[Paper Review] Functorial LCH for immersed Lagrangian cobordisms

Yu Pan, Dan Rutherford|arXiv (Cornell University)|May 21, 2019

Geometric and Algebraic Topology26 references4 citations

TL;DR

This paper extends the functoriality of Legendrian contact homology (LCH) DGA to immersed exact Lagrangian cobordisms by introducing a new framework using conical Legendrian cobordisms. It constructs an immersed DGA map via a diagram of DGA maps from the LCH DGA of the positive end to a new DGA associated to the cobordism, and from the cobordism DGA to the negative end, proving invariance under isotopy and providing examples of augmentations realizable only via immersed fillings with a single double point.

ABSTRACT

For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in \cite{EHK}, to a class of immersed exact Lagrangian cobordisms by considering their Legendrian lifts as conical Legendrian cobordisms. To a conical Legendrian cobordism $Σ$ from $Λ_-$ to $Λ_+$, we associate an immersed DGA map, which is a diagram $$\alg(Λ_+) \stackrel{f}{ ightarrow} \alg(Σ) \stackrel{i}{\hookleftarrow} \alg(Λ_-), $$ where $f$ is a DGA map and $i$ is an inclusion map. This construction gives a functor between suitably defined categories of Legendrians with immersed Lagrangian cobordisms and DGAs with immersed DGA maps. In an algebraic preliminary, we consider an analog of the mapping cylinder construction in the setting of DG-algebras and establish several of its properties. As an application we give examples of augmentations of Legendrian twist knots that can be induced by an immersed filling with a single double point but cannot be induced by any orientable embedded filling.

Motivation & Objective

• To extend the functoriality of Legendrian contact homology (LCH) DGA from embedded to immersed exact Lagrangian cobordisms.
• To define a new DGA, denoted $\mathcal{A}(\Sigma)$, associated to a conical Legendrian cobordism $\Sigma$ from $\Lambda_{-}$ to $\Lambda_{+}$, invariant under conical Legendrian isotopy.
• To construct an immersed DGA map $\mathcal{A}(\Lambda_{+}) \xrightarrow{f} \mathcal{A}(\Sigma) \xleftarrow{i} \mathcal{A}(\Lambda_{-})$, where $f$ is a DGA map and $i$ is an inclusion, generalizing the embedded case.
• To establish that this construction defines a functor from a category of Legendrians with immersed Lagrangian cobordisms to a category of DGAs with immersed DGA maps.
• To provide examples of augmentations of Legendrian twist knots that arise from immersed fillings with a single double point but are not realizable via any orientable embedded filling.

Proposed method

• The construction lifts an immersed exact Lagrangian cobordism $L$ to a conical Legendrian cobordism $\Sigma$ in $J^1(\mathbb{R}_{>0} \times M)$ via a symplectomorphism $\Phi: \mathrm{Symp}(J^1M) \to T^*(\mathbb{R}_{>0} \times M)$.
• The DGA $\mathcal{A}(\Sigma)$ is defined as the free product $\mathcal{A}(\Lambda_{-}) * \mathcal{A}(L)$ over $\mathbb{Z}/2$, with a differential $\partial_\Sigma$ counting rigid holomorphic disks with punctures at Reeb chords of $\Lambda_{-}$ and double points of $L$.
• The DGA map $f: \mathcal{A}(\Lambda_{+}) \to \mathcal{A}(\Sigma)$ is defined by counting rigid holomorphic disks with boundary on $\mathbb{R} \times \Lambda_{+}$ and punctures at double points of $L$ or Reeb chords of $\Lambda_{-}$, extended via the Leibniz rule.
• The differential and map are shown to be invariant under conical Legendrian isotopy of $\Sigma$ with a fixed metric, using compactness and gluing results from relative SFT and Legendrian contact homology.
• An equivalence is established between the holomorphic disk definition and the gradient flow tree (GFT) definition of the DGA and map for a restricted class of almost complex structures.
• The construction is shown to agree with the standard DGA map in the embedded case, confirming consistency with prior work.

Experimental results

Research questions

• RQ1 Can the functoriality of Legendrian contact homology DGA be extended from embedded to immersed exact Lagrangian cobordisms?
• RQ2 What is the correct algebraic structure to capture the DGA map induced by an immersed Lagrangian cobordism?
• RQ3 How can the DGA of a conical Legendrian cobordism $\Sigma$ be defined so that it is invariant under isotopy and recovers the standard DGA on the ends?
• RQ4 Are there augmentations of Legendrian knots that can be realized via immersed fillings with a single double point but not via any orientable embedded filling?
• RQ5 Does the holomorphic disk definition of the DGA and DGA map agree with the gradient flow tree definition in the immersed case?

Key findings

• The paper constructs a well-defined DGA $\mathcal{A}(\Sigma)$ for a conical Legendrian cobordism $\Sigma$ from $\Lambda_{-}$ to $\Lambda_{+}$, invariant under conical Legendrian isotopy with a fixed metric.
• The induced DGA map $f: \mathcal{A}(\Lambda_{+}) \to \mathcal{A}(\Sigma)$ and inclusion $i: \mathcal{A}(\Lambda_{-}) \to \mathcal{A}(\Sigma)$ form an immersed DGA map diagram, generalizing the embedded case.
• The construction defines a contravariant functor from the category of Legendrians with immersed Lagrangian cobordisms to the category of DGAs with immersed DGA maps.
• For a restricted class of almost complex structures, the holomorphic disk and gradient flow tree definitions of the DGA and map $f$ agree.
• The paper provides explicit examples of augmentations of Legendrian twist knots that are realizable via an immersed filling with a single double point but not via any orientable embedded filling.
• The map $f_{\Sigma}$ agrees with the standard DGA map induced by embedded cobordisms when $L$ is embedded, confirming consistency with prior results.