The Dehn twist on a connected sum of two homology tori
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916631544856576 |
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| author | Qiu, Haochen |
| author_facet | Qiu, Haochen |
| contents | Kronheimer-Mrowka shows that the Dehn twist along a $3$-sphere in the neck of two $K3$ surfaces is not smoothly isotopic to the identity. Their result requires that the manifolds are simply connected and the signature of one of them is $16 \mod 32$. We generalize the Pin$(2)$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if $X_1,X_2$ are two homology tori such that the determinants $r_1,r_2$ of them are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\# X_2$ is not smoothly isotopic to the identity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02461 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dehn twist on a connected sum of two homology tori Qiu, Haochen Geometric Topology Kronheimer-Mrowka shows that the Dehn twist along a $3$-sphere in the neck of two $K3$ surfaces is not smoothly isotopic to the identity. Their result requires that the manifolds are simply connected and the signature of one of them is $16 \mod 32$. We generalize the Pin$(2)$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if $X_1,X_2$ are two homology tori such that the determinants $r_1,r_2$ of them are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\# X_2$ is not smoothly isotopic to the identity. |
| title | The Dehn twist on a connected sum of two homology tori |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2410.02461 |