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2020 Augmentations are sheaves
Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, Eric Zaslow
Geom. Topol. 24(5): 2149-2286 (2020). DOI: 10.2140/gt.2020.24.2149

Abstract

We show that the set of augmentations of the Chekanov–Eliashberg algebra of a Legendrian link underlies the structure of a unital A–category. This differs from the nonunital category constructed by Bourgeois and Chantraine (J. Symplectic Geom. 12 (2014) 553–583), but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by Shende, Treumann and Zaslow (Invent. Math. 207 (2017) 1031–1133), who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x–line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.

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Lenhard Ng. Dan Rutherford. Vivek Shende. Steven Sivek. Eric Zaslow. "Augmentations are sheaves." Geom. Topol. 24 (5) 2149 - 2286, 2020. https://doi.org/10.2140/gt.2020.24.2149

Information

Received: 29 September 2017; Revised: 7 November 2019; Accepted: 7 December 2019; Published: 2020
First available in Project Euclid: 5 January 2021

MathSciNet: MR4194293
Digital Object Identifier: 10.2140/gt.2020.24.2149

Subjects:
Primary: 53D42
Secondary: 53D37

Rights: Copyright © 2020 Mathematical Sciences Publishers

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