The Proportionality Principle of Simplicial Volume

Abstract

The simplicial volume is a homotopy invariant of oriented closed connected manifolds measuring the efficiency of representing the fundamental class by singular chains with real coefficients. Despite of its topological nature, the simplicial volume is linked to Riemannian geometry in various ways, e.g., by the proportionality principle. The proportionality principle of simplicial volume states that the simplicial volume and the Riemannian volume are proportional for oriented closed connected Riemannian manifolds sharing the same universal Riemannian covering. Thurston indicated a proof of the proportionality principle using his (smooth) measure homology. It is the purpose of this diploma thesis to provide a full proof of the proportionality principle based on Thurston's approach. In particular, it is shown that (smooth) measure homology and singular homology are isometrically isomorphic for all smooth manifolds. This implies that the simplicial volume indeed can be computed in terms of measure homology.