Macroscopic scalar curvature and codimension 2 width

Abstract

We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following"macroscopic curvature"assumptions: every ball of radius $10$ in $M$ has volume at most $4$, and every loop in every ball of radius $1$ in $M$ is null-homologous in the concentric ball of radius $2$.