Bilinearized Legendrian contact homology and the augmentation category

Abstract

In this paper, we construct an $\mathcal{A}_{\infty}$-category associated to a Legendrian submanifold of a jet space. 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 bilinearized Legendrian contact homology. Those are constructed as a generalization of linearized Legendrian contact homology using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order composition maps in the category and generalizes the idea of G. Civan, P. Koprowski, J. Etnyre, J.M. Sabloff and A. Walker, Product structures for Legendrian contact homology, 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 bilinearized cohomology groups are efficient to distinguish those equivalences classes.We also generalize the duality exact sequence from T. Ekholm, J. Etnyre and M. Sullivan in our context, and interpret geometrically the bilinearized homology in terms of the Floer homology of Lagrangian fillings (following T. Ekholm, Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology).