On weakly exact Lagrangians in Liouville bi-fillings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908463071756288 |
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| author | Rizell, Georgios Dimitroglou |
| author_facet | Rizell, Georgios Dimitroglou |
| contents | Here we study several questions concerning Liouville domains that are diffeomorphic to cylinders, so called trivial bi-fillings, for which the Liouville skeleton moreover is smooth and of codimension one; we also propose the notion of a Liouville-Hamiltonian structure, which encodes the symplectic structure of a hypersurface tangent to the Liouville flow, e.g. the skeleta of certain bi-fillings. We show that the symplectic homology of a bi-filling is non-trivial, and that a connected Lagrangian inside a bi-filling whose boundary lives in different components of the contact boundary automatically has non-vanishing wrapped Floer cohomology. We also prove geometric vanishing and non-vanishing criteria for the wrapped Floer cohomology of an exact Lagrangian with disconnected cylindrical ends. Finally, we give homotopy-theoretic restrictions on the closed weakly exact Lagrangians in the McDuff and torus bundle Liouville domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On weakly exact Lagrangians in Liouville bi-fillings Rizell, Georgios Dimitroglou Symplectic Geometry 53D05, 53D12, 53D40 Here we study several questions concerning Liouville domains that are diffeomorphic to cylinders, so called trivial bi-fillings, for which the Liouville skeleton moreover is smooth and of codimension one; we also propose the notion of a Liouville-Hamiltonian structure, which encodes the symplectic structure of a hypersurface tangent to the Liouville flow, e.g. the skeleta of certain bi-fillings. We show that the symplectic homology of a bi-filling is non-trivial, and that a connected Lagrangian inside a bi-filling whose boundary lives in different components of the contact boundary automatically has non-vanishing wrapped Floer cohomology. We also prove geometric vanishing and non-vanishing criteria for the wrapped Floer cohomology of an exact Lagrangian with disconnected cylindrical ends. Finally, we give homotopy-theoretic restrictions on the closed weakly exact Lagrangians in the McDuff and torus bundle Liouville domains. |
| title | On weakly exact Lagrangians in Liouville bi-fillings |
| topic | Symplectic Geometry 53D05, 53D12, 53D40 |
| url | https://arxiv.org/abs/2412.20941 |