Sharp $hat A$-Genus Bound for the $kappa$-Invariant

Definitions and mathematical statement

Let $\mathcal C_m$ denote the class of compact spin manifolds of dimension $4m$ to which Theorem 4 of the cited Oberwolfach report applies. The complete hypotheses of that theorem and the admissible range of $m$ are not reproduced in the supplied record. For a closed smooth spin manifold $M$, its $\hat A$-genus is the characteristic number \[ \hat A(M)=\langle \hat A(TM),[M]\rangle, \] where $\hat A(TM)$ is the total $\hat A$-characteristic class of the tangent bundle and the pairing takes its component of degree $\dim M$. The source uses an invariant denoted by $\kappa(M)$, but the supplied record does not define this invariant.

The record reports the estimate \[ \forall M\in\mathcal C_m,\qquad |\hat A(M)|\leq 2^{2m-1}\kappa(M). \] The question is whether, for every admissible $m$, the stronger universal estimate \[ \forall M\in\mathcal C_m,\qquad |\hat A(M)|\leq 2^m\kappa(M) \] holds with the same hypotheses and the same invariant.

This formulation retains unresolved source dependencies: a definition of $\kappa$, the full hypotheses of Theorem 4, and the range of $m$ must be supplied before it is a standalone mathematical statement. In particular, the record says only "compact" and does not specify whether boundary is excluded. The displayed definition of $\hat A(M)$ applies to closed manifolds; if manifolds with boundary are intended, the necessary additional data and conventions are absent.

Short English statement

The supplied question asks whether the coefficient $2^{2m-1}$ in the reported bound for the absolute $\hat A$-genus can be replaced universally by $2^m$, under exactly the hypotheses of the source's Theorem 4. The definition of $\kappa$ and the complete hypotheses remain unresolved.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30002633$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2014.
• Supplied primary-source citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2015), Oberwolfach Reports, pp. 1991--2046, DOI 10.4171/owr/2014/36. https://doi.org/10.4171/owr/2014/36. The original question refers to Theorem 4. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Bär--Dahl conformal Laplacian paper. https://doi.org/10.1007/s00039-003-0419-6. The record supplies no fuller bibliographic entry or definition of $\kappa$.
• Supplied literature source: original OWR report. https://doi.org/10.4171/owr/2014/36.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open, attributes the coefficient $2^{2m-1}$ bound and the sharper-coefficient question to the primary source, and reports that no later proof of the sharper universal bound was located. This is supplied dated literature triage, not an independently verified assessment; its status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports checking it against the official Oberwolfach report to restore the source's hypotheses and notation. That verification claim is supplied metadata and has not been independently verified in this draft.