Foliations and Diffeomorphism Groups — Question 72

Definitions and mathematical statement

Let $X=B\overline{\Gamma}^{\omega}_{1}$, using the notation of the supplied source, and let $S^{1}=\{z\in{\mathbb{C}}:|z|=1\}$. A loop in $X$ is a continuous map $\ell:S^{1}\to X$. It is homotopically nontrivial if it is not homotopic to a constant map. Two loops $\ell,\tau:S^{1}\to X$ are freely homotopic if there exists a continuous map $H:S^{1}\times[0,1]\to X$ such that $H(z,0)=\ell(z)$ and $H(z,1)=\tau(z)$ for every $z\in S^{1}$; no basepoint is required to remain fixed.

Let $\mathcal T$ denote the collection of loops in $X$ that are transverse to the $\Gamma^{\omega}_{1}$-structure intended by the source. This is a source-dependent predicate: the supplied record does not define the structure, its realization on $X$, or the regularity and transversality conditions imposed on these loops.

Question 72 (Nariman and Yazdi, 2025). Does there exist a continuous loop $\ell:S^{1}\to X$ such that \[ \ell\text{ is not null-homotopic} \quad\text{and}\quad \forall\tau\in\mathcal T,\quad \ell\text{ is not freely homotopic to }\tau? \]

The source precedes this question with the assertion that a loop in $B\Gamma_H$ which is freely homotopic to a loop transverse to the $\Gamma^{\omega}_{1}$-structure represents a nontrivial element of infinite order in \[ \pi_1\left(B\overline{\Gamma}^{\omega}_{1}\right)=\Gamma_H. \] This assertion is retained as supplied background, without independent verification. The record does not define $\Gamma_H$, specify the basepoint for this fundamental group, or explain the relationship between $B\Gamma_H$, used in the background assertion, and $B\overline{\Gamma}^{\omega}_{1}$, used in the question. Consequently, the displayed question is a formulation relative to the source's intended spaces and transversality predicate; those geometric identifications remain unresolved.

Short English statement

The question asks whether $B\overline{\Gamma}^{\omega}_{1}$ contains a non-null-homotopic loop whose free homotopy class has no transverse representative. The supplied background says that classes admitting transverse representatives have infinite order, but it uses a different space notation whose relationship to the question's space is unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $800131$, source problem code catalogue entry, title: Foliations and Diffeomorphism Groups --- Question 72. The registry attributes the problem to Nariman and Yazdi, with proposed year 2025, and labels it open; this is a supplied status label, not an independently verified assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 72. 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 details, not a claim of access or verification for this draft.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.