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Augmentations are sheaves

Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek and Eric Zaslow

Geometry & Topology 24 (2020) 2149–2286

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.

Legendrian knots, Legendrian contact homology, augmentations, constructible sheaves
Mathematical Subject Classification 2010
Primary: 53D42
Secondary: 53D37
Received: 29 September 2017
Revised: 7 November 2019
Accepted: 7 December 2019
Published: 29 December 2020
Proposed: Yasha Eliashberg
Seconded: Ciprian Manolescu, Paul Seidel
Lenhard Ng
Department of Mathematics
Duke University
Durham, NC
United States
Dan Rutherford
Department of Mathematical Sciences
Ball State University
Muncie, IN
United States
Vivek Shende
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Steven Sivek
Department of Mathematics
Imperial College London
United Kingdom
Eric Zaslow
Department of Mathematics
Northwestern University
Evanston, IL
United States