Obstructions to Lagrangian Cobordisms between Legendrians via Generating Families

TL;DR

This paper develops a generating family approach to obstruct and analyze Lagrangian cobordisms between Legendrian submanifolds, revealing new invariants and dualities that connect topology, Floer theory, and Legendrian invariants.

Contribution

It introduces a generating family-based TQFT-like framework that produces obstructions and structural invariants for Lagrangian cobordisms and fillings, extending Floer theory techniques.

Findings

Obstructions to certain Lagrangian cobordisms are derived using generating families.

Generating family cohomology satisfies a form of Alexander duality and Poincaré duality.

The approach proves the Arnold Conjecture for Legendrians with linear-at-infinity 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 Poincar\'e 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.