Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications

Abstract

Let \mathbb R \times (P \times \mathbb R) be the symplectisation of the contactisation of an exact symplectic manifold P , and let \mathbb R \times \Lambda be a cylinder over a Legendrian submanifold of the contactisation. We show that a pseudo-holomorphic polygon in P having boundary on the projection of \Lambda can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on \mathbb R \times \Lambda . It follows that Legendrian contact homology may be equivalently defined by counting either of these objects. Using this result, we give a proof of Seidel's isomorphism of the linearised Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling.