Equivariant Nullhomotopy of the Regular-Map Derivative

Definitions and mathematical statement

Conjecture C concerns every positive odd integer $n$. Let \[ R=\operatorname{reg}(D^n\times D^2,D^2) \] be the space of regular smooth maps equal to the projection near the boundary. In the usual rank convention, regular means the derivative is surjective everywhere. Use the source's smooth topology, boundary/corner convention and standard tangent trivializations. The derivative takes values in the space of surjective maps ${\mathbb{R}}^{n+2}\to{\mathbb{R}}^2$, which retracts to the two-frame Stiefel space \[ V=O(n+2)/O(n). \] Since the derivative is the fixed projection near the boundary, collapsing the domain boundary gives the based derivative map \[ \delta:R\longrightarrow\Omega^{n+2}V. \] This explains the target, conditional on the indicated trivializations and disk-product identification, without replacing the source's exact model.

The distinguished subspace is \[ R_1=\operatorname{reg}_1(D^n\times D^2,D^2) =\{f\in R:q_2\circ f=q_{n+2}\}. \] The maps $q_2,q_{n+2}$ and their common codomain are not defined in the record; their compatibility with the boundary projection cannot be checked from their symbols alone. In particular no norm-square or quotient-map interpretation is guessed.

The conjecture asks for a rational nullhomotopy of $\delta$ which both satisfies the stated derived $SO(2)$-equivariance condition and extends the preferred rational nullhomotopy on $R_1$. In an appropriate rationalized target $Y_{{\mathbb{Q}}}$ for $Y=\Omega^{n+2}V$, the underlying homotopy would have \[ H(f,0)=r\delta(f),\qquad H(f,1)=*,\qquad H|_{R_1\times[0,1]}=H_{\mathrm{pref}}, \] where $r$ and the preferred homotopy are those of the source's rational framework. Exact restriction versus relative homotopy and component/localization conventions are not supplied. Vanishing induced rational homotopy maps is not silently substituted for existence of this relative nullhomotopy.

The $SO(2)$ actions and the word derived are essential: strict equivariance, homotopy-coherent equivariance and a condition after homotopy orbits/fixed points are different notions. Neither a nonequivariant nullhomotopy nor an arbitrary choice on $R_1$ satisfies the full requested statement. Their precise definitions and $H_{\mathrm{pref}}$ are absent.

The literature triage checked 21 August 2026 reports no authoritative resolution of the extracted claim. It supplies no further theorem, construction or definitions; its dated status is not independently verified here.

Short English statement

For odd $n$, nullhomotope the derivative map of boundary-standard regular maps after rationalization, with the source's derived rotation equivariance and a prescribed extension on its distinguished subspace. Both extra conditions matter, but the $q$-maps, actions and preferred homotopy are not supplied.

Sources

• Local UnsolvedMath registry, permanent ID 30000587, exact source code catalogue entry; Equivariant Nullhomotopy of the Regular-Map Derivative. Proposed year 2006, status open. The corrected_verified statement restores Conjecture C and both additional nullhomotopy conditions; claimed inspection is metadata.
• Supplied report citation: Singularities (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/42, pp. 2505--2566, https://doi.org/10.4171/owr/2006/42. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2006/42, and reports no authoritative resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no separate detailed construction reference is supplied.