Gromov positive scalar curvature conjecture and rationally inessential macroscopically large manifolds

TL;DR

This paper constructs new examples of manifolds that are rationally inessential yet macroscopically large, providing counterexamples to a rationality conjecture and supporting the Gromov positive scalar curvature conjecture.

Contribution

It introduces the first examples of such manifolds, using small covers and surgery, and demonstrates their properties related to scalar curvature and fundamental groups.

Findings

Counterexamples to Dranishnikov rationality conjecture

Existence of manifolds that are macroscopically large but rationally inessential

Some manifolds do not admit positive scalar curvature metrics

Abstract

We give the first examples of rationally inessential but macroscopically large manifolds. Our manifolds are counterexamples to the Dranishnikov rationality conjecture. For some of them we prove that they do not admit a metric of positive scalar curvature, thus satisfy the Gromov positive scalar curvature conjecture. Fundamental groups of our manifolds are finite index subgroups of right angled Coxeter groups. The construction uses small covers of convex polyhedrons (or alternatively Davis complexes) and surgery.