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

TL;DR

This paper demonstrates how pseudo-holomorphic polygons in a symplectic manifold can be lifted to the symplectisation, enabling an alternative approach to defining Legendrian contact homology and proving Seidel's isomorphism.

Contribution

It establishes a method to lift pseudo-holomorphic polygons to the symplectisation, linking Legendrian contact homology to the homology of Lagrangian fillings.

Findings

Pseudo-holomorphic polygons can be lifted to the symplectisation.

Legendrian contact homology can be defined via these lifted objects.

Proof of Seidel's isomorphism relating contact homology and Lagrangian filling homology.

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 in 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.