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Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 435–474 | Cite as

(Bounded) continuous cohomology and Gromov’s proportionality principle

  • Roberto FrigerioEmail author
Article

Abstract

Let X be a topological space, and let C*(X) be the complex of singular cochains on X with coefficients in \({\mathbb{R}}\) . We denote by \({C^{\ast}_{c}(X) \subseteq C^{\ast}(X)}\) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We prove that at least for “reasonable” spaces the inclusion \({C^{\ast}_{c}(X) \hookrightarrow C^{\ast}(X)}\) induces an isomorphism in cohomology, thus answering a question posed by Mostow. We also prove that this isomorphism is isometric with respect to the L -norm on cochains defined by Gromov. As an application, we clarify some details of Gromov’s original proof of the proportionality principle for the simplicial volume of Riemannian manifolds, also providing a self-contained exposition of Gromov’s argument.

Mathematics Subject Classification (2000)

55N10 (53C23, 55N40, 57N65) 

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References

  1. 1.
    Bucher-Karlsson M.: The proportionality constant for the simplicial volume of locally symmetric spaces. Colloq. Math. 111, 183–198 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bott, R.: Some remarks on continuous cohomology, Manifolds—Tokyo 1973. In: Proceedings of International Conference, Tokyo, 1973, vol. 8, pp. 161–170. University of Tokyo Press, Tokyo (1975)Google Scholar
  3. 3.
    Bourbaki, N.: Intégration vii et viii Élements de mathématique Actualits Sci. Ind. no. 1306. Hermann, Paris (1963)Google Scholar
  4. 4.
    Benedetti R., Petronio C.: Lectures on Hyperbolic Geometry, Universitext. Springer, Berlin (1992)Google Scholar
  5. 5.
    Bredon G.E.: Sheaf theory, Graduate Texts in Mathematics, no. 170. Springer, New York (1997)Google Scholar
  6. 6.
    Dugundji J.: Topology. Allyn and Bacon, Inc., Boston, Mass (1966)zbMATHGoogle Scholar
  7. 7.
    Gromov M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1982)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Hu S.-T.: Cohomology theory. Markham Publishing Company, Chicago (1968)zbMATHGoogle Scholar
  9. 9.
    Ivanov N.V.: Foundation of the theory of bounded cohomology. J. Soviet. Math. 37, 1090–1114 (1987)zbMATHCrossRefGoogle Scholar
  10. 10.
    Lee J.M.: Introduction to smooth manifolds, Graduate Texts in Mathematics, no. 218. Springer, New York (2003)Google Scholar
  11. 11.
    Strohm, C.: The proportionality principle of simplicial volume. Diploma Thesis, Universität Münster (2004). http://arxiv.org/abs/math/0504106
  12. 12.
    Löh C.: Measure homology and singular homology are isometrically isomorphic. Math. Z. 253, 197–218 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mdzinarishvili L.: Continuous singular cohomology. Georgian Math. J. 16, 321–342 (2009)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Monod N.: Continuous bounded cohomology of locally compact groups, Lecture notes in Mathematics, no. 1758. Springer, Berlin (2001)CrossRefGoogle Scholar
  15. 15.
    Mostow M.A.: Continuous cohomology of spaces with two topologies. Mem. Am. Math. Soc. 7, 1–142 (1976)MathSciNetGoogle Scholar
  16. 16.
    Sauer, R.: l 2-invariants of groups and discrete measured groupoids. Ph.D. thesis (2002). http://www.math.uni-muenster.de/u/lueck/homepages/roman_sauer/publ/sauer.pdf.
  17. 17.
    Stone A.H.: A note on paracompactness and normality of mapping spaces. Proc. Am. Math. Soc. 14, 81–83 (1963)zbMATHCrossRefGoogle Scholar
  18. 18.
    Thurston, W.P.: The Geometry and Topology of 3-manifolds. In: Mimeographed Notes. Princeton (1979)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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