Barcode of a pair of compact exact Lagrangians in a punctured exact two-dimensional symplectic manifold
| Autor: | Pasquer, Tangi |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this article, we modify the classical Floer complex $CF(L_0,L_1)$ of a pair of two compact exact Lagrangian submanifolds $L_0,L_1$ of an exact symplectic 2-manifold $M$ into a $\mathbb{Z}_2[T]$-complex $CF_h(L_0,L_1)$, whose differential keeps track of how many times a pseudo-holomorphic strip passes through a distinguished point $h\in M$. We show that this complex is invariant under Hamiltonian isotopy, and we prove that its barcode, if it exists, is the same as both barcodes $\mathcal{B}(CF(L_0,L_1;M))$ and $\mathcal{B}(CF(L_0,L_1;M\setminus \{h \}))$. This allows us to extend a conjecture of Viterbo, which states that for every Hamiltonian isotopy $\phi^1_H(L)$ in $D^* L$, the spectral norm $\gamma(L,\phi^1_H(L))$ remains bounded independently of $H$, to the case of $D^* L$ with a point removed. Comment: 27 pages, 4 figures |
| Databáze: | arXiv |
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