Bilinearised Legendrian contact homology and the augmentation category

By Frederick J. Bourgeois III, Baptiste Chantraine. 2012

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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]).

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/99c797930046659a2ddfdc590025de468a150357

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Citations in the corpus

Duality between Lagrangian and Legendrian invariants (2017, background): “Remark 5 In case Γ is empty, the A∞ -algebra LA ∗ is obtained from the construction in [19] and [10], known as Aug− category, by adding a copy of k making it unital.”
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors (2017, background): “rised Legendrian contact homology groups is independent of the choice of Jand is a Legendrian isotopy invariant by the adaptation of Chekanov’s argument from [11] due to the ﬁrst author and Bourgeois [8]. 4. Floer homology for exact Lagrangian immersions In this section we deﬁne a version of Floer h”
Obstructions to Lagrangian concordance (2014, methodology): “Chantraine proved in [9], using the augmentation category of Bourgeois–Chantraine [4], that there is a Lagrangian slice knot Λ of type 946 (the mirror of 946) such that Λ 6≺ U .”