(Bounded) continuous cohomology and Gromov’s proportionality principle

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.