Lagrangian Cobordisms via Generating Families: Constructions and Geography

TL;DR

This paper develops methods to construct Lagrangian cobordisms between Legendrian submanifolds using generating families, and demonstrates how these techniques realize various algebraic invariants as geometric objects.

Contribution

It introduces new constructions for Lagrangian cobordisms via generating families and applies these to solve a non-classical geography problem in Legendrian contact homology.

Findings

Constructed Lagrangian cobordisms from isotopy, spinning, and handle attachment.

Showed any Legendrian with a generating family admits an immersed Lagrangian filling.

Realized any suitable graded duality group as a generating family homology.

Abstract

Embedded Lagrangian cobordisms between Legendrian submanifolds are produced from isotopy, spinning, and handle attachment constructions that employ the technique of generating families. Moreover, any Legendrian with a generating family has an immersed Lagrangian filling with a compatible generating family. These constructions are applied in several directions, in particular to a non-classical geography question: any graded group satisfying a duality condition can be realized as the generating family homology of a connected Legendrian submanifold in R^{2n+1} or in the 1-jet space of any compact n-manifold with n at least 2.