Continuous Cohomology and Gromov Proportionality Principle

Let X be a topological space, and let C(X) be the complex of singular cochains on X with coefficients in R. We denote by C c (X) (resp. C ∗ b (X)) the subcomplex of C(X) given by continuous (resp. locally bounded Borelian) cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous (resp. locally bounded Borelian) real function. It is folklore that at least for “reasonable” spaces the inclusion C c (X) →֒ C (X) induces an isomorphism in cohomology. We prove that this is true if X is paracompact, locally path connected and admits a contractible universal covering. Moreover, we prove that for every paracompact, second countable and locally contractible X, the inclusion C b (X) →֒ C (X) induces an isomorphism in cohomology. Similar statements about bounded cohomology are also proved. As an application, we discuss a cohomological proof of Gromov proportionality principle for the simplicial volume of Riemannian manifolds. 1. Preliminaries and statements Let X be a topological space. We denote by C∗(X) (resp. by C (X)) the usual complex of singular chains (resp. cochains) on X with coefficients in R. For i ∈ N, we let Si(X) be the set of singular i-simplices in X, and we endow Si(X) with the compact-open topology (see Appendix A for basic definitions and results about the compact-open topology). We also regard Si(X) as a subset of the singular chains Ci(X) on X, so that for any cochain φ ∈ C (X) it makes sense to consider its restriction φ|Si(X). For every φ ∈ C i(X), we set ||φ|| = ||φ||∞ = sup {|φ(σ)|, σ ∈ Si(X)} ∈ [0,∞]. We denote by Ĉ(X) the submodule of bounded cochains, i.e. we set Ĉ(X) = {φ ∈ C(X) | ||φ|| < ∞}. Since the differential takes bounded cochains into bounded cochains, Ĉ(X) is a subcomplex of C(X). We say that a cochain φ ∈ Sq(X) is locally bounded if every simplex s ∈ Sq(X) has a neighbourhood Us ⊆ Sq(X) such that φ|Us is bounded (note that this condition does not imply in general that every 2000 Mathematics Subject Classification. 55N10 (55N40, 57N65).