Positive Scalar Curvature Through Odd-Order Finite Covers

Definitions and mathematical statement

A closed smooth manifold means a compact smooth manifold without boundary. A Riemannian metric $g$ on a smooth manifold $X$ has positive scalar curvature if its scalar curvature satisfies $\operatorname{Scal}_g(x)>0$ for every $x\in X$.

For every integer $n\geq 5$, let $M$ be a closed connected smooth $n$-manifold, choose a base point $m_0\in M$, and suppose that $\pi_1(M,m_0)$ is finite with odd order. Let $p\colon\widetilde M\to M$ be its universal covering, with the induced smooth structure. Is the following implication always true? \[ \left(\exists\text{ a smooth Riemannian metric }\widetilde g\text{ on }\widetilde M\quad\forall x\in\widetilde M,\quad\operatorname{Scal}_{\widetilde g}(x)>0\right) \Longrightarrow \left(\exists\text{ a smooth Riemannian metric }g\text{ on }M\quad\forall y\in M,\quad\operatorname{Scal}_{g}(y)>0\right). \] The metric $\widetilde g$ in the hypothesis is not assumed invariant under deck transformations, and the conclusion does not require $p^*g=\widetilde g$.

The restriction $n\geq5$ follows the supplied corrected statement. The original transcription omitted a dimension restriction; the supplied verification note says that this hypothesis was restored after checking the official report and that the analogous assertion fails in dimension four. These are supplied source-verification claims, not checks performed for this draft. No additional orientability or spin hypothesis is specified.

Short English statement

For every closed connected smooth manifold of dimension at least five with finite fundamental group of odd order, does the existence of a positive scalar curvature metric on its universal cover imply the existence of such a metric on the manifold itself? This is a statement-only draft requiring human mathematical review; the supplied record labels the problem partially solved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004388$, title: "Positive Scalar Curvature Through Odd-Order Finite Covers," in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. The record lists proposed year $2020$ and supplies no individual proposer.
• Supplied source citation: Manifolds and Groups (2021), Oberwolfach Reports, DOI $10.4171/owr/2020/7$, pp. 417--463. Source URL and supplied literature reference "Oberwolfach report": https://doi.org/10.4171/owr/2020/7. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference "Hanke 2021": https://doi.org/10.2140/gt.2021.25.497. No article title or more detailed attribution is supplied in the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and states that the odd-order finite-fundamental-group positive-scalar-curvature conjecture is proved in dimensions $1$--$3$ and several higher-dimensional classes, but remains open in general dimensions at least $5$. This is supplied dated curation, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked, restoring the missing dimension hypothesis and noting failure of the analogous assertion in dimension four. This label and verification account are supplied provenance, not independent verification by this draft.