Duplicate Reduced-Structure-Group Lower Bound

Definitions and mathematical statement

The supplied record poses the following lower-bound question. Let $M$ be a compact oriented manifold of dimension $4k-1$, with $k>1$, and write $G=\pi_1(M)$. Is it always true that \[ \operatorname{rank}\widetilde S(M)\geq N_{\mathrm{fin}}(G)? \] Here $G$ denotes the fundamental group of $M$. The supplied record does not define the reduced structure group $\widetilde S(M)$, its rank convention, or the invariant $N_{\mathrm{fin}}(G)$. If the intended structure group is abelian, its usual rank would be $\dim_{{\mathbb{Q}}}(\widetilde S(M)\otimes_{{\mathbb{Z}}}{\mathbb{Q}})$, but applicability of that convention is unresolved. The record also does not specify the manifold category, whether boundary is allowed, connectedness or basepoint conventions, or explicitly the domain of $k$. These omissions prevent a self-contained precise formulation; the displayed inequality preserves the supplied question without supplying those missing hypotheses or definitions.

The extraction is marked unrecoverable: its supplied verification note identifies it as a duplicate of the reduced-structure-group rank conjecture represented canonically, with definitions, by OWR-13686-009. Those definitions are not included in this record, and no separate clean statement is promoted here.

The supplied dated literature assessment labels the problem partially solved and says that the conjectured lower bound is proved under a residually finite quotient with torsion-free kernel, while the general conjecture remains unproved. The precise additional hypotheses and theorem statement are not supplied, so this qualification cannot be formulated here as a precise theorem. This is supplied curation, not an independently verified mathematical or literature conclusion.

Short English statement

The question asks whether the rank of a reduced structure group of a compact oriented $(4k-1)$-manifold, for $k>1$, is bounded below by an unspecified finite-torsion invariant of its fundamental group. The record is a duplicate of OWR-13686-009; missing definitions and manifold conventions require review. Its partially solved label and special-case result are supplied dated assessments.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30002903$, exact source code catalogue entry, title Duplicate Reduced-Structure-Group Lower Bound, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2015; no individual proposer is supplied. The supplied verification note points to OWR-13686-009 as the canonical formulation with definitions.
• Supplied source citation: Noncommutative Geometry (2016), Oberwolfach Reports, pp. 1631--1700, DOI 10.4171/owr/2015/29. Original Oberwolfach report: https://doi.org/10.4171/owr/2015/29. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's year label and the year in the DOI differ; this has not been resolved.
• Supplied literature source labeled "Weinberger and Yu primary paper": https://doi.org/10.2140/gt.2015.19.2767. No paper title, theorem number, or precise special-case hypotheses are supplied.
• Literature assessment date: 2026-08-21. The supplied dated curation reports partial resolution under a residually finite quotient with torsion-free kernel and says the general conjecture remains unproved. It describes this assessment as literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims. This assessment is preserved as supplied dated curation, not independently verified; neither linked source was visited for this draft.