(Bounded) continuous cohomology and Gromov proportionality principle

TL;DR

This paper proves that for certain spaces, continuous cochains and all singular cochains have the same cohomology, which is also isometric under the L-infinity norm, and applies this to Gromov's proportionality principle.

Contribution

It establishes an isomorphism between continuous and singular cohomology for reasonable spaces, confirming a question by Mostow and linking it to Gromov's proportionality principle.

Findings

Continuous cohomology is isomorphic to singular cohomology for reasonable spaces.

The isomorphism preserves the L-infinity norm, making it isometric.

Provides a cohomological proof of Gromov's 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.