we-solve (beta)

Math-2468 · mathematics · Topology

Positive Scalar Curvature Obstruction for Aspherical Manifolds

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every closed smooth aspherical manifold $M$ of dimension $n\geq 4$, $M$ admits no Riemannian metric of positive scalar curvature.

Formulation references: [S1], [S2].

Short English statement

For every closed smooth aspherical manifold $M$ of dimension $n\geq 4$, $M$ admits no Riemannian metric of positive scalar curvature.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2468-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2468

The aspherical corollary claims nonexistence of positive-scalar-curvature metrics on every closed smooth aspherical manifold, including nonorientable ones. In particular it covers the stated dimension-$n\ge4$ obstruction.

Positive Scalar Curvature Obstruction for Aspherical Manifolds

OpenAI's claim

Integral negative scalar curvature: For each $n\ge3$, the manuscript claims a dimension-only positive lower bound on the integral of $(\mathrm{Scal}^-)^{n/2}$ in terms of real simplicial volume; the bound is uniform over closed oriented manifolds and their metrics.

Positive scalar curvature and inessentiality: OpenAI claims that positive scalar curvature forces the rational fundamental class to vanish under the classifying map, without spin or group restrictions; consequences include the aspherical obstruction, flatness rigidity at nonnegative scalar curvature, and zero real simplicial volume in the nonnegative case.

Mathematical statement of the claimed result

Mathematical claim 1.
For each integer $n\ge3$, some dimension-dependent $a_n>0$ is claimed to satisfy \[\int_M\left(\max\{0,-\operatorname{Scal}_g\}\right)^{n/2}\,dV_g\ge a_n\|M\|\] for each smooth metric $g$ on a closed smooth $n$-manifold $M$ which is oriented and connected. Here $\|M\|$ is the infimum of the sum of absolute coefficients over finite real singular cycles representing its fundamental class. Thus nonnegative scalar curvature forces $\|M\|=0$, but this implication alone does not treat all aspherical manifolds with vanishing simplicial volume.
Mathematical claim 2.
For a closed connected oriented smooth $n$-manifold, $n\ge2$, admitting a metric with $\operatorname{Scal}>0$, the classifying map $c_M:M\to B\pi_1(M)$ is claimed to satisfy \[(c_M)_*[M]=0\in H_n(B\pi_1(M);\mathbb Q).\] Its explicit aspherical corollary rules out positive scalar curvature on every closed smooth aspherical manifold, including nonorientable ones by passage to an orientable cover.

Outline of the argument

Integral negative scalar curvature: The companion vanishing result forces a negative Yamabe minimizer, which is normalized to scalar curvature minus one; Hölder then reduces the desired integral estimate to a volume bound. A bounded cocycle is represented through an equivariant probability map from the cover. Weighted graph deformations increase map energy while retaining the volume budget, and a coordinate-by-coordinate degree estimate pays for the detecting cocycle, yielding the uniform comparison.

Positive scalar curvature and inessentiality: An essential rational class is detected on an auxiliary cover at a finite word-metric scale. Compactly supported cohomology and relative duality produce a finite-dimensional comparison with a nonzero local degree. Transverse realization and scalar-curvature reduction contradict that degree; the nonnegative aspherical rigidity statement then uses splitting after the positive-curvature obstruction is established.

Formal verification

Integral negative scalar curvature: The pinned catalogue lists no Lean formalization for this result.

Positive scalar curvature and inessentiality: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →