Bilinearized Legendrian contact homology and the augmentation category

Abstract

In this paper, we construct an A ∞ -category associated to a Legendrian submanifold of a jet space. Objects of the category are augmentations of the Chekanov algebra A(Λ) 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 [6] where an A ∞ -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 [12] in our context, and interpret geometrically the bilinearized homology in terms of the Floer homology of Lagrangian fillings (following [8]).

Floer theory for Lagrangian cobordisms

“…If Σ = R × Λ, then H • (Σ) = H • (Λ), and hence the above long exact sequence recovers the duality long exact sequence for Legendrian contact homology, which was proved by Sabloff in [58] for Legendrian knots and later generalised to arbitrary Legendrian submanifolds in [36] by Ekholm, Etnyre and Sabloff. In the bilinearised setting, the duality long exact sequence was introduced by Bourgeois and the first author in [8]. In Section 8.4 we use Exact Sequence (2) to prove that the fundamental class in LCH defined by Sabloff in [58] and Ekholm, Etnyre and Sabloff in [36] is functorial with respect to the maps induced by exact Lagrangian cobordisms.…”

“…as follows from [53, Theorem 1.35 (8)], using the fact that π is infinite. The long weakly exact sequence of a pair (Proposition 8.15(1)) immediately implies the base case dim ℓ 2 H…”

The minimal length of a Lagrangian cobordism between Legendrians

“…Remark 4.3. The grading convention in the definition of A ∞ algebra above is not standard: one usually wants m k to be a map of degree 2 − k. As remarked in [10] and as carried out in [2], one could fix this by working with the grading-shifted complex A * Λ [1]. Further, we avoid the usual sign conventions for the A ∞ relation as we are working over F 2 .…”

“…For a connected Legendrian submanifold Λ ⊂ R 2n+1 with augmentation ε, there is a distinguished class λ ∈ LCH n (Λ, ε) called the fundamental class. 2 To define λ, we use the duality exact sequence of [15]:…”

The augmentation category map induced by exact Lagrangian cobordisms

“…Analogous to the derived Fukaya category of exact Lagrangian compact submanifolds introduced in [NZ09], the augmentation category is an A ∞ category of Legendrian knots in (R 3 , ker α). Bourgeois and Chantraine first introduced a non-unital A ∞ category in [BC14] and then Ng, Rutherford, Sivek, Shende and Zaslow introduced a unital version in [NRS + 15]. We will focus on the latter one.…”