Gromov's macroscopic dimension conjecture

TL;DR

This paper constructs a specific 4-manifold with a torsion-free fundamental group whose universal cover has a macroscopic dimension of 3, providing a counterexample to Gromov's conjecture on macroscopic dimension.

Contribution

It presents the first known counterexample to Gromov's conjecture by explicitly constructing a 4-manifold with the described properties.

Findings

Counterexample to Gromov's conjecture established

Universal cover of the constructed manifold has macroscopic dimension 3

Manifold has torsion-free fundamental group

Abstract

In this note we construct a closed 4-manifold having torsion-free fundamental group and whose universal covering is of macroscopic dimension 3. This yields a counterexample to Gromov's conjecture about the falling of macroscopic dimension.