Piecewise straightening and Lipschitz simplicial volume

Abstract

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above. We obtain also yet another proof of the proportionality principle in the compact case by a direct approximation of the smearing map.Comment: v2: small changes in the introduction, references and section 2.3 (one minor mistake corrected) v3: corrected sections 2.1, 2.2 v4: new section 2.1 about 'exponential' neighbourhoods, more details about the procedure itself in section 2.3, barycentric C^1 homology corrected to piecewise C^1 in section