Duality between Lagrangian and Legendrian invariants

Abstract

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, $C{E}^{\ast }(\Lambda )$, with coefficients in chains of the based loop space of $\Lambda$ and study its relation to the Floer cohomology $C{F}^{\ast }(L)$ of $L$. Using the augmentation induced by $L$, $C{E}^{\ast }(\Lambda )$ can be expressed as the Adams cobar construction $\Omega$ applied to a Legendrian coalgebra, $L{C}_{\ast }(\Lambda )$. We define a twisting cochain:\[\mathfrak{t} \colon LC_{\ast}(Λ) \to \mathrm{B} (CF^*(L))^\#\]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 $C{E}^{\ast }(\Lambda )$ and $C{F}^{\ast }(L)$ are Koszul dual. In particular, $\mathfrak{t}$ induces a quasi-isomorphism between $C{E}^{\ast }(\Lambda )$ and the cobar of the Floer homology of $L$, $\Omega C{F}_{\ast }(L)$. We use the duality result to show that under certain connectivity and locally finiteness assumptions, $C{E}^{\ast }(\Lambda )$ is quasi-isomorphic to $C_{-\ast }(\Omega L)$ 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 $C{E}^{\ast }(\Lambda )$ 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.

arXiv comment126 pages, 20 figures. Substantial overall revision based on referee's comments. The main results remain the same but the exposition has been improved

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