The augmentation category map induced by exact Lagrangian cobordisms

TL;DR

This paper investigates the augmentation category map induced by exact Lagrangian cobordisms between Legendrian knots, establishing a long exact sequence and deriving new obstructions to cobordism existence.

Contribution

It introduces a long exact sequence relating cohomologies of augmentation categories and proves injectivity of the functor on object classes, providing new obstructions.

Findings

Long exact sequence relating cohomologies of augmentation categories

Injectivity of the functor on equivalence classes of objects

New obstructions to exact Lagrangian cobordisms using contact homology and ruling polynomials

Abstract

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.