Lagrangian concordance is not a partial order in high dimensions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913680705191936 |
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| author | Golovko, Roman |
| author_facet | Golovko, Roman |
| contents | In this short note we provide the examples of pairs of closed, connected Legendrian non-isotopic Legendrian submanifolds $(Λ_{-}, Λ_{+})$ of the $(4n+1)$-dimensional contact vector space, $n>1$, such that there exist Lagrangian concordances from $Λ_-$ to $Λ_+$ and from $Λ_+$ to $Λ_-$. This contradicts anti-symmetry of the Lagrangian concordance relation, and, in particular, implies that Lagrangian concordances with connected Legendrian ends do not define a partial order in high dimensions. In addition, we explain how to get the same result for the relation given by exact Lagrangian cobordisms with connected Legendrian ends in the $(2n+1)$-dimensional contact vector space, $n>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lagrangian concordance is not a partial order in high dimensions Golovko, Roman Symplectic Geometry 53D12, 53D42 In this short note we provide the examples of pairs of closed, connected Legendrian non-isotopic Legendrian submanifolds $(Λ_{-}, Λ_{+})$ of the $(4n+1)$-dimensional contact vector space, $n>1$, such that there exist Lagrangian concordances from $Λ_-$ to $Λ_+$ and from $Λ_+$ to $Λ_-$. This contradicts anti-symmetry of the Lagrangian concordance relation, and, in particular, implies that Lagrangian concordances with connected Legendrian ends do not define a partial order in high dimensions. In addition, we explain how to get the same result for the relation given by exact Lagrangian cobordisms with connected Legendrian ends in the $(2n+1)$-dimensional contact vector space, $n>1$. |
| title | Lagrangian concordance is not a partial order in high dimensions |
| topic | Symplectic Geometry 53D12, 53D42 |
| url | https://arxiv.org/abs/2411.12114 |