Geometry & Topology

Quasimorphisms on contactomorphism groups and contact rigidity

Matthew Strom Borman and Frol Zapolsky

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We build homogeneous quasimorphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental’s nonlinear Maslov index and a contact reduction technique for quasimorphisms. We show how these quasimorphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.

Article information

Geom. Topol., Volume 19, Number 1 (2015), 365-411.

Received: 15 August 2013
Revised: 21 January 2014
Accepted: 30 January 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 53D12: Lagrangian submanifolds; Maslov index 53D20: Momentum maps; symplectic reduction

quasimorphism contactomorphism contact rigidity nonlinear Maslov index prequantization toric


Borman, Matthew Strom; Zapolsky, Frol. Quasimorphisms on contactomorphism groups and contact rigidity. Geom. Topol. 19 (2015), no. 1, 365--411. doi:10.2140/gt.2015.19.365.

Export citation


  • J F Aarnes, Quasistates and quasimeasures, Adv. Math. 86 (1991) 41–67
  • M Abreu, M S Borman, D McDuff, Displacing Lagrangian toric fibers by extended probes, Algebr. Geom. Topol. 14 (2014) 687–752
  • M Abreu, L Macarini, Contact homology of good toric contact manifolds, Compos. Math. 148 (2012) 304–334
  • M Abreu, L Macarini, Remarks on Lagrangian intersections in toric manifolds, Trans. Amer. Math. Soc. 365 (2013) 3851–3875
  • P Albers, U Frauenfelder, Erratum to: A variational approach to Givental's nonlinear Maslov index, Geom. Funct. Anal. 23 (2013) 482–499
  • P Albers, W J Merry, Orderability, contact nonsqueezing and Rabinowitz Floer homology
  • G Alston, L Amorim, Floer cohomology of torus fibers and real Lagrangians in Fano toric manifolds, Int. Math. Res. Not. 2012 (2012) 2751–2793
  • A Banyaga, The structure of classical diffeomorphism groups, Math. and its Applications 400, Kluwer Academic Publ. Group (1997)
  • V V Batyrev, Toric Fano $3$–folds, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981) 704–717, 927
  • V V Batyrev, On the classification of toric Fano $4$–folds, J. Math. Sci. $($New York$)$ 94 (1999) 1021–1050
  • C Bavard, Longueur stable des commutateurs, Enseign. Math. 37 (1991) 109–150
  • G Ben Simon, The nonlinear Maslov index and the Calabi homomorphism, Commun. Contemp. Math. 9 (2007) 769–780
  • G Ben Simon, The geometry of partial order on contact transformations of prequantization manifolds, from: “Arithmetic and geometry around quantization”, (Ö Ceyhan, Y I Manin, M Marcolli, editors), Progr. Math. 279, Birkhäuser, Basel (2010) 37–64
  • G Ben Simon, T Hartnick, Quasitotal orders and translation numbers
  • G Ben Simon, T Hartnick, Reconstructing quasimorphisms from associated partial orders and a question of Polterovich, Comment. Math. Helv. 87 (2012) 705–725
  • P Biran, O Cornea, A Lagrangian quantum homology, 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) 1–44
  • P Biran, O Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009) 2881–2989
  • P Biran, M Entov, L Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math. 6 (2004) 793–802
  • W M Boothby, H C Wang, On contact manifolds, Ann. of Math. 68 (1958) 721–734
  • M S Borman, Symplectic reduction of quasimorphisms and quasistates, J. Symplectic Geom. 10 (2012) 225–246
  • M S Borman, Quasistates, quasimorphisms and the moment map, Int. Math. Res. Not. 2013 (2013) 2497–2533
  • M Brandenbursky, Quasimorphisms and $L\sp p$–metrics on groups of volume-preserving diffeomorphisms, J. Topol. Anal. 4 (2012) 255–270
  • M Brandenbursky, E Shelukhin, On the large-scale geometry of the ${L}\sp p$–metric on the symplectomorphism group of the two-sphere
  • L Buhovsky, M Entov, L Polterovich, Poisson brackets and symplectic invariants, Selecta Math. 18 (2012) 89–157
  • D Calegari, scl, MSJ Memoirs 20, Math. Soc. Japan, Tokyo (2009)
