Topological properties of asymptotic invariants and universal volume bounds

In this thesis, we prove that many asymptotic invariants of closed
manifolds depend only on the image of the fundamental class under
the classifying map of the universal covering. Examples include
numerical invariants that reflect the asymptotic behaviour of the
universal covering, like the minimal volume entropy and the
spherical volume, as well as properties that are qualitative
measures for the largeness of a manifold and its coverings, like
enlargeability and hypersphericity. Another important class of
invariants that share the above invariance property originates from
universal volume bounds. The main example is the systolic constant,
which encodes the relation between short noncontractible loops and
the volume of a manifold. Further interesting examples are provided
by the optimal constants in Gromov's filling inequalities, for
which we show that they depend only on the dimension and
orientability. Considering higher-dimensional generalizations of
the systolic constant, a complete answer to the question about the
existence of stable systolic inequalities is given. In the spirit
of the results mentioned already, we also prove that the stable
systolic constant depends only on the image of the fundamental
class in a suitable Eilenberg-Mac Lane space.