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arXiv · 0803.2455

A Duality Exact Sequence for Legendrian Contact Homology

Tobias Ekholm·John B. Etnyre·Joshua M. Sabloff
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Abstract

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class \alpha in CH^* such that \alpha(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6].

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Tobias Ekholm, John B. Etnyre, Joshua M. Sabloff. 2008-03-17. A Duality Exact Sequence for Legendrian Contact Homology. https://doi.org/10.1215/00127094-2009-046

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