Foliations and Diffeomorphism Groups — Question 3

Definitions and mathematical statement

The supplied record asks for two properties of an object denoted by $Z$, but supplies no definition of $Z$, its ambient parameter space, or its corners. Consequently, it does not determine a complete mathematical proposition.

The requested assertions can be recorded schematically as follows. Write $\mathcal C(Z)$ for the intended set of corners, $\mathcal R$ for the intended set of rational points, and $\mathcal S$ for the parameter points intended by the phrase "$S^1\times S^2$ surgeries." These symbols are placeholders for missing source definitions. The requested assertions are \[ \mathcal C(Z)\subseteq\mathcal R, \qquad \operatorname{Acc}_{X}\left(\mathcal C(Z)\right)=\mathcal S, \] where $X$ would have to be the intended ambient topological space. With the usual meaning of accumulation point, \[ \operatorname{Acc}_{X}(A) = \left\{x\in X:\text{every open neighborhood }U\text{ of }x \text{ satisfies }U\cap\left(A\setminus\{x\}\right)\ne\varnothing\right\}. \] Thus the second assertion requires both that every accumulation point of the corners correspond to such a surgery and that every such surgery correspond to an accumulation point.

Here $S^n=\{x\in{\mathbb{R}}^{n+1}:\sum_{i=1}^{n+1}x_i^2=1\}$ denotes the standard sphere. However, the source record does not specify the surgery construction, the initial manifold or other surgery data, or how surgeries are identified with points of $X$. It also does not specify coordinates defining rationality, a definition of corner, or whether accumulation is taken in a parameter space, its closure, or a compactification. These unresolved choices must be recovered from the source before the schematic assertions can become a precise statement.

Short English statement

The source asks whether every corner of $Z$ is rational and whether its corner accumulation points are exactly the points corresponding to $S^1\times S^2$ surgeries. The supplied excerpt omits the definitions and parameter space needed to formulate these assertions precisely.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800062$, exact source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 3." The registry labels this problem open; that label is supplied metadata, not an independently verified mathematical or literature assessment. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 3, as attributed in the supplied record. 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 access or verification.