On macroscopic dimension of universal coverings of closed manifolds

Abstract

We give a homological characterization of $n$-manifolds whose universal covering $\Wi M$ has Gromov's macroscopic dimension $\dim_{mc}\Wi M<n$. As the result we distinguish $\dim_{mc}$ from the macroscopic dimension $\dim_{MC}$ defined by the author \cite{Dr}. We prove the inequality $\dim_{mc}\Wi M<\dim_{MC}\Wi M=n$ for every closed $n$-manifold $M$ whose fundamental group $\pi$ is a geometrically finite amenable duality group with the cohomological dimension $cd(\pi)> n$.