Geometry & Topology

Quasimorphisms on contactomorphism groups and contact rigidity

Matthew Strom Borman and Frol Zapolsky

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Abstract

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

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

Dates
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
https://projecteuclid.org/euclid.gt/1510858683

Digital Object Identifier
doi:10.2140/gt.2015.19.365

Mathematical Reviews number (MathSciNet)
MR3318754

Zentralblatt MATH identifier
1312.53109

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

Keywords
quasimorphism contactomorphism contact rigidity nonlinear Maslov index prequantization toric

Citation

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. https://projecteuclid.org/euclid.gt/1510858683


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References

  • 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 http://search.proquest.com/docview/304469611 {\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