On four-dimensional Dehn twists and Milnor fibrations
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914951861370880 |
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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 study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More precisely, we establish infinite order non-triviality results for boundary Dehn twists on indefinite symplectic fillings of links of minimally elliptic surface singularities. Using this, we exhibit a wide variety of new phenomena in dimension four: (1) smoothings of isolated complex surface singularities whose Milnor fibration has monodromy with infinite order as a diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3) compactly supported exotic diffeomorphisms of exotic $\mathbb{R}^4$'s and contractible manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On four-dimensional Dehn twists and Milnor fibrations Konno, Hokuto Lin, Jianfeng Mukherjee, Anubhav Muñoz-Echániz, Juan Geometric Topology Algebraic Geometry Differential Geometry Symplectic Geometry 57K41, 57R58 We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More precisely, we establish infinite order non-triviality results for boundary Dehn twists on indefinite symplectic fillings of links of minimally elliptic surface singularities. Using this, we exhibit a wide variety of new phenomena in dimension four: (1) smoothings of isolated complex surface singularities whose Milnor fibration has monodromy with infinite order as a diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3) compactly supported exotic diffeomorphisms of exotic $\mathbb{R}^4$'s and contractible manifolds. |
| title | On four-dimensional Dehn twists and Milnor fibrations |
| topic | Geometric Topology Algebraic Geometry Differential Geometry Symplectic Geometry 57K41, 57R58 |
| url | https://arxiv.org/abs/2409.11961 |