The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912126095851520 |
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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 prove that the monodromy diffeomorphism of a complex 2-dimensional isolated hypersurface singularity of weighted-homogeneous type has infinite order in the smooth mapping class group of the Milnor fiber, provided the singularity is not a rational double point. This is a consequence of our main result: the boundary Dehn twist diffeomorphism of an indefinite symplectic filling of the canonical contact structure on a negatively-oriented Seifert-fibered rational homology 3-sphere has infinite order in the smooth mapping class group. Our techniques make essential use of analogues of the contact invariant in the setting of $\mathbb{Z}/p$-equivariant Seiberg--Witten--Floer homology of 3-manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_12202 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory Konno, Hokuto Lin, Jianfeng Mukherjee, Anubhav Muñoz-Echániz, Juan Geometric Topology Algebraic Geometry Symplectic Geometry 57K41, 57R58, 32S25 We prove that the monodromy diffeomorphism of a complex 2-dimensional isolated hypersurface singularity of weighted-homogeneous type has infinite order in the smooth mapping class group of the Milnor fiber, provided the singularity is not a rational double point. This is a consequence of our main result: the boundary Dehn twist diffeomorphism of an indefinite symplectic filling of the canonical contact structure on a negatively-oriented Seifert-fibered rational homology 3-sphere has infinite order in the smooth mapping class group. Our techniques make essential use of analogues of the contact invariant in the setting of $\mathbb{Z}/p$-equivariant Seiberg--Witten--Floer homology of 3-manifolds. |
| title | The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory |
| topic | Geometric Topology Algebraic Geometry Symplectic Geometry 57K41, 57R58, 32S25 |
| url | https://arxiv.org/abs/2411.12202 |