Diffeomorphism Group Homotopy in Positive Ricci Metric Spaces

Definitions and mathematical statement

For a smooth manifold $M$, let \[ \mathcal R_{\operatorname{Ric}>0}(M)=\{g\text{ a smooth Riemannian metric on }M:\operatorname{Ric}_g(v,v)>0\text{ for every }p\in M\text{ and }0\ne v\in T_pM\}. \] Let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms of $M$, acting on metrics by pullback. A space of metrics consists of the metrics themselves; its moduli space is a quotient by a specified diffeomorphism group. These are distinct possible homotopy-theoretic targets. Write $S^d$ for the smooth $d$-dimensional sphere.

Source question, with unresolved qualifications. Can some stable rational homotopy of a diffeomorphism group, described in the supplied record as known to occur in the moduli space of pinched metrics on $S^d$, also occur, for some manifold $M$, in $\mathcal R_{\operatorname{Ric}>0}(M)$?

Here "occur" retains the source's unresolved meaning: the record does not specify a map, injection, image, or other relation realizing the diffeomorphism-group classes in the positive-Ricci metric space. It also does not specify the diffeomorphism group, stabilization, stable degree range, sphere dimension, pinching condition, basepoints, or topologies. No compactness, connectedness, dimension, boundary hypothesis, or boundary condition for $M$ is supplied. Consequently this is a qualified existential research question, not a uniquely specified homotopy-theoretic assertion. In particular, replacing the target by a moduli space, or replacing the diffeomorphism group by its classifying space, would require additional information and may change the question.

Short English statement

The question asks whether some stable rational homotopy associated with diffeomorphisms, reportedly visible in a moduli space of pinched sphere metrics, can also be realized in a space of positive-Ricci-curvature metrics on some manifold. The realization mechanism and several essential hypotheses remain unspecified. The supplied open status is dated curation, not an independently verified mathematical conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003363$, source problem code catalogue entry, title "Diffeomorphism Group Homotopy in Positive Ricci Metric Spaces," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The proposed year is recorded as 2017.
• Supplied report citation: Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics (2018), Oberwolfach Reports, pp. 133--166, DOI 10.4171/owr/2017/3. https://doi.org/10.4171/owr/2017/3. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied literature list also labels this reference "Oberwolfach discussion of positive-Ricci moduli."
• Supplied literature reference: The space of positive Ricci curvature metrics on spin manifolds. https://arxiv.org/abs/2009.06199.
• Supplied literature reference: Topology of the space of Ricci-positive metrics. https://mural.maynoothuniversity.ie/id/eprint/15508/1/DW-positivericci-2020.pdf.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and reports nontrivial path components and some higher-topological results for positive-Ricci metric spaces on special manifolds, while reporting no theorem realizing the stable rational homotopy from diffeomorphism groups requested here. This is supplied dated curation derived from the OWR record, not independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that notation, typography, and sentence structure were repaired without changing mathematical content. This supplied editorial label does not resolve the mathematical ambiguities above.