Lagrangian cobordism functor in microlocal sheaf theory I
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916760605687808 |
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| author | Li, Wenyuan |
| author_facet | Li, Wenyuan |
| contents | Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $Λ_-$ to $Λ_+$, we construct a functor $Φ_L^*: Sh^c_{Λ_+}(M) \rightarrow Sh^c_{Λ_-}(M) \otimes_{C_{-*}(Ω_*Λ_-)} C_{-*}(Ω_*L)$ between sheaf categories of compact objects with singular support on $Λ_\pm$ and its right adjoint on sheaf categories of proper objects, using Nadler-Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_10914 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lagrangian cobordism functor in microlocal sheaf theory I Li, Wenyuan Symplectic Geometry Geometric Topology Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $Λ_-$ to $Λ_+$, we construct a functor $Φ_L^*: Sh^c_{Λ_+}(M) \rightarrow Sh^c_{Λ_-}(M) \otimes_{C_{-*}(Ω_*Λ_-)} C_{-*}(Ω_*L)$ between sheaf categories of compact objects with singular support on $Λ_\pm$ and its right adjoint on sheaf categories of proper objects, using Nadler-Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds. |
| title | Lagrangian cobordism functor in microlocal sheaf theory I |
| topic | Symplectic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2108.10914 |