Duality between Lagrangian and Legendrian invariants
Duality between Lagrangian and Legendrian invariants
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DOI：
10.2140/gt.2023.27.2049
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发表时间：
2017-01
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影响因子：
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通讯作者：
T. Ekholm;Yankı Lekili
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作者：
T. Ekholm;Yankı Lekili
考虑一对$(X,L)$，其中$X$是一个温斯坦流形，$L$是一个精确的拉格朗日子流形，其理想接触边界为$(Y,\Lambda)$，这里$Y$是一个接触流形，$\Lambda\subset Y$是一个勒让德子流形。我们引入切卡诺夫 - 埃利亚什贝格微分分次代数$CE^{\ast}(\Lambda)$，其系数为$\Lambda$的带基点的环路空间的链，并研究它与$L$的弗洛尔上同调$CF^{\ast}(L)$的关系。利用由$L$诱导的增广，$CE^{\ast}(\Lambda)$可表示为应用于一个勒让德余代数$LC_{\ast}(\Lambda)$的亚当斯余棒构造$\Omega$。我们通过全纯曲线计数定义一个扭曲上链：
\[\mathfrak{t} \colon LC_{\ast}(\Lambda) \to \mathrm{B} (CF^*(L))^\#\]
其中$\mathrm{B}$表示棒构造，$\#$表示分次线性对偶。我们在单连通假设下证明相应的科苏尔复形是无圈的，这进而意味着$CE^*(\Lambda)$和$CF^{\ast}(L)$是科苏尔对偶的。特别地，$\mathfrak{t}$诱导了$CE^*(\Lambda)$与$L$的弗洛尔同调的余棒$\Omega CF_*(L)$之间的一个拟同构。我们利用对偶结果证明，在某些连通性和局部有限性假设下，对于$\Lambda$的任何拉格朗日填充$L$，$CE^*(\Lambda)$与$C_{-*}(\Omega L)$是拟同构的。我们的构造在拉格朗日柄附接的各种情形之后，在包裹弗洛尔上同调方面有相应的解释。特别地，我们概述一个证明，即$CE^{\ast}(\Lambda)$与通过沿着$\Lambda$将$T^{\ast}(\Lambda\times[0,\infty))$附接到$X$而得到的温斯坦区域中的一个纤维圆盘$C$的包裹弗洛尔上同调是拟同构的（或者，用arXiv:1604.02540的术语来说，是在$X$中被$\Lambda$阻止包裹的$C$的包裹弗洛尔上同调）。在此过程中，我们给出了一个无哈密顿扰动的包裹弗洛尔上同调的定义。
Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is a contact manifold and $\Lambda\subset Y$ is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, $CE^{\ast}(\Lambda)$, with coefficients in chains of the based loop space of $\Lambda$ and study its relation to the Floer cohomology $CF^{\ast}(L)$ of $L$. Using the augmentation induced by $L$, $CE^{\ast}(\Lambda)$ can be expressed as the Adams cobar construction $\Omega$ applied to a Legendrian coalgebra, $LC_{\ast}(\Lambda)$. We define a twisting cochain:\[\mathfrak{t} \colon LC_{\ast}(\Lambda) \to \mathrm{B} (CF^*(L))^\#\]via holomorphic curve counts, where $\mathrm{B}$ denotes the bar construction and $\#$ the graded linear dual. We show under simply-connectedness assumptions that the corresponding Koszul complex is acyclic which then implies that $CE^*(\Lambda)$ and $CF^{\ast}(L)$ are Koszul dual. In particular, $\mathfrak{t}$ induces a quasi-isomorphism between $CE^*(\Lambda)$ and the cobar of the Floer homology of $L$, $\Omega CF_*(L)$. We use the duality result to show that under certain connectivity and locally finiteness assumptions, $CE^*(\Lambda)$ is quasi-isomorphic to $C_{-*}(\Omega L)$ for any Lagrangian filling $L$ of $\Lambda$. Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that $CE^{\ast}(\Lambda)$ is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk $C$ in the Weinstein domain obtained by attaching $T^{\ast}(\Lambda\times[0,\infty))$ to $X$ along $\Lambda$ (or, in the terminology of arXiv:1604.02540 the wrapped Floer cohomology of $C$ in $X$ with wrapping stopped by $\Lambda$). Along the way, we give a definition of wrapped Floer cohomology without Hamiltonian perturbations.