[Paper Review] Floer homology and Lagrangian concordance

Baptiste Chantraine, Georgios Dimitroglou Rizell+2 more|arXiv (Cornell University)|Jan 17, 2015

Geometric and Algebraic Topology33 references3 citations

TL;DR

This paper establishes that Lagrangian concordances between Legendrian submanifolds admitting exact Lagrangian fillings induce isomorphisms on bilinearised Legendrian contact cohomology, proving the existence of non-invertible concordances in all dimensions. It further classifies all exact Lagrangian concordances from the Legendrian unknot to itself in the tight contact three-sphere as traces of Legendrian isotopies.

ABSTRACT

We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of bilinearised Legendrian contact cohomology. This is used to prove the existence of non-invertible exact Lagrangian concordances in all dimensions. In addition, using a result of Eliashberg-Polterovich, we completely classify exact Lagrangian concordances from the Legendrian unknot to itself in the tight contact-three sphere: every such concordance is the trace of a Legendrian isotopy. We also discuss a high dimensional topological result related to this classification.

Motivation & Objective

• To understand rigidity phenomena in Lagrangian concordances between Legendrian submanifolds in the standard contact sphere.
• To derive constraints on concordances using bilinearised Legendrian contact cohomology when the negative end admits an exact Lagrangian filling.
• To prove the existence of non-invertible exact Lagrangian concordances in all dimensions.
• To completely classify exact Lagrangian concordances from the Legendrian unknot to itself in the tight contact three-sphere.
• To establish a high-dimensional topological result related to the classification of concordances via spin constructions.

Proposed method

• Use of bilinearised Legendrian contact cohomology as an invariant for Legendrian submanifolds with augmentations induced by exact Lagrangian fillings.
• Application of Theorem 1.1, which states that a Lagrangian concordance induces an isomorphism on bilinearised cohomology for augmentations from exact fillings.
• Employment of the Künneth-type formula (Theorem 2.11) to compute cohomology of spun Legendrian submanifolds via tensor products with the cohomology of spheres.
• Construction of Legendrian submanifolds via $S^m$-spun operations on known examples, such as the Legendrian knot $\Lambda_{m(9_{46})}$, to generate higher-dimensional examples.
• Use of the $h$-principle for exact Lagrangian cobordisms when the negative end is loose, to restrict attention to non-loose fillable Legendrians.
• Application of Eliashberg-Polterovich's result to classify concordances from the Legendrian unknot to itself in dimension three as traces of isotopies.

Experimental results

Research questions

• RQ1 Can Lagrangian concordances between Legendrian submanifolds with exact fillings be non-invertible?
• RQ2 What constraints does bilinearised Legendrian contact cohomology impose on Lagrangian concordances?
• RQ3 Are all exact Lagrangian concordances from the Legendrian unknot to itself in $S^3$ the trace of a Legendrian isotopy?
• RQ4 How does the bilinearised cohomology behave under $S^m$-spun constructions of Legendrian submanifolds?
• RQ5 What topological obstructions arise in higher dimensions for the existence of inverse concordances?

Key findings

• Lagrangian concordances from Legendrian submanifolds with exact fillings induce isomorphisms on bilinearised Legendrian contact cohomology for augmentations from such fillings.
• There exist non-invertible exact Lagrangian concordances in all dimensions, as shown by constructing examples where the inverse concordance would violate cohomological constraints.
• In the tight contact three-sphere, every exact Lagrangian concordance from the Legendrian unknot to itself is the trace of a Legendrian isotopy.
• The $S^m$-spun Legendrian submanifold of the Legendrian knot $\Lambda_{m(9_{46})}$ admits a concordance to the spun unknot but not vice versa, due to non-vanishing $LCH^{-1}_{\varepsilon_0,\varepsilon_1}(\Sigma_{S^m}\Lambda) \neq 0$.
• For any $m_1, \dots, m_k \in \mathbb{N}$, there exist fillable Legendrian submanifolds $\Lambda_1, \Lambda_2$ diffeomorphic to $S^1 \times S^{m_1} \times \cdots \times S^{m_k}$ such that a concordance exists from $\Lambda_1$ to $\Lambda_2$ but not in the reverse direction.
• The bilinearised cohomology of the spun Legendrian $\Sigma_{S^m}\Lambda_{0}$ vanishes in negative degrees for all augmentation pairs, which obstructs the existence of a reverse concordance when the source has non-trivial negative cohomology.