The Gromov-Lawson conjecture: closed aspherical manifolds admit no metric of positive scalar curvature
A closed manifold is aspherical if its universal cover is contractible. Gromov and Lawson (1983), with Dirac-operator methods, and Schoen and Yau, with minimal hypersurfaces, obstructed positive scalar curvature on tori and enlargeable manifolds; the conjecture that no closed aspherical manifold carries a metric of positive scalar curvature was open beyond dimension five (Chodosh-Li, Gromov). Hanke (2011, Conjecture 2.9) formulated the stronger homological form: if a closed oriented -manifold admits positive scalar curvature, the classifying map sends its rational fundamental class to zero in . Does any closed aspherical manifold admit a metric of positive scalar curvature, and are positive-scalar-curvature manifolds always rationally inessential?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Scalar curvature; topology of manifolds
- Posed by
- M. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. IHES 58 (1983); homological form: B. Hanke (2011), Conjecture 2.9
- Year posed
- 1983
- Years open
- 43y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 60 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if a closed connected oriented smooth , , admits positive scalar curvature then in . Corollaries: no closed aspherical manifold (orientable or not) admits positive scalar curvature; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat; a closed oriented manifold mapping with nonzero degree to an aspherical one has no positive scalar curvature; and nonnegative scalar curvature forces zero real simplicial volume. Prior cases: dimensions 4 and 5 (Chodosh-Li, Gromov). It does not give the quantitative scalar-curvature/simplicial-volume inequality (that is the companion entry).
What the AI did
Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.
Verification
No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and the aspherical corollaries of the TeX source, read against Hanke's Conjecture 2.9 and the aspherical conjecture as cited. Theorem 1.1 states rational inessentiality for every closed connected oriented smooth manifold of dimension at least 2 with positive scalar curvature, with no spin, fundamental-group or dimension hypothesis; the aspherical corollary covers nonorientable manifolds via the double cover. The proof uses the local torical theorem of Cecchini-Schick as its terminal input and a new coordinate-shortening scheme with circle stabilization. Not refereed here. No Lean formalization.