Augmentations and immersed Lagrangian fillings

Abstract

For a Legendrian link $\Lambda \subset J^{1}M$ with $M=\mathbb{R}$ or $S^1$, immersed exact Lagrangian fillings $L \subset \mbox{Symp}(J^1M) \cong T^*(\mathbb{R}_{>0} \times M)$ of $\Lambda$ can be lifted to conical Legendrian fillings $\Sigma \subset J^1({\mathbb{R}}_{>0}\times M)$ of $\Lambda$. When $\Sigma$ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [30], for each augmentation $\alpha :\mathcal{A}(\Sigma )\to \mathbb{Z}/2$ of the LCH algebra of $\Sigma$, there is an induced augmentation ${\epsilon }_{(\Sigma ,\alpha )}:\mathcal{A}(\Lambda )\to \mathbb{Z}/2$. With $\Sigma$ fixed, the set of homotopy classes of all such induced augmentations, $I_{\Sigma }\subset Aug(\Lambda )/\sim$, is a Legendrian isotopy invariant of $\Sigma$. We establish methods to compute $I_{\Sigma }$ based on the correspondence between Morse complex families and augmentations. This includes developing a functoriality for the cellular DGA from [31] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary $n\ge 1$, we give examples of Legendrian torus knots with $2n$ distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when $\rho \ne 1$ and $\Lambda \subset J^1\mathbb{R}$ every $\rho$-graded augmentation of $\Lambda$ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of $\rho$-graded augmented Legendrian cobordism.

51 pages, 22 figures. Accepted version to appear in Journal of Topology. Version 2 is shorter than Version 1 with more efficient exposition. In places, readers desiring more details are referred to Version 1

References in corpus (2)

• Functorial LCH for immersed Lagrangian cobordisms
• Ruling polynomials and augmentations for Legendrian tangles

Cited by in corpus (4)

• Functorial LCH for immersed Lagrangian cobordisms
• The persistence of a relative Rabinowitz-Floer complex
• Non-fillable augmentations of twist knots
• Augmented Legendrian cobordism in $J^1{S}^1$