Foliations and Diffeomorphism Groups — Question 64 (Hurder-Vogt [hurder1993product,hurder2005problem,langevin1992list])

Definitions and mathematical statement

Let $q$ denote the codimension parameter in the source question; the supplied record does not specify its allowed range. Write \[ SO(q)=\{A\in M_q({\mathbb{R}}):A^{\mathsf T}A=I_q,\ \det A=1\}, \] with its canonical action on ${\mathbb{R}}^q$ given by $(A,v)\mapsto Av$. A codimension-$q$ microfoliation on a manifold $M$ is a germ, along the zero section of a rank-$q$ vector bundle over $M$, of a foliation transverse to the bundle fibers.

Let $\gamma$ be the $\bar\Gamma_q$-structure on $SO(q)$ described in the source: its microfoliation is the suspension of this canonical action. Let \[ c_\gamma\colon SO(q)\longrightarrow B\bar\Gamma_q \] be a classifying map for this structure. The question is whether $c_\gamma$ is null-homotopic. Explicitly, does there exist a point $b\in B\bar\Gamma_q$ and a continuous map \[ H\colon SO(q)\times[0,1]\longrightarrow B\bar\Gamma_q \] such that, for every $A\in SO(q)$, \[ H(A,0)=c_\gamma(A),\qquad H(A,1)=b? \] The notation $\bar\Gamma_q$, its regularity and normal-bundle conventions, and the suspension construction are retained from the source. The supplied record does not define these conventions or provide enough construction details to specify them independently. The question is therefore conditional on those intended conventions; no additional choices are imposed here.

Short English statement

Does the structure on $SO(q)$ obtained by suspending its standard action on ${\mathbb{R}}^q$ have a classifying map homotopic to a constant map in $B\bar\Gamma_q$? The intended range of $q$ and the precise structure and suspension conventions require clarification from the source.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $800123$, exact source problem code catalogue entry. The registry labels the problem open; this is supplied curation, not an independently verified mathematical status.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 64. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source verification.
• Supplied attribution/context: Hurder--Vogt, with reference labels hurder1993product, hurder2005problem, and langevin1992list. The supplied record does not include their full bibliographic entries or separate URLs.