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 3 (see [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]