Manuscripta Mathematica

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

(Bounded) continuous cohomology and Gromov’s proportionality principle

  • Roberto FrigerioEmail author


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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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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