Dax invariants, light bulbs, and isotopies of symplectic structures
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| Format: | Preprint |
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2025
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| _version_ | 1866908801218641920 |
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| author | Lin, Jianfeng Wu, Weiwei Xie, Yi Zhang, Boyu |
| author_facet | Lin, Jianfeng Wu, Weiwei Xie, Yi Zhang, Boyu |
| contents | This paper addresses several isotopy problems on $4$-manifolds. First, we classify the isotopy classes of embeddings of $Σ$ in $Σ\times S^2$ that are geometrically dual to $\{\mbox{pt}\}\times S^2$, where $Σ$ is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic, thereby answering a question of Gabai.
By combining this construction with techniques from symplectic topology, we also answer Problem 2(a) in McDuff-Salamon's problem list and a question of Cieliebak-Eliashberg-Mishachev, which concern the uniqueness and $h$-principle of symplectic structures on closed $4$-manifolds. We answer these questions by establishing the following results: (1) The space of symplectic forms on every irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components; (2) There exist infinitely many symplectic forms on every irrational ruled surface that are formally homotopic, cohomologous, but not homotopic to each other. Both are the first such examples for closed $4$-manifolds.
The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. In the course of the proof, we obtain several properties of the smooth mapping class group of $Σ\times S^2$, which may be of independent interest. For example, we show that there exists a surjective homomorphism from $π_0\operatorname{Diff}(Σ\times S^2)$ to $\mathbb{Z}^\infty$, such that its restriction to the subgroup of elements pseudo-isotopic to the identity is of infinite rank. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_16083 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dax invariants, light bulbs, and isotopies of symplectic structures Lin, Jianfeng Wu, Weiwei Xie, Yi Zhang, Boyu Geometric Topology 57R52, 57R50 This paper addresses several isotopy problems on $4$-manifolds. First, we classify the isotopy classes of embeddings of $Σ$ in $Σ\times S^2$ that are geometrically dual to $\{\mbox{pt}\}\times S^2$, where $Σ$ is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic, thereby answering a question of Gabai. By combining this construction with techniques from symplectic topology, we also answer Problem 2(a) in McDuff-Salamon's problem list and a question of Cieliebak-Eliashberg-Mishachev, which concern the uniqueness and $h$-principle of symplectic structures on closed $4$-manifolds. We answer these questions by establishing the following results: (1) The space of symplectic forms on every irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components; (2) There exist infinitely many symplectic forms on every irrational ruled surface that are formally homotopic, cohomologous, but not homotopic to each other. Both are the first such examples for closed $4$-manifolds. The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. In the course of the proof, we obtain several properties of the smooth mapping class group of $Σ\times S^2$, which may be of independent interest. For example, we show that there exists a surjective homomorphism from $π_0\operatorname{Diff}(Σ\times S^2)$ to $\mathbb{Z}^\infty$, such that its restriction to the subgroup of elements pseudo-isotopic to the identity is of infinite rank. |
| title | Dax invariants, light bulbs, and isotopies of symplectic structures |
| topic | Geometric Topology 57R52, 57R50 |
| url | https://arxiv.org/abs/2501.16083 |