Mathematische Annalen

, Volume 353, Issue 2, pp 523–544 | Cite as

The norm of the Euler class

Article

Abstract

We prove that the norm of the Euler class \({\mathcal {E}}\) for flat vector bundles is 2n (in even dimension n, since it vanishes in odd dimension). This shows that the Sullivan–Smillie bound considered by Gromov and Ivanov–Turaev is sharp. In the course of the proof, we construct a new cocycle representing \({\mathcal {E}}\) and taking only the two values ±2n. Furthermore, we establish the uniqueness of a canonical bounded Euler class.

Mathematics Subject Classification (1991)

57R20 22E41 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams S., Elliott G.A., Giordano T.: Amenable actions of groups. Trans. Am. Math. Soc. 344(2), 803–822 (1994)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bucher M., Gelander T.: Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes. C. R. Math. Acad. Sci. Paris 34(611–12), 661–666 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bucher, M., Gelander, T.: The generalized Chern conjecture for manifolds that are locally a product of surfaces. (Preprint) (2009). arXiv:0902.1215.Google Scholar
  4. 4.
    Burger M., Iozzi A.: Bounded Kähler class rigidity of actions on Hermitian symmetric spaces. Ann. Sci. École Norm. Sup. (4) 37(1), 77–103 (2004)MathSciNetMATHGoogle Scholar
  5. 5.
    Burger M., Monod N.: Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. (JEMS) 1(2), 199–235 (1999)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Burger M., Monod N.: Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12(2), 219–280 (2002)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bucher, M.: Characteristic classes and bounded cohomology. PhD thesis, ETHZ Dissertation Number 15636 (2004)Google Scholar
  8. 8.
    Bucher M.: Finiteness properties of characteristic classes of flat bundles. Enseign. Math. (2) 53(1–2), 33–66 (2007)MathSciNetMATHGoogle Scholar
  9. 9.
    Bucher M.: The simplicial volume of closed manifolds covered by \({\mathbb H^2\times\mathbb H^2}\) . J. Topol. 1(3), 584–602 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Clerc J.-L., Ørsted B.: The Gromov norm of the Kaehler class and the Maslov index. Asian J. Math. 7(2), 269–295 (2003)MathSciNetMATHGoogle Scholar
  11. 11.
    Domic A., Toledo D.: The Gromov norm of the Kaehler class of symmetric domains. Math. Ann. 276(3), 425–432 (1987)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Dupont, J.L.: Bounds for characteristic numbers of flat bundles. In: Algebraic Topology, Aarhus 1978 (Proceedings Symposium of University of Aarhus, Aarhus, 1978), pp. 109–119. Springer, Berlin (1979)Google Scholar
  13. 13.
    Furstenberg H.: A Poisson formula for semi-simple Lie groups. Ann. Math. 77, 335–386 (1963)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ghys, É.: Groupes d’homéomorphismes du cercle et cohomologie bornée. In: The Lefschetz Centennial Conference, Part III (Mexico City, 1984), pp. 81–106. American Mathematical Society, Providence (1987)Google Scholar
  15. 15.
    Gromov M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. (1982) 56, 5–99 (1983)MathSciNetGoogle Scholar
  16. 16.
    Hirzebruch F.: Automorphe Formen und der Satz von Riemann-Roch, Symposium internacional de topologí a algebraica. In: International Symposium on Algebraic Topology, pp. 129–144. Universidad Nacional Autónoma de México and UNESCO, Mexico (1958)Google Scholar
  17. 17.
    Ivanov N.V., Turaev Vladimir G.: The canonical cocycle for the Euler class of a flat vector bundle. Dokl. Akad. Nauk SSSR 265(3), 521–524 (1985)Google Scholar
  18. 18.
    Milnor J.: On the existence of a connection with curvature zero. Comment. Math. Helv. 32, 215–223 (1958)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Monod N.: Continuous bounded cohomology of locally compact groups. Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)CrossRefGoogle Scholar
  20. 20.
    Monod, N.: An invitation to bounded cohomology. In: Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures, pp. 1183–1211. European Mathematical Society, Zürich (2006)Google Scholar
  21. 21.
    Monod N.: Vanishing up to the rank in bounded cohomology. Math. Res. Lett. 14(4), 681–687 (2007)MathSciNetMATHGoogle Scholar
  22. 22.
    Monod N.: On the bounded cohomology of semi-simple groups, S-arithmetic groups and products. J. Reine Angew. Math. 640, 167–202 (2010)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Simonnet, M.: Measures and probabilities. Universitext, Springer, New York (1996) (with a foreword by Charles-Michel Marle)Google Scholar
  24. 24.
    Smillie, J.: The Euler characteristic of flat bundles (unpublished manuscript)Google Scholar
  25. 25.
    Sullivan D.: A generalization of Milnor’s inequality concerning affine foliations and affine manifolds. Comment. Math. Helv. 51(2), 183–189 (1976)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Thurston W.P.: Geometry and Topology of 3-Manifolds. Princeton Notes, Princeton (1978)Google Scholar
  27. 27.
    Wigner D.: Algebraic cohomology of topological groups. Trans. Am. Math. Soc. 178, 83–93 (1973)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Wood J.W.: Bundles with totally disconnected structure group. Comment. Math. Helv. 46, 257–273 (1971)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Université de GenèveGenevaSwitzerland
  2. 2.EPFLLausanneSwitzerland

Personalised recommendations