Common Models for Whitney-Reversibility Counterexample Classes

Determine, for each \(i\in\{3,4,5\}\), whether the class \(\mathcal{C}_i\) of continua witnessing failure of the Whitney-reversible property for \(\mathcal{Q}_i\), or the class \(\mathcal{C}'_i\) witnessing failure of the strong Whitney-reversible property for \(\mathcal{Q}_i\), admits a common model.

Background

For each i∈{3,4,5}i\in\{3,4,5\}, the properties Qi\mathcal{Q}_i concern the existence of a block point, a non-shore point, or a strong center point, respectively. The paper proves that the classes witnessing failure of their sequential strong Whitney-reversibility properties admit no common model. It also explains that the corresponding common-model question for Whitney-reversibility and strong Whitney-reversibility remains unresolved, whereas the analogous questions for Q1\mathcal{Q}_1, Q2\mathcal{Q}_2, and Q6\mathcal{Q}_6 are already settled.

References

Thus, among $\mathcal{Q}_1, \ldots, \mathcal{Q}_6$, the question of whether the witnessing class for the Whitney-reversible property (resp., strong Whitney-reversible property) admits a common model remains open solely for $\mathcal{Q}_3, \mathcal{Q}_4$, and $\mathcal{Q}_5$. Hence, we naturally pose the following question. \begin{problem} For each $i \in {3, 4, 5}$, let $\mathcal{C}_i$ (resp., $\mathcal{C}_i'$) be the class of continua witnessing the failure of the Whitney-reversible property (resp., strong Whitney-reversible property) for $\mathcal{Q}_i$. Does $\mathcal{C}_i$ (or $\mathcal{C}_i'$) admit a common model?

— Whitney Properties and Whitney reversible properties of Cut and Non-Cut Points  (2609.31034 - Matsuhashi, 25 Sep 2026) in Section 5, Problem \ref{problemcommon}, following the paragraph beginning “Recall that $\mathcal{Q}_1$ and $\mathcal{Q}_2$...”