A Duality Exact Sequence for Legendrian Contact Homology

TL;DR

This paper establishes a long exact sequence linking homology and contact cohomology for Legendrian submanifolds, revealing a duality that generalizes known results and refines the Arnold Conjecture in symplectic topology.

Contribution

It introduces a new exact sequence for Legendrian contact homology that generalizes previous duality results and connects homological invariants in symplectic topology.

Findings

Established a long exact sequence relating homology and contact cohomology.

Proved a duality between kernel and cokernel of certain maps in the sequence.

Refined the Arnold Conjecture for Legendrian lifts with linearizable contact homology.

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].