# Quantum Topology

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**Volume 7, Issue 1, 2016, pp. 29–105**

**DOI: 10.4171/QT/73**

Published online: 2016-02-08

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

Georgios Dimitroglou Rizell^{[1]}(1) University of Cambridge, UK

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.

*Keywords: *Legendrian contact homology, wrapped Floer homology, exact Lagrangian fillings

Dimitroglou Rizell Georgios: Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications. *Quantum Topol.* 7 (2016), 29-105. doi: 10.4171/QT/73