(Bounded) continuous cohomology and Gromov proportionality principle
Abstract
Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(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 of Cc(X) in C(X) induces an isomorphism in cohomology, thus answering a question posed by Mostow. We also prove that such isomorphism is isometric with respect to the L^infty-norm on cohomology defined by Gromov. As an application, we discuss a cohomological proof of Gromov's proportionality principle for the simplicial volume of Riemannian manifolds.
Keywords
Cite
@article{arxiv.0903.4412,
title = {(Bounded) continuous cohomology and Gromov proportionality principle},
author = {Roberto Frigerio},
journal= {arXiv preprint arXiv:0903.4412},
year = {2010}
}
Comments
An important improvement with respect to the preceding version: we are now able to show that continuous cohomology is isomorphic to singular cohomology even for (a large class of) spaces with non-contractible universal covering. Therefore, the definition of locally bounded Borelian cohomology is not needed any more.