Algebraic & Geometric Topology

Generating families and augmentations for Legendrian surfaces

Dan Rutherford and Michael G Sullivan

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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.

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 3 (2018), 1675-1731.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1524708103

Digital Object Identifier
doi:10.2140/agt.2018.18.1675

Mathematical Reviews number (MathSciNet)
MR3784016

Zentralblatt MATH identifier
06866410

Subjects
Primary: 53D42: Symplectic field theory; contact homology

Keywords
Legendrian surfaces augmentations generating families

Citation

Rutherford, Dan; Sullivan, Michael G. Generating families and augmentations for Legendrian surfaces. Algebr. Geom. Topol. 18 (2018), no. 3, 1675--1731. doi:10.2140/agt.2018.18.1675. https://projecteuclid.org/euclid.agt/1524708103


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References

  • M Aganagic, T Ekholm, L Ng, C Vafa, Topological strings, D-model, and knot contact homology, Adv. Theor. Math. Phys. 18 (2014) 827–956
  • V I Arnol'd, S M Guseĭn-Zade, A N Varchenko, Singularities of differentiable maps, I: The classification of critical points, caustics and wave fronts, Monographs in Mathematics 82, Birkhäuser, Boston (1985)
  • R Casals, E Murphy, Differential algebra of cubic graphs, preprint (2017)
  • Y V Chekanov, Critical points of quasifunctions, and generating families of Legendrian manifolds, Funktsional. Anal. i Prilozhen. 30 (1996) 56–69 In Russian; translated in Funct. Anal. Appl. 30 (1996) 118–128
  • Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441–483
  • G Dimitroglou Rizell, Knotted Legendrian surfaces with few Reeb chords, Algebr. Geom. Topol. 11 (2011) 2903–2936
  • G Dimitroglou Rizell, Lifting pseudo-holomorphic polygons to the symplectisation of $P\times\mathbb{R}$ and applications, Quantum Topol. 7 (2016) 29–105
  • T Ekholm, Morse flow trees and Legendrian contact homology in $1\mkern-1.5mu$–jet spaces, Geom. Topol. 11 (2007) 1083–1224
  • T Ekholm, Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology, from “Perspectives in analysis, geometry, and topology” (I Itenberg, B Jöricke, M Passare, editors), Progr. Math. 296, Springer (2012) 109–145
  • T Ekholm, J B Etnyre, L Ng, M G Sullivan, Knot contact homology, Geom. Topol. 17 (2013) 975–1112
  • T Ekholm, J Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005) 453–532
  • T Ekholm, J Etnyre, M Sullivan, Legendrian contact homology in $P\times\mathbb R$, Trans. Amer. Math. Soc. 359 (2007) 3301–3335
  • D Fuchs, Chekanov–Eliashberg invariant of Legendrian knots: existence of augmentations, J. Geom. Phys. 47 (2003) 43–65
  • D Fuchs, T Ishkhanov, Invariants of Legendrian knots and decompositions of front diagrams, Mosc. Math. J. 4 (2004) 707–717
  • D Fuchs, D Rutherford, Generating families and Legendrian contact homology in the standard contact space, J. Topol. 4 (2011) 190–226
  • A Hatcher, J Wagoner, Pseudo-isotopies of compact manifolds, Astérisque 6, Soc. Math. France, Paris (1973)
  • M B Henry, Connections between Floer-type invariants and Morse-type invariants of Legendrian knots, Pacific J. Math. 249 (2011) 77–133
  • M B Henry, D Rutherford, Equivalence classes of augmentations and Morse complex sequences of Legendrian knots, Algebr. Geom. Topol. 15 (2015) 3323–3353
  • F Laudenbach, On the Thom–Smale complex Appendix to J-M Bismut, W Zhang, An extension of a theorem by Cheeger and Müller, Astérisque 205, Soc. Math. France, Paris (1992)
  • J Milnor, Lectures on the h-cobordism theorem, Princeton Univ. Press (1965)
  • E Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, preprint (2012)
  • L Ng, Framed knot contact homology, Duke Math. J. 141 (2008) 365–406
  • L Ng, D Rutherford, V Shende, S Sivek, E Zaslow, Augmentations are sheaves, preprint (2015)
  • P E Pushkar', Y V Chekanov, Combinatorics of fronts of Legendrian links, and Arnol'd's $4$–conjectures, Uspekhi Mat. Nauk 60 (2005) 99–154 In Russian; translated in Russian Math. Surveys 60 (2005) 95–149
  • D Rutherford, HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links, Quantum Topol. 2 (2011) 183–215
  • D Rutherford, M G Sullivan, Cellular Legendrian contact homology for surfaces, I, preprint (2016)
  • D Rutherford, M Sullivan, Cellular Legendrian contact homology for surfaces, II, preprint (2016)
  • T L Saaty, P C Kainen, The four-color problem: assaults and conquest, 2nd edition, Dover, New York (1986)
  • J M Sabloff, Augmentations and rulings of Legendrian knots, Int. Math. Res. Not. 2005 (2005) 1157–1180
  • J M Sabloff, L Traynor, Obstructions to Lagrangian cobordisms between Legendrians via generating families, Algebr. Geom. Topol. 13 (2013) 2733–2797
  • V Shende, Generating families and constructible sheaves, preprint (2015)
  • V Shende, D Treumann, E Zaslow, Legendrian knots and constructible sheaves, Invent. Math. 207 (2017) 1031–1133
  • L Traynor, Generating function polynomials for Legendrian links, Geom. Topol. 5 (2001) 719–760
  • D Treumann, E Zaslow, Cubic planar graphs and legendrian surface theory, preprint (2017)