Skip to main content
SpringerLink
Log in

A \(C^0\) counterexample to the Arnold conjecture

  • Published:
Inventiones mathematicae Aims and scope

Abstract

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold \((M, \omega )\) must have at least as many fixed points as the minimal number of critical points of a smooth function on M. It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher. More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. M is said to be symplectically aspherical if \(\omega \) and \(c_1\), the first Chern class of M, both vanish on \(\pi _2(M)\).

  2. This is precisely the class of area preserving homeomorphisms with vanishing mean rotation vector.

  3. This fact is an immediate consequence of Floer’s proof of the Arnold conjecture; see also [17].

  4. It can be shown that this property holds for closed symplectic surfaces, as well as for the standard \(\mathbb {C}P^2\) and monotone \(S^2 \times S^2\).

  5. \(A \leqslant \frac{\pi }{2}\) is sufficiently small for our purposes.

References

  1. Barannikov, S.A.: The framed Morse complex and its invariants. In: Arnold, V.I. (ed.) Singularities and Bifurcations. Transl. ed. by Sossinsky, A.B., vol. 21 of Advances in Soviet Mathematics, pp. 93–115. American Mathematical Society, Providence, RI (1994)

  2. Buhovsky, L.: The 2/3-convergence rate for the Poisson bracket. Geom. Funct. Anal. 19(6), 1620–1649 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buhovsky, L., Opshtein, E.: Some quantitative results in \(C^0\) symplectic geometry. Invent. Math. 205(1), 1–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buhovsky, L., Seyfaddini, S.: Uniqueness of generating Hamiltonians for topological Hamiltonian flows. J. Symplectic Geom. 11(1), 37–52 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cardin, F., Viterbo, C.: Commuting Hamiltonians and Hamilton–Jacobi multi-time equations. Duke Math. J. 144(2), 235–284 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Conley, C.C., Zehnder, E.: The Birkhoff–Lewis fixed point theorem and a conjecture of V. I. Arnol\({}^{\prime }\) d. Invent. Math. 73(1), 33–49 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Eliashberg, Y.: Estimates on the Number of Fixed Points of Area Preserving Transformations. Syktyvkar University, Syktyvkar (1979). (preprint)

    Google Scholar 

  8. Entov, M., Polterovich, L.: \(C^0\)-rigidity of Poisson brackets. In: Symplectic Topology and Measure Preserving Dynamical Systems, vol. 512 of Contemporary Mathematics, pp. 25–32. American Mathematical Society, Providence, RI (2010)

  9. Floer, A.: Proof of the Arnol\({}^{\prime }\) d conjecture for surfaces and generalizations to certain Kähler manifolds. Duke Math. J. 53(1), 1–32 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. 41, 775–813 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Floer, A.: Cuplength estimates on Lagrangian intersections. Commun. Pure Appl. Math. 42(4), 335–356 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120(4), 575–611 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fortune, B.: A symplectic fixed point theorem for \(\mathbf{C}{{\rm P}}^{n}\). Invent. Math. 81(1), 29–46 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fortune, B., Weinstein, A.: A symplectic fixed point theorem for complex projective spaces. Bull. Am. Math. Soc. (N.S.) 12(1), 128–130 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Franks, J.: Rotation vectors and fixed points of area preserving surface diffeomorphisms. Trans. Am. Math. Soc. 348(7), 2637–2662 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ginzburg, V.L.: The Conley conjecture. Ann. Math. (2) 172(2), 1127–1180 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gromov, M.: Soft and hard symplectic geometry. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2 (Berkeley, CA, 1986), pp. 81–98. American Mathematical Society, Providence, RI (1987)

  19. Hingston, N.: Subharmonic solutions of Hamiltonian equations on tori. Ann. Math. (2) 170(2), 529–560 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hofer, H.: Lusternik–Schnirelman-theory for Lagrangian intersections. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(5), 465–499 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Howard, W.: Action selectors and the fixed point set of a Hamiltonian diffeomorphism. ArXiv:1211.0580

