Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known "elementary" building blocks for Lagrangian cobordisms that are smoothly the attachment of 0-and 1-handles. An important question is whether every pair of nonempty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is "decomposable" into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.
We construct a product on the Floer complex associated to a pair of Lagrangian cobordisms. More precisely, given three exact transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map m 2 by a count of rigid pseudo-holomorphic disks with boundary on the cobordisms and having punctures asymptotic to intersection points and Reeb chords of the negative Legendrian ends of the cobordisms. More generally, to a (d + 1)-tuple of exact transverse Lagrangian cobordisms we associate a map m d such that the family (m d ) d≥1 are A∞-maps. Finally, we extend the Ekholm-Seidel isomorphism to an A∞-morphism, giving in particular that it is a ring isomorphism. r∈Σ1∩Σ3 δ ′ 1
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.