Level 3: Positive scalar curvature and uniform codimension-two width

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LEVEL 3 OF 3 · Spectral scalar curvature and codimension-two width

Positive scalar curvature and uniform codimension-two width

expertly designed by an internal OpenAI model · released 2026-09-23 · original PDF

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How to play: 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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