Abstract
Let M be a weakly monotone symplectic manifold and H be a time-dependent 1-periodic Hamiltonian; we assume that the 1-periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M and derive from it new lower bounds for the number of 1-periodic orbits. Using these invariants we prove in particular that if π 1(M) is finite and solvable or simple, then the number of 1-periodic orbits is not less than the minimal number of generators of π 1(M). For a general closed symplectic manifold with infinite fundamental group, we show the existence of 1-periodic orbit of Conley–Zehnder index 1 − n for any non-degenerate 1-periodic Hamiltonian system.
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Notes
- 1.
- 2.
There is an approach to construct Hamiltonian Floer complex with integer coefficients for non-degenerate periodic Hamiltonian systems on arbitrary closed symplectic manifold [9]. Since the details have not been carried out, we restrict ourselves to the class of weakly monotone symplectic manifolds.
- 3.
In [17], [ω] is a rational cohomology class, i.e., an integral cohomology class after multiplying a suitable integer. Any symplectic form is approximated by rational symplectic forms and the estimate for the number of fixed points of a non-degenerate Hamiltonian diffeomorphism is reduced to the one on a closed symplectic manifolds with rational symplectic class.
- 4.
The symbol \(\lceil z\rceil \) denotes the minimal integer k ≥ z.
- 5.
Our terminology here differs from that of Swan’s paper [23].
References
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1978). Translated from the 1974 Russian edition by K. Vogtman and A. Weinstein
Arnold, V.I. (ed.): Arnold Problems. Springer, Berlin (2004)
Barraud, J.-F.: A Floer Fundamental group. Preprint (2014) arXiv:1404.3266
Cieliebak, K., Frauenfelder, U.: Morse homology on noncompact manifolds. J. Korean Math. Soc. 48, 749–774 (2011)
Damian, M.: On the stable Morse number of a closed manifold. Bull. Lond. Math. Soc. 34, 420–430 (2002)
Fine, J., Panov, D.: The diversity of symplectic Calabi-Yau six-manifolds preprint (2011). arXiv:1108.5944
Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120,575–611 (1989)
Fukaya, K., Ono, K.: Arnold conjecture and Gromov-Witten invariant. Topology 38, 933–1048 (1999)
Fukaya, K., Ono, K.: Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds - summary. Adv. Stud. Pure Math. 31, 75–91 (2001)
Fukaya, K., Oh, Y-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I and II. AMS/IP Studies in Advanced Mathematics, vols. 46–1 and 46–2. American Mathematical Society/International Press, Providence (2009)
Gruenberg, K.W.: Über die Relationenmodule einer endlichen Gruppe. Math. Z. 118, 30–33 (1970)
Hofer, H., Salamon, D.: Floer homology and Novikov ring. In: Hofer, H., et al. (eds.) The Floer Memorial Volume, pp. 483–524. Birkhauser, Basel (1995)
Le, H.-V., Ono, K.: Symplectic fixed points, the Calabi invariant and Novikov homology. Topology 34, 155–176 (1995)
Liu, G., Tian, G.: Floer homology and Arnold conjecture. J. Differ. Geom. 49, 1–74 (1998)
Lucchini, A.: The minimal number of generators and the presentation rank. Rend. Circ. Mat. Palermo (2) 23, 221–226 (1990)
Lucchini, A.: A bound on the presentation rank of a finite group. Bull. Lond. Math. Soc. 29, 389–394 (1997)
Ono, K.: On the Arnold Conjecture for weakly monotone symplectic manifolds. Invent. Math. 119, 519–537 (1995)
Ono, K.: Floer-Novikov cohomology and the flux conjecture. Geom. Funct. Anal. 16,981–1020 (2006)
Piunikhin, P., Salamon, D., Schwarz, M.: Symplectic Floer–Donaldson theory and quantum cohomology. In: Contact and Symplectic Geometry (Cambridge, 1994), pp. 171–200. Cambridge University Press, Cambridge (1996)
Roggenkamp, K.: Integral representations and presentations of finite groups. In: Integral Representations. Lecture Notes in Mathematics, vol. 744, pp. 144–275. Springer, Heidelberg (1979)
Sharko, V.: Functions on Manifolds, Algebraic and Topological Aspects. Translations of Mathematical Monographs, vol. 131. American Mathematical Society, Providence (1993). Translated from the Russian by V. V. Minachin
Smale, S.: On the structure of manifolds. Am. J. Math. 84, 387–399 (1962)
Swan, R.: Minimal resolutions for finite groups. Topology 4, 193–208 (1965)
Acknowledgements
The authors started this work during the visit of the second author to RIMS, Kyoto University, in May 2013, and continued it during the visit of the first author to the Nantes University in November 2013. It was finalized during the visit of the second author to the Kavli IPMU, the University of Tokyo, in April 2014.
The first author thanks Laboratoire Jean Leray, Université de Nantes, for the financial support and its hospitality. He is partly supported by JSPS Grant-in-Aid for Scientific Research # 21244002 and # 26247006. The second author gratefully acknowledges the support of RIMS, Kyoto University, the Program for Leading Graduate Schools, MEXT, Japan, and of the Kavli IPMU, the University of Tokyo, and thanks RIMS, Kyoto University, and the Kavli IMPU for their hospitality. The second author thanks A. Lucchini for sending the papers [15, 16].
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Ono, K., Pajitnov, A. (2016). On the Fixed Points of a Hamiltonian Diffeomorphism in Presence of Fundamental Group. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_10
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