Obstructions to Lagrangian Cobordisms between Legendrians via Generating Families

Abstract

The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yield structural information about invariants of Legendrian submanifolds. For example, the obstructions devised in this paper show that there is no generating family compatible Lagrangian cobordism between the Chekanov-Eliashberg Legendrian $m(5_2)$ knots. Further, the generating family cohomology groups of a Legendrian submanifold restrict the topology of a Lagrangian filling. Structurally, the generating family cohomology of a Legendrian submanifold satisfies a type of Alexander duality that, when the Legendrian is null-cobordant, can be seen as Poincaré duality of the associated Lagrangian filling. This duality implies the Arnold Conjecture for Legendrian submanifolds with linear-at-infinity generating families. The results are obtained by developing a generating family version of wrapped Floer cohomology and establishing long exact sequences that arise from viewing the the spaces underlying these cohomology groups as mapping cones.

56 pages, 10 figures; changes to the introduction and notation

References in corpus (4)

• Generating family invariants for Legendrian links of unknots
• A note on Lagrangian cobordisms between Legendrian submanifolds of R^{2n+1}
• A bordered Chekanov-Eliashberg algebra
• Monopole Floer homology and Legendrian knots

Cited by in corpus (9)

• Lagrangian Cobordisms via Generating Families: Constructions and Geography
• Obstructions to Lagrangian concordance
• Generating families and augmentations for Legendrian surfaces
• A note on the front spinning construction
• The augmentation category map induced by exact Lagrangian cobordisms
• Equivalence classes of augmentations and Morse complex sequences of Legendrian knots
• On Newton polytopes of Lagrangian augmentations
• Non-Orientable Lagrangian Cobordisms between Legendrian Knots
• Homological invariants of codimension 2 contact submanifolds