A Duality Exact Sequence for Legendrian Contact Homology

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 [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].