Volume and macroscopic scalar curvature

TL;DR

This paper establishes macroscopic analogs of key conjectures relating scalar curvature bounds to topological and geometric invariants, replacing pointwise bounds with volume bounds on universal covers.

Contribution

It proves the macroscopic versions of three major conjectures linking scalar curvature to simplicial volume, essential manifolds, and $L^2$-Betti numbers.

Findings

Proves a bound on simplicial volume under volume constraints.

Shows rationally essential manifolds cannot have metrics with positive scalar curvature.

Establishes bounds on $L^2$-Betti numbers for aspherical manifolds.

Abstract

We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of $L^2$-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of $1$-balls in the universal cover.