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Augmentations are
sheaves
Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek and Eric Zaslow |
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Geometry & Topology 24 (2020) 2149–2286
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Abstract |
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We show that the set of augmentations of the Chekanov–Eliashberg algebra of a Legendrian link underlies the structure of a unital –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 –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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Keywords
Legendrian knots, Legendrian contact homology,
augmentations, constructible sheaves
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Mathematical Subject Classification 2010
Primary: 53D42
Secondary: 53D37
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References |
Publication
Received: 29 September 2017
Revised: 7 November 2019
Accepted: 7 December 2019
Published: 29 December 2020
Proposed: Yasha Eliashberg
Seconded: Ciprian Manolescu, Paul Seidel
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Authors |
| Lenhard Ng | |
| Department of Mathematics Duke University Durham, NC United States |
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| https://services.math.duke.edu/~ng | |
| Dan Rutherford | |
| Department of Mathematical
Sciences Ball State University Muncie, IN United States |
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| Vivek Shende | |
| Department of Mathematics University of California, Berkeley Berkeley, CA United States |
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| https://math.berkeley.edu/~vivek | |
| Steven Sivek | |
| Department of Mathematics Imperial College London London United Kingdom |
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| http://wwwf.imperial.ac.uk/~ssivek | |
| Eric Zaslow | |
| Department of Mathematics Northwestern University Evanston, IL United States |
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| https://sites.math.northwestern.edu/~zaslow | |
