Duality between Lagrangian and Legendrian invariants

Ekholm, Tobias¹ ; Lekili, Yankı²

¹ Department of Mathematics, Uppsala University, Uppsala, Sweden, Insitut Mittag-Leffler, Djursholm, Sweden
² Department of Mathematics, Imperial College London, South Kensington, London, United Kingdom

Geometry & topology, Tome 27 (2023) no. 6, pp. 2049-2179.

Résumé

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, ${\mathrm{CE}}^{\ast }(\Lambda )$, with coefficients in chains of the based loop space of $\Lambda$, and study its relation to the Floer cohomology ${\mathrm{CF}}^{\ast }(L)$ of $L$. Using the augmentation induced by $L$, ${\mathrm{CE}}^{\ast }(\Lambda )$ can be expressed as the Adams cobar construction $\mathrm{\Omega }$ applied to a Legendrian coalgebra, ${\mathrm{LC}}_{\ast }(\Lambda )$. We define a twisting cochain $\mathfrak{𝔱}:{\mathrm{LC}}_{\ast }(\Lambda )\to B{({\mathrm{CF}}^{\ast }(L))}^{\#}$ via holomorphic curve counts, where $B$ denotes the bar construction and $\#$ the graded linear dual. We show under simple-connectedness assumptions that the corresponding Koszul complex is acyclic, which then implies that ${\mathrm{CE}}^{\ast }(\Lambda )$ and ${\mathrm{CF}}^{\ast }(L)$ are Koszul dual. In particular, $\mathfrak{𝔱}$ induces a quasi-isomorphism between ${\mathrm{CE}}^{\ast }(\Lambda )$ and $\mathrm{\Omega }{\mathrm{CF}}_{\ast }(L)$, the cobar of the Floer homology of $L$.

This generalizes the classical Koszul duality result between $C^{\ast }(L)$ and $C_{-\ast }(\mathrm{\Omega }L)$ for $L$ a simply connected manifold, where $\mathrm{\Omega }L$ is the based loop space of $L$, and provides the geometric ingredient explaining the computations given by Etgü and Lekili (2017) in the case when $X$ is a plumbing of cotangent bundles of $2$–spheres (where an additional weight grading ensured Koszulity of $\mathfrak{𝔱}$).

We use the duality result to show that under certain connectivity and local-finiteness assumptions, ${\mathrm{CE}}^{\ast }(\Lambda )$ is quasi-isomorphic to $C_{-\ast }(\mathrm{\Omega }L)$ 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 ${\mathrm{CE}}^{\ast }(\Lambda )$ 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 Sylvan (2019), the wrapped Floer cohomology of $C$ in $X$ with wrapping stopped by $\Lambda$). Along the way, we give a definition of wrapped Floer cohomology via holomorphic buildings that avoids the use of Hamiltonian perturbations, which might be of independent interest.

Détail

DOI : 10.2140/gt.2023.27.2049

Classification : 57R17
Keywords: Lagrangian, Legendrian, Floer cohomology, Chekanov–Eliashberg dg–algebra

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