The Gromov–Lawson aspherical conjecture

The Gromov–Lawson aspherical conjecture asserts that no closed aspherical manifold admits a Riemannian metric of positive scalar curvature. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims the conjecture in every dimension, as a consequence of a homological statement: every closed oriented manifold with positive scalar curvature is rationally inessential. A companion manuscript dated 5 October 2026 uses its vanishing theorem for simplicial volume to claim Gromov's 1986 integral inequality, which bounds the simplicial volume of a closed manifold by the integral of the negative part of its scalar curvature. The collection's contents file lists both manuscripts under the result Gromov's integral scalar-curvature bound for simplicial volume. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the introductions of the manuscripts:
  • A closed connected oriented manifold is rationally essential if the classifying map of its universal cover sends its rational fundamental class to a nonzero class in the rational homology of the classifying space of its fundamental group, and rationally inessential otherwise. Closed oriented aspherical manifolds are rationally essential, because for them the classifying map is a homotopy equivalence.
  • Schoen and Yau's minimal-hypersurface method of 1979 gives obstructions to positive scalar curvature without a spin assumption in dimensions at most seven, including for manifolds that map with nonzero degree to tori. Gromov and Lawson's Dirac-operator obstructions of 1983, based on enlargeability, have no dimension bound but use spin coverings.
  • Chodosh and Li proved the aspherical conjecture in dimensions four and five using generalized soap bubbles (Annals of Mathematics 199, 2024). Gromov independently treated closed five-dimensional manifolds (arXiv:2009.05332).
  • Hanke formulated the conjecture that rationally essential manifolds admit no metric of positive scalar curvature (Conjecture 2.9 of Positive scalar curvature, K-area and essentialness, in Global Differential Geometry, 2012). The manuscript also identifies this statement with the classifying-map form of a conjecture in Gromov's Four lectures on scalar curvature and with a problem in Yau's 2000 survey Review of geometry and analysis.
  • Gromov introduced simplicial volume in 1982. In Large Riemannian manifolds (Lecture Notes in Mathematics 1201, 1986; Conjecture 3.A, Remark (B), Equation (8′)) he asked for a dimension-dependent bound of the simplicial volume by the integral of (Scal⁻)^n/2, where Scal⁻ is the negative part of the scalar curvature.
  • Ma, Wang, Xie, Yu and Zhu proved that nonnegative scalar curvature forces zero simplicial volume for closed oriented manifolds with spin universal cover (arXiv:2606.29135v4). Min, Zheng and Zhu proved the integral inequality for every metric on every closed Kähler surface, with coefficient 27/2 in the upper bound for simplicial volume; their Lemma 2.1 states the equivalence of the integral form and the uniform-volume form through the Yamabe problem (arXiv:2608.17335v1).

The original results

• Positive scalar curvature forces rational inessentiality (23 September 2026). Theorem 1.1 of the manuscript states that every closed connected oriented smooth manifold of dimension n ≥ 2 admitting a metric of strictly positive scalar curvature is rationally inessential. The manuscript notes that the theorem has no spin, universal-cover spin or fundamental-group hypothesis and no upper dimension bound. Its corollaries state:
  • the Gromov–Lawson aspherical conjecture: no closed smooth aspherical manifold, orientable or not, admits a metric of positive scalar curvature;
  • every metric of nonnegative scalar curvature on a closed aspherical manifold is flat;
  • a closed connected oriented manifold that maps with nonzero degree to a closed connected oriented aspherical manifold of the same dimension admits no metric of positive scalar curvature, the oriented form of Conjecture 18 in Chodosh and Li's survey Recent results concerning topological obstructions to positive scalar curvature; in particular no connected sum N # P with N aspherical admits such a metric;
  • every closed oriented smooth manifold of positive dimension admitting a metric of nonnegative scalar curvature has zero real simplicial volume.
  • Method, as described: a comparison at a large finite scale turns the detecting cohomology class into a map of nonzero local degree on a cover; graph deformations with warped circles shorten this map while the scalar-curvature loss grows only logarithmically in the number of coordinates; the local torical theorem of Cecchini and Schick supplies the final contradiction.
  • Source: the manuscript Positive scalar curvature forces rational inessentiality.
• An integral scalar curvature bound for real simplicial volume (5 October 2026). Theorem 1.1 of the manuscript states that for every integer n ≥ 3 there is a constant a_n > 0, depending only on n, such that every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g on M satisfy ∫_M (Scal_g⁻)^n/2 dV_g ≥ a_n ‖M‖, where Scal_g⁻ = max{0, −Scal_g} and ‖M‖ is the real simplicial volume.
  • The manuscript states that it uses the simplicial-volume vanishing theorem of the companion manuscript, and that a conformal reduction through the Yamabe problem leaves the bound ‖M‖ ≤ C_n Vol_g(M) for metrics of scalar curvature −1.
  • Method, as described: a bounded cocycle realized as a differential form on a simplex of probability vectors; joint deformations of the metric and an equivariant probability map, controlled by Perelman's weighted scalar curvature; degree estimates through μ-bubbles for measured families; and an adaptation of Gaifullin's reflected permutohedral realization of cycles.
  • The theorem asserts the existence of a_n. The introduction describes a lower bound required in the proof as bounded by an exponential tower whose fixed height depends only on n.
  • Source: the manuscript An integral scalar curvature bound for real simplicial volume.
• Formalization. The collection has no Lean scope note (lean/docs/) or Comparator challenge for these manuscripts.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 18 October 2026)
2026-10-16	J. Jardine, Lattice24, referee review of Positive scalar curvature forces rational inessentiality in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	The referee review reports no gap after reading all eight sections, and states that the scalar-curvature identity of Proposition 4.1 was checked symbolically and numerically. Its two minor notes say that the argument ends in the local torical inequality of Cecchini and Schick, applied in unbounded dimension, whose proof above dimension eight rests on Schoen and Yau's 2017 minimal slicing theory, and that the existence argument for the graph equation on noncompact covers (Theorem 5.2) is only sketched.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.
2026-10-16	J. Jardine, Lattice24, referee review of An integral scalar curvature bound for real simplicial volume in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	The referee review reports no gap after reading all six sections. Its two minor notes say that the argument depends on the companion manuscript's vanishing theorem, which it cites rather than proves, and that the quantitative Gaifullin-type resolution (Lemma 6.4) and the compactness and selection arguments for singular μ-bubbles (Section 5.4) were checked only in outline, with no specific failure found.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.