Abstract
We investigate the properties of a two-cocycle on the group of symplectic diffeomorphisms of an exact symplectic manifold defined by Ismagilov, Losik, and Michor. We provide both vanishing and nonvanishing results and applications to foliated symplectic bundles and to Hamiltonian actions of finitely generated groups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brown K.S.: Cohomology of groups. Graduate Texts in Mathematics, vol. 87. Springer-Verlag, New York (1982)
Gal Ś.R., Kędra J.: A cocycle on the group of symplectic diffeomorphisms. Adv. Geom. 11(1), 73–88 (2011)
Gompf R.E., Stipsicz A.I.: 4-manifolds and Kirby calculus. Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence (1999)
Guichardet A., Wigner D.: Sur la cohomologie réelle des groupes de Lie simples réels. Ann. Sci. École Norm. Sup. (4) 11(2), 277–292 (1978)
Ismagilov R.S., Losik M., Michor P.W.: A 2-cocycle on a symplectomorphism group. Mosc. Math. J. 6(2), 307–315, 407 (2006)
Kędra J.: Symplectically hyperbolic manifolds. Differential Geom. Appl. 27(4), 455–463 (2009)
Kędra J., Rudyak Y., Tralle A.: Symplectically aspherical manifolds. J. Fixed Point Theory Appl. 3(1), 1–21 (2008)
McDuff D.: Monodromy in Hamiltonian Floer theory. Comment. Math. Helv. 85(1), 95–133 (2010)
McDuff D., Salamon D.: Introduction to symplectic topology, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1998)
McDuff D., Salamon D.: J-holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications, vol. 2. American Mathematical Society, Providence, RI (2004)
Milnor J.: Construction of universal bundles. I. Ann. Math. 63 (2) 272–284 (1956)
Milnor J.: Construction of universal bundles. II. Ann. Math. 63 (2) 430–436 (1956)
Morita, S.: Geometry of characteristic class. Translations of Mathematical Monographs, vol. 199. American Mathematical Society, Providence, RI, 2001. Translated from the 1999 Japanese original, Iwanami Series in Modern Mathematics
Polterovich L.: Growth of maps, distortion in groups and symplectic geometry. Invent. Math. 150(3), 655–686 (2002)
Schwarz M.: On the action spectrum for closed symplectically aspherical manifolds. Pac. J. Math. 193(2), 419–461 (2000)
Vinberg, E. B. (ed.): Lie Groups and Lie Algebras. II, of Encyclopaedia of Mathematical Sciences, vol. 21. Springer, Berlin (2000). Discrete Subgroups of Lie Groups and Cohomologies of Lie Groups and Lie Algebras, A translation of Current problems in mathematics. Fundamental directions, vol. 21 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1988 [MR 89f:22001], Translated by John Danskin, Translation edited by Onishchik A. L., Vinberg E. B.
Wienhard, A.K.: Bounded Cohomology and Geometry. Bonner Mathematische Schriften [Bonn Mathematical Publications], 368. Universität Bonn Mathematisches Institut, Bonn, 2004. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (2004)
Acknowledgments
We warmly thank Dusa McDuff for explaining to us the proof of Theorem 1.4 and Example 5.6 (4). We thank Dieter Kotschick for drawing our attention to the paper of Ismagilov, Losik, and Michor. And, last but not least, the anonymous referee for helpful remarks improving the final exposition.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Ś. Gal was partially supported by Polish MNiSW grant N N201 541738 and Swiss NSF Sinergia Grant CRSI22-130435.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Gal, Ś., Kędra, J. A two-cocycle on the group of symplectic diffeomorphisms. Math. Z. 271, 693–706 (2012). https://doi.org/10.1007/s00209-011-0884-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0884-9