Normal invariant of nearby Lagrangians via twisted derivative
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912664572133376 |
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| author | Abouzaid, Mohammed Álvarez-Gavela, Daniel Courte, Sylvain Kragh, Thomas |
| author_facet | Abouzaid, Mohammed Álvarez-Gavela, Daniel Courte, Sylvain Kragh, Thomas |
| contents | Let $L$ and $M$ be closed, connected, smooth manifolds and let $L \hookrightarrow T^*M$ be an exact Lagrangian embedding. The induced map $L \to M$ is known by earlier work to be a homotopy equivalence. We show that the associated normal invariant $M \to G/O$ factors through a map $B(\mathcal{T},\mathcal{Q}) \to G/O$ which is a twisted version of the Waldhausen derivative $\mathcal{T} \to G$ on the space $\mathcal{T}$ of tubes. Further, we show that this twisted derivative map itself factors though a map $B(G/O) \to G/O$ which is a twisted version of the $S$-duality map $BG \to G$. In particular we deduce that the normal invariant of the homotopy equivalence $L \to M$ is 2-torsion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normal invariant of nearby Lagrangians via twisted derivative Abouzaid, Mohammed Álvarez-Gavela, Daniel Courte, Sylvain Kragh, Thomas Symplectic Geometry 53D12 Let $L$ and $M$ be closed, connected, smooth manifolds and let $L \hookrightarrow T^*M$ be an exact Lagrangian embedding. The induced map $L \to M$ is known by earlier work to be a homotopy equivalence. We show that the associated normal invariant $M \to G/O$ factors through a map $B(\mathcal{T},\mathcal{Q}) \to G/O$ which is a twisted version of the Waldhausen derivative $\mathcal{T} \to G$ on the space $\mathcal{T}$ of tubes. Further, we show that this twisted derivative map itself factors though a map $B(G/O) \to G/O$ which is a twisted version of the $S$-duality map $BG \to G$. In particular we deduce that the normal invariant of the homotopy equivalence $L \to M$ is 2-torsion. |
| title | Normal invariant of nearby Lagrangians via twisted derivative |
| topic | Symplectic Geometry 53D12 |
| url | https://arxiv.org/abs/2505.12515 |