Homological invariance for asymptotic invariants and systolic inequalities

We show that the systolic constant, the minimal entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value on the real projective space) and derive an inequality between the minimal entropy and the systolic constant.