Abstract:
Let us call a seminorm defined on real singular homology of topological spaces natural if the homology mappings induced by continuous mappings of topological spaces do not increase this seminorm. An example of such a seminorm, called Gromov's simplicial volume has been known for a long time. This seminorm is defined by the $l_1$-norm on the complex of real singular chains of a space.
We discuss in the talk two other seminorms of purely geometric nature. They are defined respectively by the systolic volume and the volume entropy of Riemannian manifolds. Miraculously, all three listed semi-norms turn out to be equivalent in the sense of the geometry of semi-Banach spaces.
