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, yields 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
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. These results are obtained by developing a generating family version of
wrapped Floer cohomology and establishing long exact sequences that arise
from viewing the spaces underlying these cohomology groups as mapping
cones.
