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

### Georgios Dimitroglou Rizell

University of Cambridge, UK

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## 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.

## Cite this article

Georgios Dimitroglou Rizell, Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications. Quantum Topol. 7 (2016), no. 1, pp. 29–105

DOI 10.4171/QT/73