Foliations and Diffeomorphism Groups — Question 68

Definitions and mathematical statement

Let $D\subseteq{\mathbb{Z}}_{\geq 0}$ denote the intended range of dimensions, which the supplied record does not specify, and write \[ X_n=\overline{\mathrm{B}\Gamma}_n^{\mathrm{PL}}\qquad(n\in D) \] for the spaces appearing in the source. The record does not define these spaces, the meaning of the overline, or the relevant piecewise-linear classifying-space construction; these definitions must be recovered from the original source before the question is fully specified.

A space $X$ is $(n+1)$-connected if it is nonempty and path-connected and, for every base point $x\in X$, \[ \pi_i(X,x)=0\qquad\text{for every integer }1\leq i\leq n+1. \] The question asks whether \[ \text{for every }n\in D,\qquad \overline{\mathrm{B}\Gamma}_n^{\mathrm{PL}} \text{ is }(n+1)\text{-connected}. \] The unspecified range $D$ and the missing construction of $X_n$ are unresolved source qualifications, not additional mathematical hypotheses supplied by this draft.

The source motivates the question through perfectness of groups of piecewise-linear transformations. A group $G$ is perfect if $G=[G,G]$, where $[G,G]$ is generated by commutators $[a,b]=aba^{-1}b^{-1}$. It is uniformly perfect if there is an integer $N\geq 0$ such that every element of $G$ is a product of at most $N$ commutators. For surfaces $\Sigma$, the source suggests that work of Bowden--Hensel--Webb could establish the anticipated implication \[ \operatorname{Homeo}_0(\Sigma,\mathrm{rel}\,\partial) \text{ is not uniformly perfect} \quad\Longrightarrow\quad \mathrm{PL}_{\circ}(\Sigma,\mathrm{rel}\,\partial) \text{ is not uniformly perfect}. \] This is expressed as a likely consequence, not an established result in the supplied record. The source then describes an $h$-principle approach to perfectness of the latter group and reports that an algebraic proof and a higher-dimensional generalization of the approach are not known to its speaker. The surface hypotheses, the precise relative-boundary convention, and the meaning of the subscript $\circ$ are not supplied; no identification of that subscript with an identity component is imposed here.

Short English statement

Question 68 asks whether the spaces $\overline{\mathrm{B}\Gamma}_n^{\mathrm{PL}}$ are $(n+1)$-connected in every intended dimension. The supplied excerpt leaves both the construction of these spaces and the dimension range unspecified. Its surface-group motivation includes an explicitly tentative claim about failure of uniform perfectness and a reported lack of an algebraic proof or higher-dimensional extension.

Sources

• Local UnsolvedMath registry, permanent ID $800127$, exact source problem code catalogue entry, titled "Foliations and Diffeomorphism Groups --- Question 68." The registry attributes the problem to Nariman and Yazdi, with proposed year 2025, and labels it open; this label is supplied metadata, not an independently verified assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 68. Supplied source URL: https://arxiv.org/abs/2503.18274. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of independent access or verification for this draft.
• The excerpt mentions work of Bowden--Hensel--Webb but supplies no title, bibliographic reference, or link identifying that work.