2006/12/01 by Tobias Ekholm, Ekholm, Tobias
Mathematics · #53D12 #53D40 #57R17 #57R58 #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.SG #msc:53D12 #msc:53D40 #msc:57R17 #msc:57R58
paper · pdf · doi:10.48550/arxiv.math/0612029
arxiv created 2006/12/01 · openalex publication_date 2006/12/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a version of rational Symplectic Field Theory for pairs (X,L), where X is an exact symplectic manifold, where L⊂ X is an exact Lagrangian submanifold with components subdivided into k subsets, and where both X and L have cylindrical ends. The theory associates to (X,L) a \Z-graded chain complex of vector spaces over \Z2, filtered with k filtration levels. The corresponding k-level spectral sequence is invariant under deformations of (X,L) and has the following property: if (X,L) is obtained by joining a negative end of a pair (X',L') to a positive end of a pair (X'',L''), then there are natural morphisms from the spectral sequences of (X',L') and of (X'',L'') to the spectral sequence of (X,L). As an application, we show that if Λ⊂ Y is a Legendrian submanifold of a contact manifold then the spectral sequences associated to (Y×\R,Λks×\R), where Y×\R is the symplectization of Y and where Λks⊂ Y is the Legendrian submanifold consisting of s parallel copies of Λ subdivided into k subsets, give Legendrian isotopy invariants of Λ.