Positive Scalar Curvature for Spin Manifolds with Odd Fundamental Group

Definitions and mathematical statement

Let $M$ be a closed (compact without boundary), connected, smooth spin manifold of dimension $d\geq 5$, and let $G=\pi_1(M)$ be a finite abelian group of odd order. For a prime $p$, call $M$ $p$-atoral if, for every integer $\ell\geq 1$ and every $d$-tuple $c_1,\ldots,c_d\in H^1(M;{\mathbb{Z}}/p^\ell)$, \[ \langle c_1\smile\cdots\smile c_d,[M]_{p^\ell}\rangle=0\quad\text{in }{\mathbb{Z}}/p^\ell, \] where $[M]_{p^\ell}\in H_d(M;{\mathbb{Z}}/p^\ell)$ is the reduction of the fundamental class determined by the spin orientation. Assume this condition for every prime $p\mid |G|$. If $G$ is trivial, this collection of conditions is vacuous.

Using the standard real convention, write \[ \alpha(M)=\operatorname{Ind}_{C^*_{{\mathbb{R}}}G}\left(D_M\otimes\mathcal L_G\right)\in KO_d(C^*_{{\mathbb{R}}}G) \] for the Rosenberg invariant: the real higher index of the spin Dirac operator $D_M$, twisted by the flat bundle $\mathcal L_G=\widetilde M\times_G C^*_{{\mathbb{R}}}G$. Here $C^*_{{\mathbb{R}}}G$ is the real group $C^*$-algebra; its maximal and reduced completions agree for finite $G$.

The proposed statement is that, for every such spin manifold, \[ \alpha(M)=0\quad\Longrightarrow\quad \text{there exists a smooth Riemannian metric }g\text{ on }M \text{ with }\operatorname{Scal}_g(x)>0\text{ for every }x\in M. \] The supplied original formulation asks whether a non-spin positive-scalar-curvature existence theorem extends to this spin setting. The supplied cleaned statement identifies the invariant as the Rosenberg invariant; the real convention above makes that interpretation explicit, although the record does not specify its coefficient convention. This is a proposed implication requiring mathematical review, not an asserted theorem.

Short English statement

Does every closed connected smooth spin manifold of dimension at least five, with finite abelian fundamental group of odd order and the stated vanishing of all top-degree products of degree-one classes over ${\mathbb{Z}}/p^\ell$ for each relevant prime and every $\ell\geq1$, admit a metric of everywhere positive scalar curvature whenever its Rosenberg invariant vanishes? The supplied record labels this spin extension open.

Sources

• Local UnsolvedMath registry, permanent ID $30004389$, source problem code catalogue entry, titled "Positive Scalar Curvature for Spin Manifolds with Odd Fundamental Group," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied primary citation: Manifolds and Groups (2021), Oberwolfach Reports, pp. 417--463, DOI 10.4171/owr/2020/7. https://doi.org/10.4171/owr/2020/7. The record says the official report was checked and labels the statement corrected_verified; these are supplied verification claims, not independent verification for this draft.
• Supplied literature source labeled "Hanke 2021": https://doi.org/10.2140/gt.2021.25.497. No article title or precise theorem locator is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes the spin extension with vanishing alpha-invariant as conjectural, and identifies connective K-homology as the missing method. This assessment is supplied dated curation derived from the Oberwolfach material, not independently verified here; its time-sensitive status and cited claims require independent checking.