Algebraic & Geometric Topology

Obstructions to Lagrangian cobordisms between Legendrians via generating families

Joshua M Sabloff and Lisa Traynor

Full-text: Open access


The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1–jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yields structural information about invariants of Legendrian submanifolds. For example, the obstructions devised in this paper show that there is no generating family compatible Lagrangian cobordism between the Chekanov–Eliashberg Legendrian m(52) knots. Further, the generating family cohomology groups of a Legendrian submanifold restrict the topology of a Lagrangian filling. Structurally, the generating family cohomology of a Legendrian submanifold satisfies a type of Alexander duality that, when the Legendrian is null-cobordant, can be seen as Poincaré duality of the associated Lagrangian filling. This duality implies the Arnold Conjecture for Legendrian submanifolds with linear-at-infinity generating families. These results are obtained by developing a generating family version of wrapped Floer cohomology and establishing long exact sequences that arise from viewing the spaces underlying these cohomology groups as mapping cones.

Article information

Algebr. Geom. Topol., Volume 13, Number 5 (2013), 2733-2797.

Received: 1 September 2012
Revised: 26 March 2013
Accepted: 27 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D12: Lagrangian submanifolds; Maslov index 57R17: Symplectic and contact topology
Secondary: 57Q60: Cobordism and concordance

Lagrangian cobordism Legendrian generating family duality


Sabloff, Joshua M; Traynor, Lisa. Obstructions to Lagrangian cobordisms between Legendrians via generating families. Algebr. Geom. Topol. 13 (2013), no. 5, 2733--2797. doi:10.2140/agt.2013.13.2733.

Export citation


  • A Abbondandolo, M Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006) 254–316
  • M Abouzaid, P Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010) 627–718
  • V I Arnol'd, Lagrange and Legendre cobordisms, I, Funktsional. Anal. i Prilozhen. 14 (1980) 1–13, 96
  • V I Arnol'd, Lagrange and Legendre cobordisms, II, Funktsional. Anal. i Prilozhen. 14 (1980) 8–17, 95
  • A Banyaga, D E Hurtubise, A proof of the Morse–Bott lemma, Expo. Math. 22 (2004) 365–373
  • F Bourgeois, A Oancea, An exact sequence for contact- and symplectic homology, Invent. Math. 175 (2009) 611–680
  • F Bourgeois, J M Sabloff, L Traynor, Lagrangian cobordisms via generating families with applications to Legendrian geography and botany, in preparation
  • B Chantraine, Lagrangian concordance of Legendrian knots, Algebr. Geom. Topol. 10 (2010) 63–85
  • Y V Chekanov, Critical points of quasifunctions, and generating families of Legendrian manifolds, Funktsional. Anal. i Prilozhen. 30 (1996) 56–69, 96
  • W Chongchitmate, L Ng, An atlas of Legendrian knots, Exp. Math. 22 (2013) 26–37
  • G Civan, P Koprowski, J B Etnyre, J M Sabloff, A Walker, Product structures for Legendrian contact homology, Math. Proc. Cambridge Philos. Soc. 150 (2011) 291–311
  • P Eiseman, J D Lima, J M Sabloff, L Traynor, A partial ordering on slices of planar Lagrangians, J. Fixed Point Theory Appl. 3 (2008) 431–447
  • T Ekholm, Rational symplectic field theory over $\Bbb Z\sb 2$ for exact Lagrangian cobordisms, J. Eur. Math. Soc. $($JEMS$)$ 10 (2008) 641–704
  • T Ekholm, A version of rational SFT for exact Lagrangian cobordisms in $1$–jet spaces, from: “New perspectives and challenges in symplectic field theory”, (M Abreu, F Lalonde, L Polterovich, editors), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 173–199
  • 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, New York (2012) 109–145
  • T Ekholm, J B Etnyre, J M Sabloff, A duality exact sequence for Legendrian contact homology, Duke Math. J. 150 (2009) 1–75
  • T Ekholm, J B Etnyre, M Sullivan, Non-isotopic Legendrian submanifolds in $\Bbb R\sp {2n+1}$, J. Differential Geom. 71 (2005) 85–128
  • T Ekholm, J B Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005) 453–532
  • T Ekholm, K Honda, T Kálmán, Legendrian knots and exact Lagrangian cobordisms
  • Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560–673
  • J B Etnyre, L Ng, V Vertesi, Legendrian and transverse twist knots, to apear in J. Eur. Math. Soc.
  • A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513–547
  • 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, 783
  • D Fuchs, D Rutherford, Generating families and Legendrian contact homology in the standard contact space, J. Topol. 4 (2011) 190–226
  • K Fukaya, P Seidel, I Smith, The symplectic geometry of cotangent bundles from a categorical viewpoint, from: “Homological mirror symmetry”, (A Kapustin, M Kreuzer, K-G Schlesinger, editors), Lecture Notes in Phys. 757, Springer, Berlin (2009) 1–26
  • R Golovko, A note on Lagrangian cobordisms between Legendrian submanifolds of $\mathbb{R}^{2n+1}$, Pacific J. Math. 261 (2013) 101–116
  • J E Grigsby, D Ruberman, S Strle, Knot concordance and Heegaard Floer homology invariants in branched covers, Geom. Topol. 12 (2008) 2249–2275
  • 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, A combinatorial DGA for Legendrian knots from generating families, Comm. Contemp. Math.
  • J Jordan, L Traynor, Generating family invariants for Legendrian links of unknots, Algebr. Geom. Topol. 6 (2006) 895–933
  • P B Kronheimer, T S Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994) 797–808
  • P Melvin, S Shrestha, The nonuniqueness of Chekanov polynomials of Legendrian knots, Geom. Topol. 9 (2005) 1221–1252
  • J Milnor, Morse theory, Annals of Math. Studies 51, Princeton Univ. Press (1963) Based on lecture notes by M Spivak and R Wells
  • 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
  • G Rizell, Lifting pseudo-holomorphic polygons to the symplectisation of $P\times R$ and applications
  • J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827–844
  • L Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995) 155–163
  • J M Sabloff, Augmentations and rulings of Legendrian knots, Int. Math. Res. Not. 2005 (2005) 1157–1180
  • J M Sabloff, Duality for Legendrian contact homology, Geom. Topol. 10 (2006) 2351–2381
  • J M Sabloff, L Traynor, Obstructions to the existence and squeezing of Lagrangian cobordisms, J. Topol. Anal. 2 (2010) 203–232
  • S Sivek, A bordered Chekanov–Eliashberg algebra, J. Topol. 4 (2011) 73–104
  • S Sivek, Monopole Floer homology and Legendrian knots, Geom. Topol. 16 (2012) 751–779
  • E H Spanier, Algebraic topology, Springer, New York (1981)
  • D Théret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology Appl. 96 (1999) 249–266
  • D Théret, A Lagrangian camel, Comment. Mathem. Helv. 74 (1999) 591–614
  • L Traynor, Generating function polynomials for Legendrian links, Geom. Topol. 5 (2001) 719–760
  • C Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992) 685–710