} \times S^{2}$. The main ingredients of the proofs are Y. Kato's 10/8-type inequality for involutions and a refinement of it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11631 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Dehn twists and the Nielsen realization problem for spin 4-manifolds Konno, Hokuto Geometric Topology Differential Geometry We prove that, for a closed oriented smooth spin 4-manifold $X$ with non-zero signature, the Dehn twist about a $(+2)$- or $(-2)$-sphere in $X$ is not homotopic to any finite order diffeomorphism. In particular, we negatively answer the Nielsen realization problem for each group generated by the mapping class of a Dehn twist. We also show that there is a discrepancy between the Nielsen realization problems in the topological category and smooth category for connected sums of copies of $K3$ and $S^{2} \times S^{2}$. The main ingredients of the proofs are Y. Kato's 10/8-type inequality for involutions and a refinement of it. |
| title | Dehn twists and the Nielsen realization problem for spin 4-manifolds |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2203.11631 |