Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912685079134208 |
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| author | Konno, Hokuto Lin, Jianfeng Mukherjee, Anubhav Muñoz-Echániz, Juan |
| author_facet | Konno, Hokuto Lin, Jianfeng Mukherjee, Anubhav Muñoz-Echániz, Juan |
| contents | We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplectomorphisms which are smoothly non-trivial. In particular, their symplectic Torelli mapping class group is not generated by squared Dehn-Seidel twists on Lagrangian spheres -- providing a negative answer to a question of Donaldson. (2) We provide the first examples of irreducible closed $4$-manifolds (both symplectic and non-symplectic) that admit exotic diffeomorphisms given by Seifert-fibered Dehn twist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory Konno, Hokuto Lin, Jianfeng Mukherjee, Anubhav Muñoz-Echániz, Juan Geometric Topology Algebraic Geometry Differential Geometry Symplectic Geometry We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplectomorphisms which are smoothly non-trivial. In particular, their symplectic Torelli mapping class group is not generated by squared Dehn-Seidel twists on Lagrangian spheres -- providing a negative answer to a question of Donaldson. (2) We provide the first examples of irreducible closed $4$-manifolds (both symplectic and non-symplectic) that admit exotic diffeomorphisms given by Seifert-fibered Dehn twist. |
| title | Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory |
| topic | Geometric Topology Algebraic Geometry Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2511.00601 |