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Lifting pseudo-holomorphic polygons to the symplectisation of \(P \times \mathbb{R}\) and applications. (English) Zbl 1346.53074
The author studies the lifting problem of pseudoholomorphic polygons on an exact symplectic manifold \(P\) to the symplectization of the contactization, i.e., to \(\mathbb R \times (P \times \mathbb R)\). Denote by \(\Pi(\Lambda)\) the \(P\)-projection of a given closed Legendrian submanifold \(\Lambda\), which defines an exact Lagrangian immersion in \(P\). Let \(\mathbb R \times \Lambda \subset \mathbb R \times (P \times \mathbb R)\) be the corresponding Lagrangian cylinder. Then the author proves that a pseudoholomorphic polygon in \(P\) having boundary on \(\Pi(\Lambda)\) can be lifted to a pseudoholomorphic disc in the \(\mathbb R \times (P \times \mathbb R)\) having boundary on \(\mathbb R \times \Lambda\). As a result it is shown that Legendrian contact homology may be equivalently defined by counting either of these objects. Using the result, the author gives a proof that the linearized Legendrian contact homology induced by an exact Lagrangian filling is isomorphic to the singular homology of the filling. Such an isomorphism was first observed by Seidel and its proof was previously outlined by T. Ekholm [Prog. Math. 296, 109–145 (2012; Zbl 1254.57024)]. The proof is based on some vanishing result of wrapped Floer cohomology of the pair \((L,L')\) of exact Lagrangian fillings in the symplectization \(\mathbb R \times (P \times \mathbb R)\), which in turn is a consequence of Hamiltonian displacement of one from the other.

MSC:
53D42 Symplectic field theory; contact homology
53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
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