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.