Lifting pseudo-holomorphic polygons to the symplectization of $P \times \R$ and applications

We show that pseudo-holomorphic polygons in a Liouville-domain can be lifted to the symplectization of its contactization. In particular, Legendrian contact homology may equivalently be defined by counting either of these objects. We use this fact to prove an isomorphism between the linearized Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling, a result which was first conjectured by Seidel.