Duality between Lagrangian and Legendrian invariants

Ekholm, Tobias; Lekili, Yankı

doi:10.2140/gt.2023.27.2049

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Duality between Lagrangian and Legendrian invariants

Tobias Ekholm and Yankı Lekili

Geometry & Topology 27 (2023) 2049–2179

DOI: 10.2140/gt.2023.27.2049

Abstract

Consider a pair $(X, L)$ of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$ , with ideal contact boundary $(Y, Λ)$ , where $Y$ is a contact manifold and $Λ \subset Y$ is a Legendrian submanifold. We introduce the Chekanov–Eliashberg DG–algebra, ${CE}^{*} (Λ)$ , with coefficients in chains of the based loop space of $Λ$ , and study its relation to the Floer cohomology ${CF}^{*} (L)$ of $L$ . Using the augmentation induced by $L$ , ${CE}^{*} (Λ)$ can be expressed as the Adams cobar construction $Ω$ applied to a Legendrian coalgebra, ${LC}_{*} (Λ)$ . We define a twisting cochain $𝔱 : {LC}_{*} (Λ) \to B {({CF}^{*} (L))}^{#}$ 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 ${CE}^{*} (Λ)$ and ${CF}^{*} (L)$ are Koszul dual. In particular, $𝔱$ induces a quasi-isomorphism between ${CE}^{*} (Λ)$ and $Ω {CF}_{*} (L)$ , the cobar of the Floer homology of $L$ .

This generalizes the classical Koszul duality result between $C^{*} (L)$ and $C_{- *} (Ω L)$ for $L$ a simply connected manifold, where $Ω L$ 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 $𝔱$ ).

We use the duality result to show that under certain connectivity and local-finiteness assumptions, ${CE}^{*} (Λ)$ is quasi-isomorphic to $C_{- *} (Ω L)$ for any Lagrangian filling $L$ of $Λ$ .

Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that ${CE}^{*} (Λ)$ is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk $C$ in the Weinstein domain obtained by attaching $T^{*} (Λ \times [0, \infty))$ to $X$ along $Λ$ (or, in the terminology of Sylvan (2019), the wrapped Floer cohomology of $C$ in $X$ with wrapping stopped by $Λ$ ). 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.

Keywords

Lagrangian, Legendrian, Floer cohomology, Chekanov–Eliashberg dg–algebra

Mathematical Subject Classification 2010

Primary: 57R17

References

Publication

Received: 3 December 2019

Revised: 16 September 2021

Accepted: 18 December 2021

Published: 25 August 2023

Proposed: Yakov Eliashberg

Seconded: Leonid Polterovich, Paul Seidel

Authors

Tobias Ekholm
Department of Mathematics Uppsala University Uppsala Sweden	Insitut Mittag-Leffler Djursholm Sweden

Yankı Lekili
Department of Mathematics Imperial College London South Kensington London United Kingdom

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Volume 27, issue 6 (2023)