2009/03/25 by Roberto Frigerio, Frigerio, Roberto · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.0903.4412
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 Linfty-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.