On macroscopic dimension of universal coverings of closed manifolds

We give a homological characterization of $n$-manifolds whose universal covering $\widetilde{M}$ has Gromov’s macroscopic dimension $\mathrm{dim}_{mc}\widetilde{M}<n$. As the result we distinguish $\mathrm{dim}_{mc}$ from the macroscopic dimension $\mathrm{dim}_{MC}$ defined by the author [7]. We prove the inequality $\mathrm{dim}_{mc}\widetilde{M}<\mathrm{dim}_{MC}\widetilde{M}=n$ for every closed $n$-manifold $M$ whose fundamental group $\pi$ is a geometrically finite amenable duality group with the cohomological dimension $cd(\pi)>n$. References: 14 entries.