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The contact homology of Legendrian submanifolds in \(\mathbb R^{2n+1}\). (English) Zbl 1103.53048

The authors construct a contact homology theory for Legendrian submanifolds in \(\mathbb R^{2n+1}\) equipped with the standard contact structure.
Let \(L\in \mathbb R^{2n+1}\) be a Legendrian submanifold. A Reeb chord of \(L\) is a segment of the flow line of the Reeb vector field starting and ending in \(L\). Let \(\Pi: \mathbb R^{2n+1}=\mathbb C^n\times \mathbb R\to \mathbb C^n\) be the projection. Then \(\Pi(L)\subset \mathbb C^n\) is the image of a Lagrangian immersion and the image of a chord is a double point of \(\Pi(L)\). The submanifold \(L\) is called chord generic if \(\Pi(L)\) has only finitely many transverse double points. In this case, the authors define a differential graded algebra \(\mathcal A(L)\) freely generated over \(\mathbb Z_2[H_1(L)]\) by the Reeb chords. The grading is defined with the use of the Maslov index.
The differential is defined as follows. Let \(\beta\) be a collection of cords and \(\mathcal M_A(c,\beta)\) be the moduli space of certain holomorphic disks in \(\mathbb C^n\) with punctures in double points corresponding to \(c\) and \(\beta\). Then \[ \partial (c) := \sum _{\dim \mathcal M_A(c,{\beta})=0} (\# \mathcal M_A(a,{\beta}))A{\beta} \] is a differential of \(\mathcal A(L)\). The contact homology of \(L\) is defined to be the homology of \((\mathcal A,\partial)\).
The rest of the paper is devoted to the proof that the above homology is well defined invariant of Legendrian isotopy.

MSC:

53D12 Lagrangian submanifolds; Maslov index
53D35 Global theory of symplectic and contact manifolds
57R10 Smoothing in differential topology
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