A duality exact sequence for legendrian contact homology

Abstract

We establish a long exact sequence for Legendrian submanifolds $L\subset P \times \mathbb{R}$, 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 $H_\ast$ maps to linearized contact cohomology $CH^{\ast}$, which maps to linearized contact homology $CH_\ast$, which maps to singular homology. In particular, the sequence implies a duality between ${\rm Ker}(CH_{\ast}\to H_\ast)$ and $CH^{\ast}/{\rm Im}(H_\ast)$. 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 $a\in CH_\ast$ maps to a class $\alpha\in CH^{\ast}$ such that $\alpha(a)=1$.

The exact sequence generalizes the duality for Legendrian knots in $\mathbb{R}^3$ (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]