Bilinearised Legendrian contact homology and the augmentation category

TL;DR

This paper develops a new $ ext{A}_ ext{infty}$-category for Legendrian submanifolds, using bilinearised Legendrian contact homology to create refined invariants and analyze augmentation equivalences.

Contribution

It introduces a bilinearised Legendrian contact homology framework within an $ ext{A}_ ext{infty}$-category, extending previous linearised theories and providing new tools for Legendrian invariants.

Findings

Defines a new invariant called bilinearised Legendrian contact homology.

Establishes an equivalence notion for augmentations over any field.

Shows bilinearised cohomology distinguishes augmentation classes effectively.

Abstract

In this paper we construct an $\mathcal{A}_\infty$-category associated to a Legendrian submanifold of jet spaces. Objects of the category are augmentations of the Chekanov algebra $\mathcal{A}(\Lambda)$ and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearised Legendrian contact homology. Those are constructed as a generalisation of linearised Legendrian contact homology using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order compositions map in the category and generalises the idea of [6] where an $\mathcal{A}_\infty$-algebra was constructed from one augmentation. This category allows us to define a notion of equivalence of augmentations when the coefficient ring is a field regardless of its characteristic. We use simple examples to show that bilinearised cohomology groups are efficient to distinguish those equivalences classes. We also generalise the duality exact sequence from [12] in our contest, and interpret geometrically the bilinearised homology in terms of the Floer homology of Lagrangian fillings (following [8]).