  22. Humilière, V., Leclercq, R., Seyfaddini, S.: Coisotropic rigidity and \(C^0\)-symplectic geometry. Duke Math. J. 164(4), 767–799 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Humilière, V., Leclercq, R., Seyfaddini, S.: New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians. Comment. Math. Helv. 90(1), 1–21 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Humilière, V., Leclercq, R., Seyfaddini, S.: Reduction of symplectic homeomorphisms. Ann. Sci. Éc. Norm. Supér. (4) 49(3), 633–668 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Le Calvez, P.: Une version feuilletée équivariante du théorème de translation de Brouwer. Publ. Math. Inst. Hautes Études Sci. 102, 1–98 (2005)

    Article  MATH  Google Scholar 

  26. Le Calvez, P.: Periodic orbits of Hamiltonian homeomorphisms of surfaces. Duke Math. J. 133(1), 125–184 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Le Peutrec, D., Nier, F., Viterbo, C.: Precise Arrhenius law for \(p\)-forms: the Witten Laplacian and Morse–Barannikov complex. Ann. Henri Poincaré 14(3), 567–610 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Matsumoto, S.: Arnold conjecture for surface homeomorphisms. In: Proceedings of the French–Japanese Conference “Hyperspace Topologies and Applications” (La Bussière, 1997), vol. 104, pp. 191–214 (2000)

  29. Oh, Y.-G.: Construction of Spectral Invariants of Hamiltonian Paths on Closed Symplectic Manifolds. The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol. 232, pp. 525–570. Birkhauser, Boston (2005)

    Google Scholar 

  30. Oh, Y.-G., Müller, S.: The group of Hamiltonian homeomorphisms and \(C^0\)-symplectic topology. J. Symplectic Geom. 5(2), 167–219 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Opshtein, E.: \(C^0\)-rigidity of characteristics in symplectic geometry. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 857–864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Polterovich, L., Shelukhin, E.: Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules. Selecta Math. (N.S.) 22(1), 227–296 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rudyak, Y.B., Oprea, J.: On the Lusternik–Schnirelmann category of symplectic manifolds and the Arnold conjecture. Math. Z. 230(4), 673–678 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Schwarz, M.: On the action spectrum for closed symplectically aspherical manifolds. Pac. J. Math. 193(2), 419–461 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sikorav, J.-C.: Points fixes d’une application symplectique homologue à l’identité. J. Differ. Geom. 22(1), 49–79 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  36. Usher, M., Zhang, J.: Persistent homology and Floer–Novikov theory. ArXiv:1502.07928 (2015)

  37. Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292, 685–710 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Viterbo, C.: On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. 11, 1–9 (2006)

Download references

Acknowledgements

We would like to thank Yasha Eliashberg, Viktor Ginzburg, Helmut Hofer, Rémi Leclercq, Frédéric Le Roux, Patrice Le Calvez, Emmanuel Opshtein, Leonid Polterovich, and Claude Viterbo for fruitful discussions. LB: The research leading to this project began while I was a Professeur Invité at the Université Pierre et Marie Curie. I would like to express my deep gratitude to the members of the Institut Mathématique de Jussieu–Paris Rive Gauche, especially the team Analyse Algébrique, for their warm hospitality. SS: I would like to thank the School of Mathematics at the Institute for Advanced Study and the Department of Mathematics at MIT, where different parts of this project were carried out, for their hospitality. LB was partially supported by the Israel Science Foundation Grant 1380/13, by the Alon Fellowship, and by the Raymond and Beverly Sackler Career Development Chair. VH was partially supported by the Agence Nationale de la Recherche, Projects ANR-11-JS01-010-01 and ANR-12-BS020-0020. SS was partially supported by the NSF Postdoctoral Fellowship Grant No. DMS-1401569.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Lev Buhovsky

  2. Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, 75005, Paris, France

    Vincent Humilière

  3. CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, 75005, Paris, France

    Sobhan Seyfaddini

Authors
  1. Lev Buhovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vincent Humilière
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sobhan Seyfaddini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sobhan Seyfaddini.

Additional information

Lev Buhovsky: This author also uses the spelling “Buhovski” for his family name.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buhovsky, L., Humilière, V. & Seyfaddini, S. A \(C^0\) counterexample to the Arnold conjecture. Invent. math. 213, 759–809 (2018). https://doi.org/10.1007/s00222-018-0797-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-018-0797-x

Keywords

Search

Navigation