#### Volume 17, issue 3 (2017)

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The augmentation category map induced by exact Lagrangian cobordisms

### Yu Pan

Algebraic & Geometric Topology 17 (2017) 1813–1870
##### Abstract

To a Legendrian knot, one can associate an ${A}_{\infty }$ 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 this 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.

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##### Keywords
the augmentation category, Lagrangian cobordisms, wrapped Floer homology, Legendrian knots
##### Mathematical Subject Classification 2010
Primary: 53D42, 57R17
Secondary: 53D12, 57M50
##### Publication
Received: 1 August 2016
Revised: 21 December 2016
Accepted: 1 January 2017
Published: 17 July 2017
##### Authors
 Yu Pan Mathematics Department Duke University Durham, NC United States http://sites.google.com/site/yupanduke/