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Let be the symplectisation of the contactisation of an exact symplectic manifold , and let be a cylinder over a Legendrian submanifold of the contactisation. We show that a pseudo-holomorphic polygon in having boundary on the projection of can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on . 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.
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Georgios Dimitroglou Rizell, Lifting pseudo-holomorphic polygons to the symplectisation of and applications. Quantum Topol. 7 (2016), no. 1, pp. 29–105