Positive scalar curvature and uniform codimension-two width
We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n -manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most $n-2$ whose entire fibers have diameter bounded only in terms of n . We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.
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