Urysohn width of Hypersurfaces and Positive Macroscopic Scalar Curvature

We prove that if a complete Riemannian Formula: see text-manifold with non-trivial codimension 1 homology with Formula: see text-coefficients or Formula: see text-coefficients has positive macroscopic scalar curvature large enough, then it contains a non-nullhomologous hypersurface of small Urysohn Formula: see text-width. This constitutes a macroscopic analogue of a theorem by Bray–Brendle–Neves on the area of non-contractible 2-spheres in a closed Riemannian 3-manifold with positive scalar curvature. Our proof is based on an adaptation of Guth’s macroscopic version of the Schoen–Yau descent argument.