A Duality Exact Sequence for Legendrian Contact Homology
Abstract
We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L 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 the kernel of the map (CH* H*) and the cokernel of the map (H* CH*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH* maps to a class α in CH* such that α(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] 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 [6].