We establish a long exact sequence for Legendrian submanifolds , where is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of to off of itself. In this sequence, the singular homology maps to linearized contact cohomology , which maps to linearized contact homology , which maps to singular homology. In particular, the sequence implies a duality between and . Furthermore, this duality is compatible with Poincaré duality in in the following sense: the Poincaré dual of a singular class which is the image of maps to a class such that .
The exact sequence generalizes the duality for Legendrian knots in (see ) 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 
Tobias Ekholm. John B. Etnyre. Joshua M. Sabloff. "A duality exact sequence for legendrian contact homology." Duke Math. J. 150 (1) 1 - 75, 1 October 2009. https://doi.org/10.1215/00127094-2009-046