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Positive scalar curvature forces rational inessentiality
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Abstract
We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat. We also show that every closed oriented smooth manifold of positive dimension with nonnegative scalar curvature has zero real simplicial volume, proving the qualitative vanishing consequence of Gromov's conjectural comparison.
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Comments and discussion
- Roman Sauer, contribution to 100+ reactions to 100+ solutions (Proofs and Prompts, 8 October 2026). Sauer identifies the replacement of manifolds by measurable foliated objects, used in his work with Sabine Braun and subsequent work by Hannah Alpert, as a method appearing in result families 207 and 335. [family-level]
Related prior literature mentioned by others
- Sabine Braun and Roman Sauer, Volume and macroscopic scalar curvature (arXiv:2012.08999, 2020; revised 2021). Sauer's family-specific reaction identifies this work as a methodological antecedent; it proves analogues of the simplicial-volume bound and rational-essentiality obstruction with upper bounds on universal-cover unit-ball volumes replacing scalar-curvature lower bounds. [family-level]
Possibly relevant literature (found by the agent, not checked)
- M. Marcinkowski, Gromov positive scalar curvature conjecture and rationally inessential macroscopically large manifolds (arXiv:1408.0372; J. Topol. 9 (2016), 105–116). Constructs rationally inessential manifolds, some proved to admit no positive scalar curvature metric; relevant to the fact that Theorem 1.1 runs in one direction only, i.e. rational inessentiality does not imply positive scalar curvature.
- S. Chen, J. Chu and J. Zhu, Positive scalar curvature metrics and aspherical summands (arXiv:2312.04698, v3 Dec 2025; accepted by Duke Math. J. per arXiv comments). Shows that in dimensions 3–5 a connected sum of a closed aspherical manifold with an arbitrary non-compact manifold has no complete metric of nonnegative scalar curvature; a non-compact counterpart of the connected-sum statement in Corollary 1.4 (oriented target-map obstruction).
Digests and expositions
- Latent Seminar, Integral scalar curvature and simplicial volume (7 October 2026). This exposition covers both manuscripts, explaining the companion's rational-essentiality obstruction through local degree, graph shortening and a stabilized torical obstruction, and explicitly disclaims independent verification of the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 335
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026,
author = {{OpenAI}},
title = {{Positive scalar curvature forces rational inessentiality}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026}},
year = {2026}
}Family 335: Gromov’s integral scalar-curvature bound for simplicial volume
Proves $\int_M(\mathrm{Scal}_g^-)^{n/2}\,dV_g\ge a_n\lVert M\rVert$ for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with $a_n\gt 0$ depending only on dimension. Here $\mathrm{Scal}_g^-=\max\{0,-\mathrm{Scal}_g\}$ and $\lVert M\rVert$ is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.
Other papers in this family:
- An integral scalar curvature bound for real simplicial volume · October 5, 2026