2008/03/31 by Tobias Ekholm, John B. Etnyre, Joshua M. Sabloff · 1 citation
Mathematics · #math.SG #math.GT
paper · pdf · doi:10.1215/00127094-2009-046
published as Duke Math. J. 150, no. 1 (2009), 1-75 · 57 pages, 10 figures. Improved exposition and expanded analytic detail
arxiv created 2009/10/03 · arxiv updated 2019/12/19
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_*→ H_*) and the cokernel of the map (H_* → 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 αin CH^* such that α(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].