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A duality exact sequence for Legendrian contact homology. (English) Zbl 1193.53179

The authors consider a closed Legendrian submanifold \(L\) of the contactization \(\big(P\times\mathbb R,\mathrm d z-\theta\big)\) of a \(2n\)-dimensional exact symplectic manifold \((P,\mathrm d\theta)\) of finite geometry at infinity, where \(z\) denotes the variable in the \(\mathbb R\)-factor. The Lagrangian projection of the Legendrian submanifold \(L\) to \(P\) is assumed to be displaceable and to have only transverse double points. Reeb chords of \(L\) are in one to one correspondence with the double points of the Lagrangian projection and are the generators of the Legendrian contact homology differential graded algebra \(\mathcal A(L)\). Coefficients are taken in \(\mathbb Z_2\) or in \(\mathbb Z\), \(\mathbb Q\), \(\mathbb R\), or \(\mathbb Z_m\) provided \(L\) is spin. Under the assumption that the linearization \(Q(L)\) of \(\mathcal A(L)\) exists the authors prove that the sequence \( \ldots \rightarrow H_{k+1}(L) \rightarrow H^{n-k-1}\big(Q(L)\big) \rightarrow H_k\big(Q(L)\big) \rightarrow H_k(L) \rightarrow \ldots \) is exact. Moreover, in the case of field coefficients the duality pairings in singular (co-)homology and linearized contact (co-)homology correspond to each other. Included is also a refined answer to Arnold’s chord conjecture.

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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