Higher Homotopy of Positive Ricci Observer Moduli Spaces

Definitions and mathematical statement

For an integer $d\geq 2$, let $S^d$ be the smooth sphere and let \[ \mathcal R_{\operatorname{Ric}>0}(S^d) =\{g:\ g\text{ is a smooth Riemannian metric on }S^d,\ \operatorname{Ric}_g(v,v)>0\text{ for every nonzero tangent vector }v\}, \] with the $C^\infty$ topology. Fix an observer point $x_0\in S^d$, and define \[ \operatorname{Diff}_{x_0}(S^d) =\{\varphi\in\operatorname{Diff}(S^d):\varphi(x_0)=x_0,\ D\varphi_{x_0}=\operatorname{id}_{T_{x_0}S^d}\}. \] Using the standard observer convention, define the observer moduli space, with its quotient topology, by \[ \mathcal M^{\operatorname{Ric}>0}_{x_0}(S^d) =\mathcal R_{\operatorname{Ric}>0}(S^d)/\operatorname{Diff}_{x_0}(S^d), \] where diffeomorphisms act by pullback of metrics. For a chosen metric $g\in\mathcal R_{\operatorname{Ric}>0}(S^d)$, write $[g]$ for its image in this quotient.

Question, with unresolved parameters. Do the higher homotopy groups of this observer moduli space contain nonzero elements? Interpreting "higher" as degrees $k\geq 2$, the based mathematical question for a specified dimension and component is whether \[ \exists\,k\geq 2\qquad \pi_k\!\left(\mathcal M^{\operatorname{Ric}>0}_{x_0}(S^d),[g]\right)\neq 0. \] The supplied record does not specify the quantifier or range for $d$, the base component $[g]$, or whether "higher" includes $k=1$. Thus this display makes the degree-existence reading explicit for fixed parameters; it does not resolve the intended dimension or component quantifiers. The observer convention and topology above are standard definitional conventions requiring confirmation against the cited passage. The restriction $d\geq 2$ gives the usual positive-Ricci sphere setting; in dimension one the Ricci tensor vanishes.

The supplied dated assessment labels the problem partially solved and attributes infinite-order classes in stable degrees on sufficiently high odd-dimensional spheres to Botvinnik, Walsh, and Wraith, while describing the general higher-homotopy question as open. It supplies no precise stable range, dimension threshold, or base-component hypotheses. These statements are supplied literature qualifications, not independently verified results here, and they do not determine a precise remaining universal conjecture.

Short English statement

The question asks whether a sphere's space of positive-Ricci metrics, modulo diffeomorphisms fixing an observer point and its tangent data, has nonzero higher homotopy classes. The intended dimensions, degrees, and base components are unspecified. Supplied dated curation reports partial results on sufficiently high odd-dimensional spheres; the exact remaining target requires review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003347$, titled Higher Homotopy of Positive Ricci Observer Moduli Spaces, in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. The record gives proposed year 2017 and names no proposer.
• Supplied source 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. This is also the supplied literature link labeled "Oberwolfach report on observer moduli."
• Supplied literature reference: Homotopy groups of the observer moduli space of Ricci-positive metrics. https://doi.org/10.2140/gt.2019.23.3003. The supplied assessment attributes the cited partial results to Botvinnik, Walsh, and Wraith.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. It reports infinite-order classes in stable degrees for sufficiently high odd-dimensional spheres and labels the general question open and the problem partially solved. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official passage was checked and rewritten while preserving mathematical content. These are supplied provenance claims; no source was visited or independently checked for this draft.