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2018 Generating families and augmentations for Legendrian surfaces
Dan Rutherford, Michael G Sullivan
Algebr. Geom. Topol. 18(3): 1675-1731 (2018). DOI: 10.2140/agt.2018.18.1675

Abstract

We study augmentations of a Legendrian surface L in the 1 –jet space, J 1 M , of a surface M . We introduce two types of algebraic/combinatorial structures related to the front projection of L that we call chain homotopy diagrams (CHDs) and Morse complex 2 –families (MC2Fs), and show that the existence of a ρ –graded CHD or a ρ –graded MC2F is equivalent to the existence of a ρ –graded augmentation of the Legendrian contact homology DGA to 2 . A CHD is an assignment of chain complexes, chain maps, and homotopy operators to the 0 –, 1 –, and 2 –cells of a compatible polygonal decomposition of the base projection of L with restrictions arising from the front projection of L . An MC2F consists of a collection of formal handleslide sets and chain complexes, subject to axioms based on the behavior of Morse complexes in 2 –parameter families. We prove that if a Legendrian surface has a tame-at-infinity generating family, then it has a 0 –graded MC2F and hence a 0 –graded augmentation. In addition, continuation maps and a monodromy representation of π 1 ( M ) are associated to augmentations, and then used to provide more refined obstructions to the existence of generating families that (i) are linear at infinity or (ii) have trivial bundle domain. We apply our methods in several examples.

Citation

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Dan Rutherford. Michael G Sullivan. "Generating families and augmentations for Legendrian surfaces." Algebr. Geom. Topol. 18 (3) 1675 - 1731, 2018. https://doi.org/10.2140/agt.2018.18.1675

Information

Received: 5 April 2017; Revised: 17 December 2017; Accepted: 10 January 2018; Published: 2018
First available in Project Euclid: 26 April 2018

zbMATH: 06866410
MathSciNet: MR3784016
Digital Object Identifier: 10.2140/agt.2018.18.1675

Subjects:
Primary: 53D42

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 3 • 2018
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