On macroscopic dimension of rationally essential manifolds

TL;DR

This paper provides counterexamples in dimensions greater than 3 to Gromov's conjecture that the macroscopic dimension of rationally essential manifolds equals their dimension.

Contribution

It constructs explicit counterexamples to Gromov's conjecture in higher dimensions, challenging previous assumptions about macroscopic dimension.

Findings

Counterexamples in dimensions >3 to Gromov's conjecture

Shows that macroscopic dimension can be less than the manifold's dimension

Impacts understanding of large-scale geometric properties of manifolds

Abstract

We construct a counterexamples in dimensions $n>3$ to Gromov's conjecture \cite{Gr1} that the macroscopic dimension of rationally essential $n$-dimensional manifolds equals $n$.