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Math-2681 · mathematics · Geometry

Positive scalar curvature and codimension-two width

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $n\geq 4$, is the codimension-two Urysohn width of all connected complete smooth boundaryless n-manifolds with $Scal\geq 1$ bounded by one constant depending on n only? Urysohn width means the infimum of maximal fiber diameter over continuous maps to simplicial complexes of dimension $\leq n-2$.

Short English statement

For every $n\geq 4$, is the codimension-two Urysohn width of all connected complete smooth boundaryless n-manifolds with $Scal\geq 1$ bounded by one constant depending on n only? Urysohn width means the infimum of maximal fiber diameter over continuous maps to simplicial complexes of dimension $\leq n-2$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2681-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2681

Pointwise scalar curvature and width: The paper claims dimension-only control of the full fibers of a continuous map from any complete scalar-positive $n$-manifold to an $(n-2)$-complex, for $n\ge4$. It also gives the stated macroscopic-dimension consequence on universal covering spaces when the closed base admits a positive-scalar-curvature metric. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Positive scalar curvature and codimension-two width

OpenAI's claim

Pointwise scalar curvature and width: The paper claims dimension-only control of the full fibers of a continuous map from any complete scalar-positive $n$-manifold to an $(n-2)$-complex, for $n\ge4$. It also gives the stated macroscopic-dimension consequence on universal covering spaces when the closed base admits a positive-scalar-curvature metric.

Mathematical statement of the claimed result

Mathematical claim 1.
For every $n\ge4$ there is $C_n<\infty$ such that a connected complete smooth boundaryless $n$-manifold with $\operatorname{Scal}_g\ge1$ has a continuous map $f:M\to K$, where $K$ is a simplicial complex with the usual polyhedral topology and $\dim K\le n-2$, with \[\sup_{y\in K}\operatorname{diam}_g f^{-1}(y)\le C_n.\] Entire fibers, rather than their individual components, are bounded. For smooth Riemannian $(M,g)$ of dimension $n\ge4$, assume $M$ is closed and connected and $\operatorname{Scal}\ge\sigma^2>0$, the corollary gives $\operatorname{FillRad}(M,g)\le C_n/(2\sigma)$. A separate stated corollary covers all closed positive-scalar-curvature dimensions $n\ge2$: the universal-cover geometry obtained by lifting that metric has continuous macroscopic dimension at most $n-2$. The filling radius uses the integral fundamental class when the manifold is orientable and its mod-two fundamental class otherwise.

Outline of the argument

Pointwise scalar curvature and width: A separated net supplies labeled inner and outer balls in the original metric. Compatible walls and inserted cylinders are constructed while a nearby collapse map keeps track of those original positions. The final walls have no intersection of $n-1$ different labels, so their sign coordinates define an $(n-2)$-dimensional map; shared coordinates place the entire fiber in one uniformly bounded original ball.

Formal verification

Pointwise scalar curvature and width: 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

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