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Positive scalar curvature and rational essentiality

Does positive scalar curvature force every closed connected oriented smooth manifold of dimension at least two to be rationally inessential, meaning its fundamental class vanishes in rational homology of its fundamental-group classifying space?

Why it matters and what counts as progress

Why it matters: The proposed statement strengthens the aspherical positive-scalar-curvature obstruction and removes spin and fundamental-group restrictions.

What counts as progress: A general rational-inessentiality proof under these assumptions, or a rationally essential positive-scalar-curvature example; nonorientable consequences require the orientation double cover.

How the problem developed

  1. 1980 · Conjecture [1]

    Mikhael Gromov and H. Blaine Lawson Jr. Formulated the obstruction that closed aspherical manifolds should not admit positive-scalar-curvature metrics; the later rational-inessentiality statement is a stronger homological formulation.

  2. 2010 · Partial result [2]

    Bernhard Hanke Developed K-area methods linking positive scalar curvature to essentialness and rational inessentiality in restricted settings.

  3. 23 Sept 2026 · Proof claim [3]

    OpenAI Released a manuscript claiming that every closed connected oriented smooth manifold of dimension at least two with positive scalar curvature is rationally inessential, without spin or fundamental-group restrictions.

AI contribution

Sources

  1. Gromov and Lawson, Spin and scalar curvature in the presence of a fundamental group I
  2. Bernhard Hanke, Positive scalar curvature, K-area and essentialness
  3. Positive scalar curvature and rational inessentiality manuscript
  4. Positive scalar curvature forces rational inessentiality
  5. Introduction and main theorem
  6. OpenAI mathematics release
  7. Related integral scalar-curvature manuscript
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