  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 $2$.
