Archiv der Mathematik

, Volume 111, Issue 6, pp 647–656 | Cite as

Conley conjecture and local Floer homology

  • Erman ÇineliEmail author


In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit, the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold, the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.


Periodic orbits Hamiltonian diffeomorphisms Conley conjecture Floer homology 

Mathematics Subject Classification

53D40 37J10 37J45 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author is deeply grateful to Başak Gürel and Viktor Ginzburg for their numerous valuable remarks and suggestions. The author also thanks the anonymous referee for critical remarks and helpful comments. A part of this work was carried out during Hamiltonian and Reeb dynamics: New methods and Applications workshop at Lorentz Center (Leiden, Netherlands); and supported by UCSC. The author would like to thank these institutions for their support.


  1. 1.
    Abbondandolo, A., Schwarz, M.: Floer homology of cotangent bundles and the loop product. Geom. Topol. 14, 1569–1722 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cornea, O., Lupton, G., Oprea, J., Tanre, D.: Lusternik–Schnirelmann Category, Mathematical Surveys and Monographs, vol. 52. AMS, Providence (2013)zbMATHGoogle Scholar
  3. 3.
    Fukaya, K., Ono, K.: Arnold conjecture and Gromov–Witten invariant. Topology 38, 933–1048 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ginzburg, V.L.: The Conley conjecture. Ann. Math. (2) 172, 1127–1180 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ginzburg, V.L., Gürel, B.Z.: Action and index spectra and periodic orbits in Hamiltonian dynamics. Geom. Topol. 13, 2745–2805 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ginzburg, V.L., Gürel, B.Z.: Local Floer homology and the action gap. J. Sympl. Geom. 8, 323–357 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ginzburg, V.L., Gürel, B.Z.: Conley conjecture revisited. Int. Math. Res. Notices IMRN (2017).
  8. 8.
    Hofer, H., Salamon, D.: Floer homology and Novikov rings. In: The Floer Memorial Volume, pp. 483–524, Progr. Math., 133. Birkhäuser, Basel (1995)CrossRefGoogle Scholar
  9. 9.
    Long, Y.: Index Theory for Symplectic Paths with Applications. Birkhäuser, Basel (2002)CrossRefGoogle Scholar
  10. 10.
    McDuff, D., Salamon, D.: J-holomorphic Curves and Symplectic Topology, Colloquium Publications, vol. 52. AMS, Providence (2012)zbMATHGoogle Scholar
  11. 11.
    Pears, J.: Degenerate Critical Points and the Conley Index. University of Edinburg, Edinburg (1995)Google Scholar
  12. 12.
    Piunikhin, S., Salamon, D., Schwarz, M.: Floer–Donaldson, symplectic, theory and quantum cohomology. In: Contact and Symplectic Geometry (Cambridge, 1994), pp. 171–200, Publ. Newton Inst., vol. 8, Cambridge University Press, Cambridge (1996)Google Scholar
  13. 13.
    Salamon, D.A.: Lectures on Floer homology, In: Symplectic Geometry and Topology, pp. 143–229, IAS/Park City Math. Ser., vol. 7. AMS, Providence (1999)Google Scholar
  14. 14.
    Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45, 1303–1360 (1992)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Viterbo, C.: Some remarks on Massey products, tied cohomology classes, and the Lusternik-Shnirelmann category. Duke Math. J. 86, 547–564 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUC Santa CruzSanta CruzUSA

Personalised recommendations