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 $H_{*}$ maps to linearized contact cohomology ${\mathrm{CH}}^{*}$, which maps to linearized contact homology ${\mathrm{CH}}_{*}$, which maps to singular homology. In particular, the sequence implies a duality between $\mathrm{Ker}({\mathrm{CH}}_{*}\to H_{*})$ and ${\mathrm{CH}}^{*}/\mathrm{Im}(H_{*})$. 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 $a\in {\mathrm{CH}}_{*}$ maps to a class $\alpha \in {\mathrm{CH}}^{*}$ 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]