Download this article
 Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Duality between Lagrangian and Legendrian invariants

Tobias Ekholm and Yankı Lekili

Geometry & Topology 27 (2023) 2049–2179

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 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) 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 ΩCF (L), the cobar of the Floer homology of L.

This generalizes the classical Koszul duality result between C(L) and C(ΩL) for L a simply connected manifold, where ΩL 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(ΩL) 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 × [0,)) 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.

Lagrangian, Legendrian, Floer cohomology, Chekanov–Eliashberg dg–algebra
Mathematical Subject Classification 2010
Primary: 57R17
Received: 3 December 2019
Revised: 16 September 2021
Accepted: 18 December 2021
Published: 25 August 2023
Proposed: Yakov Eliashberg
Seconded: Leonid Polterovich, Paul Seidel
Tobias Ekholm
Department of Mathematics
Uppsala University
Insitut Mittag-Leffler
Yankı Lekili
Department of Mathematics
Imperial College London
South Kensington
United Kingdom

Open Access made possible by participating institutions via Subscribe to Open.