On macroscopic dimension of universal coverings of closed manifolds

Abstract:

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