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Augmentations are Sheaves

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Title: Augmentations are Sheaves
Authors: Ng, L
Rutherford, D
Shende, V
Sivek, S
Zaslow, E
Item Type: Journal Article
Abstract: We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], 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 [STZ], 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.
Issue Date: 29-Dec-2020
Date of Acceptance: 7-Dec-2019
URI: http://hdl.handle.net/10044/1/75472
DOI: 10.2140/gt.2020.24.2149
ISSN: 1364-0380
Publisher: Mathematical Sciences Publishers
Start Page: 2149
End Page: 2286
Journal / Book Title: Geometry and Topology
Volume: 24
Issue: 5
Copyright Statement: © Copyright 2020 Mathematical Sciences Publishers. All rights reserved.
Keywords: math.SG
math.SG
math.GT
math.SG
math.SG
math.GT
0101 Pure Mathematics
Geological & Geomatics Engineering
Notes: 102 pages; v2: added Legendrian mirror example in section 4.4.4, corrected typos and other minor changes
Publication Status: Published
Online Publication Date: 2020-12-29
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences
Mathematics