(Bounded) continuous cohomology and Gromov proportionality principle

Details (Bounded) continuous cohomology and Gromov proportionality principle (Bounded) continuous cohomology and Gromov proportionality principle

2009-03-25

arxiv.org/abs/0903.4412v2

Mathematics

Geometric Topology

An important improvement with respect to the preceding version: we are now able to show that continuous cohomology is isomorph

Scientific paper

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.

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