A Duality Exact Sequence for Legendrian Contact Homology

By T. Ekholm, John B. Etnyre, J. Sabloff. 2008

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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].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f43df03b58faf2617aef132d5af59fad13674664

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Citations in the corpus

Lagrangian cobordisms via generating families: Construction and geography (2014, unknown): “1; see also Ekholm, Etnyre and Sabloff [15], Fuchs and Rutherford [23], and Sabloff [37].”
Knot contact homology (2011, methodology): “Technical results needed for this were already developed in [8] and [11].”
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors (2017, background): “adiabatic limit of pseudoholomorphic discs on a Lagrangian and its pushoff from [18]; when the latter push-off becomes sufficiently small, these discs converge to pseudoholomorphic discs on the single Lagrangian with gradientflow lines attached (called generalised pseudoholomorphic discs in the same”
Effect of Legendrian surgery (2009, background): “Here it can be noted that a represents the fundamental class of the linearized Legendrian contact homology of ƒU ; see Ekholm, Etynre and Sullivan [17].”
Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology (2009, background): “13] similar to (1.1). Borrowing Seidel’s argument, we outline a proof of Conjecture 1.2 in Section 4.4 in which the Lagrangian Floer cohomology HF(X;L,L′) plays the role of wrapped Floer homology. In [6] a duality exact sequence for linearized contact homology of a Legendrian submanifold Λ ⊂Y, where”
Symplectic homology and the Eilenberg-Steenrod axioms (2015, methodology): “A combination of the tautological sequence with the exact sequence of the pair (L, ∂L) and Poincaré duality yields the Poincaré duality long exact sequence for Legendrian contact homology in [38]”
Augmentations are sheaves (2015, background): “ result, which can roughly be summarized as “homology in Aug is cohomology in Aug +.” Proposition 5.4. There is a quasi-isomorphism Hom ( 2; 1)y[ 2] !˘ Hom +( 1; 2). Proof. This proof is given in [EES09] (though not in the language stated here); we include the proof in our language for the convenien”