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Legendrian contact homology in \(P \times \mathbb{R}\). (English) Zbl 1119.53051
Let \(P\) be an exact symplectic \(2n\)-dimensional manifold. An \(n\)-dimensional submanifold \(L\subset P\times\mathbb{R}\) which is everywhere tangent to the contact structure vector \(\xi\) is called Legendrian, and a continuous one-parameter family of Legendrian submanifolds is a Legendrian isotopy. The main result of the present work is the following Theorem: The contact homology of Legendrian submanifolds of \((P\times\mathbb{R},\xi)\) is well defined. In particular the stable tame isomorphism class of the differential graded algebra associated to a Legendrian submanifold \(L\) is independent of the choice of a compatible almost complex structure and is invariant under Legendrian isotopies of \(L\). As application of this theorem, two other important results are obtained.
Corollary 1. If \(M_1\) and \(M_2\) are two isotopic submanifolds of \(\mathbb{R}^n\), then the contact homologies of \(L_{M_1}\) and \(L_{M_2}\) are isomorphic.
Corollary 2. In \(P\times \mathbb{R}\), there exist infinite families of Legendrian spheres, Legendrian tori, and (when the manifold \(P\) is four-dimensional) closed orientable Legendrian surfaces of arbitrary genus which have the same classical invariants but are pairwise non-Legendrian isotopic.

MSC:
53D10 Contact manifolds (general theory)
53D05 Symplectic manifolds (general theory)
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