A duality exact sequence for legendrian contact homology

Abstract

We establish a long exact sequence for Legendrian submanifolds L ⊂ P × R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H * maps to linearized contact cohomology CH * , which maps to linearized contact homology CH * , which maps to singular homology. In particular, the sequence implies a duality between Ker(CH * → H * ) and CH * / Im(H * ). Furthermore, this duality is compatible with Poincaré duality in L in the following sense: the Poincaré dual of a singular class which is the image of a ∈ CH * maps to a class α ∈ CH * such that α(a) = 1.The exact sequence generalizes the duality for Legendrian knots in R 3 [26] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7].

Cellular Legendrian contact homology for surfaces, part II

“…where the second equality is due to the N block in the k, k + 1 rows of A k,k+1 . Finally, using (11) and (12) compute…”

The relative Gromov width of Lagrangian cobordisms between Legendrians

“…We now will explain some important long exact sequences that involve LCH * (Λ, ε) and how the constuction of maps in this sequence implies the existence of particular J-holomorphic curves. An important structural result for the linearized Legendrian Contact Homology of a horizontally displaceable Legendrian is the duality long exact sequence [18]: 1…”

“…For a horizontally displaceable Legendrian Λ, we define the fundamental class for [m] a generator of H 0 (Λ). It was shown in [18] that when Λ is connected, the map σ * is injective on H 0 (Λ) and thus λ is non-zero. When one examines the construction of σ * at the chain level, one sees that the non-triviality of the fundamental class implies the existence of a J-holomorphic curve, for J ∈ J cyl π , that passes through an arbitrary point m ∈ L = R × Λ.…”

“…To guarantee the existence of an appropriate J-holomorphic curve, we will assume that Λ + is connected, Λ − and Λ + are horizontally displaceable, and Λ − admits an augmentation ε − ; we term such a cobordism a fundamental cobordism and define it officially in Definition 5.7. Note that a "horizontally displaceable" Legendrian Λ ⊂ C(P ) is one whose Lagrangian projection of Λ to P is displaceable by a Hamiltonian isotopy [18]; in particular, any Λ ⊂ R 2n+1 = J 1 R n is horizontally displaceable. Under these assumptions, we may relate the generator of H 0 (L) with the fundamental class of Λ + using the Generalized Duality Long Exact Sequence of [10, Theorem 1.2]; see Theorem 5.5.…”

Positive Legendrian isotopies and Floer theory

“…The homotopy type of the complex LCCε ,ε 1 pΛ, Λ 1 q can be seen to only depend on the Legendrian isotopy class of the link Λ Y Λ 1 and the augmentations chosen; see e.g. [27].…”