On macroscopic dimension of universal coverings of closed manifolds

TL;DR

This paper characterizes when the universal cover of a closed manifold has a macroscopic dimension less than its dimension, distinguishing between two types of macroscopic dimensions and relating them to properties of the fundamental group.

Contribution

It provides a homological criterion for the macroscopic dimension of universal covers and differentiates between two notions of macroscopic dimension for certain classes of manifolds.

Findings

Established a homological characterization of macroscopic dimension

Proved inequality between two macroscopic dimensions for specific manifolds

Distinguished macroscopic dimension from a previously defined notion

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 $cd(\pi)> n$.