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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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##### References:
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