  • Y V Chekanov, Critical points of quasifunctions, and generating families of Legendrian manifolds, Funktsional. Anal. i Prilozhen. 30 (1996) 56–69, 96 In Russian; translated in Fun. Anal. Appl. 30 (1996) 118–128
  • V Chernov, S Nemirovski, Nonnegative Legendrian isotopy in $ST\sp *M$, Geom. Topol. 14 (2010) 611–626
  • C-H Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not. 2004 (2004) 1803–1843
  • C-H Cho, Nondisplaceable Lagrangian submanifolds and Floer cohomology with nonunitary line bundle, J. Geom. Phys. 58 (2008) 1465–1476
  • C-H Cho, M Poddar, Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds (2014)
  • V Colin, S Sandon, The discriminant and oscillation lengths for contact and Legendrian isotopies, to appear in J. Eur. Math. Soc.
  • T Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988) 315–339
  • Y Eliashberg, New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc. 4 (1991) 513–520
  • Y Eliashberg, H Hofer, D Salamon, Lagrangian intersections in contact geometry, Geom. Funct. Anal. 5 (1995) 244–269
  • Y Eliashberg, S S Kim, L Polterovich, Geometry of contact transformations and domains: orderability versus squeezing, Geom. Topol. 10 (2006) 1635–1747
  • Y Eliashberg, L Polterovich, Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal. 10 (2000) 1448–1476
  • M Entov, Commutator length of symplectomorphisms, Comment. Math. Helv. 79 (2004) 58–104
  • M Entov, L Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 2003 (2003) 1635–1676
  • M Entov, L Polterovich, Quasistates and symplectic intersections, Comment. Math. Helv. 81 (2006) 75–99
  • M Entov, L Polterovich, Symplectic quasistates and semisimplicity of quantum homology, from: “Toric topology”, (M Harada, Y Karshon, M Masuda, T Panov, editors), Contemp. Math. 460, Amer. Math. Soc. (2008) 47–70
  • M Entov, L Polterovich, $C\sp 0$–rigidity of the double Poisson bracket, Int. Math. Res. Not. 2009 (2009) 1134–1158
  • M Entov, L Polterovich, Rigid subsets of symplectic manifolds, Compos. Math. 145 (2009) 773–826
  • M Entov, L Polterovich, $C\sp 0$–rigidity of Poisson brackets, from: “Symplectic topology and measure preserving dynamical systems”, (A Fathi, Y-G Oh, C Viterbo, editors), Contemp. Math. 512, Amer. Math. Soc. (2010) 25–32
  • M Entov, L Polterovich, P Py, On continuity of quasimorphisms for symplectic maps, from: “Perspectives in analysis, geometry, and topology”, (I Itenberg, B J öricke, M Passare, editors), Progr. Math. 296, Birkhäuser, Basel (2012) 169–197
  • M Entov, L Polterovich, F Zapolsky, Quasimorphisms and the Poisson bracket, Pure Appl. Math. Q. 3 (2007) 1037–1055
  • M Fraser, L Polterovich, D Rosen, On Sandon-type metrics for contactomorphism groups
  • K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: Anomaly and obstruction, Parts I–II, AMS/IP Studies in Adv. Math. 46, Amer. Math. Soc. (2009)
  • K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian Floer theory on compact toric manifolds II: Bulk deformations, Selecta Math. 17 (2011) 609–711
  • K Fukaya, Y-G Oh, H Ohta, K Ono, Spectral invariants with bulk quasimorphisms and Lagrangian Floer theory (2011)
  • J-M Gambaudo, É Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems 24 (2004) 1591–1617
  • H Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997) 455–464
  • É Ghys, Groups acting on the circle, Enseign. Math. 47 (2001) 329–407
  • É Ghys, Knots and dynamics, from: “International Congress of Mathematicians, I”, (M Sanz-Solé, J Soria, J L Varona, J Verdera, editors), Eur. Math. Soc. (2007) 247–277
  • A B Givental, Nonlinear generalization of the Maslov index, from: “Theory of singularities and its applications”, (V I Arnold, editor), Adv. Soviet Math. 1, Amer. Math. Soc. (1990) 71–103
  • S Guillermou, M Kashiwara, P Schapira, Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems, Duke Math. J. 161 (2012) 201–245
  • L Haug, On the quantum homology of real Lagrangians in Fano toric manifolds, Int. Math. Res. Not. 2013 (2013) 3171–3220
