$A_\infty$ Sabloff Duality via the LSFT Algebra
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909611549786112 |
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| author | Chen, Zhenyi |
| author_facet | Chen, Zhenyi |
| contents | We use Ng's LSFT algebra to upgrade Sabloff duality of Legendrian knots to a quasi-isomorphism of $A_\infty$ bimodules over the positive augmentation category $\mathcal{A}ug_+$. We also extend the Ekholm-Etnyre-Sabloff exact sequence to an exact sequence of $\mathcal{A}ug_+$-bimodules, using a quotient category $\mathcal{C}$ of short Reeb chords. In addition, we define curved augmentations of the LSFT algebra and show that they can be used to construct a homotopy inverse of the $A_\infty$ Sabloff map, together with all higher homotopies. The above results suggest a conjectural recipe for an explicit weak relative Calabi-Yau structure on the quotient $A_\infty$ functor $π:\mathcal{A}ug_+\to \mathcal{C}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20523 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $A_\infty$ Sabloff Duality via the LSFT Algebra Chen, Zhenyi Symplectic Geometry Geometric Topology 53D42 (Primary) 53D10, 53D12, 57K10, 57K33 (Secondary) We use Ng's LSFT algebra to upgrade Sabloff duality of Legendrian knots to a quasi-isomorphism of $A_\infty$ bimodules over the positive augmentation category $\mathcal{A}ug_+$. We also extend the Ekholm-Etnyre-Sabloff exact sequence to an exact sequence of $\mathcal{A}ug_+$-bimodules, using a quotient category $\mathcal{C}$ of short Reeb chords. In addition, we define curved augmentations of the LSFT algebra and show that they can be used to construct a homotopy inverse of the $A_\infty$ Sabloff map, together with all higher homotopies. The above results suggest a conjectural recipe for an explicit weak relative Calabi-Yau structure on the quotient $A_\infty$ functor $π:\mathcal{A}ug_+\to \mathcal{C}$. |
| title | $A_\infty$ Sabloff Duality via the LSFT Algebra |
| topic | Symplectic Geometry Geometric Topology 53D42 (Primary) 53D10, 53D12, 57K10, 57K33 (Secondary) |
| url | https://arxiv.org/abs/2410.20523 |