Augmentation categories in higher dimensions
Hanming Liu
TL;DR
This work extends the augmentation category for Legendrian contact homology from dimension three to higher-dimensional contactizations $V=M\times \mathbb{R}$ with exact $M$ and finite geometry at infinity, working over fields of characteristic $2$. It introduces a localization-based construction of the unital $A_\infty$-category $\mathcal{A}ug_+(V,\Lambda)$ using arbitrary generic perturbations of parallel copies and inverting a family of degree-zero cocycles, thereby achieving invariance under perturbations and Legendrian isotopy without requiring consistent perturbation sequences. An auxiliary category built from countably many pushoffs $\Lambda^i$ is used to guarantee transversality for finite data and to define the localization that yields $\mathcal{A}ug_+(V,\Lambda)$. The authors show that, whenever a consistent sequence of DGAs exists (as in NRSSZ for Legendrian links in $\mathbb{R}^3$), their localisation approach recovers the NRSSZ construction up to $A_\infty$-equivalence, linking the higher-dimensional theory to the established framework. This provides a robust higher-dimensional invariant that connects to sheaf-theoretic descriptions in 1-jet spaces and broadens the applicability of augmentation categories in contact topology.
Abstract
For an exact symplectic manifold $M$ and a Legendrian submanifold $Λ$ of the contactification $M\times \mathbb{R}$, we construct the augmentation category (over a field of characteristic 2), a unital $A_\infty$-category whose objects are augmentations of the Chekanov-Eliashberg differential graded algebra. This extends the construction of the augmentation category by Ng-Rutherford-Shende-Sivek-Zaslow to contact manifolds of dimension greater than 3.