  • H Hofer, D A Salamon, Floer homology and Novikov rings, from: “The Floer memorial volume”, (H Hofer, C H Taubes, A Weinstein, E Zehnder, editors), Progr. Math. 133, Birkhäuser, Basel (1995) 483–524
  • M Khanevsky, Hofer's metric on the space of diameters, J. Topol. Anal. 1 (2009) 407–416
  • B Khesin, S Tabachnikov, Contact complete integrability, Regul. Chaotic Dyn. 15 (2010) 504–520
  • D Kotschick, What is $\dots$ a quasimorphism?, Notices Amer. Math. Soc. 51 (2004) 208–209
  • D Kotschick, Stable length in stable groups, from: “Groups of diffeomorphisms”, (R Penner, D Kotschick, T Tsuboi, N Kawazumi, T Kitano, Y Mitsumatsu, editors), Adv. Stud. Pure Math. 52, Math. Soc. Japan, Tokyo (2008) 401–413
  • S Lanzat, Hamiltonian Floer homology for compact convex symplectic manifolds
  • S Lanzat, Quantum homology of compact convex symplectic manifolds
  • S Lanzat, Quasimorphisms and symplectic quasistates for convex symplectic manifolds, Int. Math. Res. Not. 2013 (2013) 5321–5365
  • F Laudenbach, Homotopie régulière inactive et engouffrement symplectique, Ann. Inst. Fourier $($Grenoble$)$ 36 (1986) 93–111
  • G Lu, Symplectic fixed points and Lagrangian intersections on weighted projective spaces, Houston J. Math. 34 (2008) 301–316
  • D McDuff, Displacing Lagrangian toric fibers via probes, from: “Low-dimensional and symplectic topology”, (M Usher, editor), Proc. Sympos. Pure Math. 82, Amer. Math. Soc. (2011) 131–160
  • D McDuff, D Salamon, Introduction to symplectic topology, 2nd edition, Oxford Univ. Press (1998)
  • D McDuff, S Tolman, Polytopes with mass linear functions, I, Int. Math. Res. Not. 2010 (2010) 1506–1574
  • I Milin, Orderability of contactomorphism groups of lens spaces, PhD thesis, Stanford University (2008) Available at \setbox0\makeatletter\@url {\unhbox0
  • A Monzner, N Vichery, F Zapolsky, Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization, J. Mod. Dyn. 6 (2012) 205–249
  • S Morita, Geometry of characteristic classes, Translations of Math. Monographs 199, Amer. Math. Soc. (2001)
  • M Øbro, An algorithm for the classification of smooth Fano polytopes
  • K Ono, Lagrangian intersection under Legendrian deformations, Duke Math. J. 85 (1996) 209–225
  • Y Ostrover, Calabi quasimorphisms for some nonmonotone symplectic manifolds, Algebr. Geom. Topol. 6 (2006) 405–434
  • L Polterovich, Floer homology, dynamics and groups, from: “Morse-theoretic methods in nonlinear analysis and in symplectic topology”, (P Biran, O Cornea, F Lalonde, editors), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer, Dordrecht (2006) 417–438
  • P Py, Quasimorphismes et invariant de Calabi, Ann. Sci. École Norm. Sup. 39 (2006) 177–195
  • T Rybicki, Commutators of contactomorphisms, Adv. Math. 225 (2010) 3291–3326
  • S Sandon, An integer-valued bi-invariant metric on the group of contactomorphisms of $\mathbb R\sp {2n}\times S\sp 1$, J. Topol. Anal. 2 (2010) 327–339
  • S Sandon, Contact homology, capacity and nonsqueezing in $\mathbb R\sp {2n}\times S\sp 1$ via generating functions, Ann. Inst. Fourier $($Grenoble$)$ 61 (2011) 145–185
  • S Sandon, Equivariant homology for generating functions and orderability of lens spaces, J. Symplectic Geom. 9 (2011) 123–146
  • H Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. 52 (2000) 383–413
  • E Shelukhin, The action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures, Comment. Math. Helv. 89 (2014) 69–123
  • D Tamarkin, Microlocal condition for non-displaceablility
  • M Usher, Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms, Geom. Topol. 15 (2011) 1313–1417
  • K Watanabe, M Watanabe, The classification of Fano $3$–folds with torus embeddings, Tokyo J. Math. 5 (1982) 37–48
  • G Wilson, C T Woodward, Quasimap Floer cohomology for varying symplectic quotients, Canad. J. Math. 65 (2013) 467–480
  • C T Woodward, Gauged Floer theory of toric moment fibers, Geom. Funct. Anal. 21 (2011) 680–749
  • F Zapolsky, Geometry of contactomorphism groups, contact rigidity and contact dynamics in jet spaces, Int. Math. Res. Not. 2013 (2013) 4687–4711