Generic Point Property for Exceptional Surface Homeomorphisms

Definitions and mathematical statement

Let \[ S^2=\{x\in{\mathbb{R}}^3:\|x\|=1\},\qquad \mathbb{RP}^2=S^2/(x\sim -x). \] For either surface $M$, equip the group $\operatorname{Homeo}(M)$ of self-homeomorphisms with the compact-open topology, and write $\operatorname{Homeo}_0(M)$ for its identity component. These are the conventional interpretations of the notation; the supplied extract does not explicitly specify them.

A $G$-flow is a nonempty compact Hausdorff space $X$ with a continuous action $G\times X\to X$. It is minimal if $\overline{G\cdot x}=X$ for every $x\in X$. A point $x\in X$ is generic if its orbit $G\cdot x$ is comeager in $X$, meaning that $X\setminus(G\cdot x)$ is a countable union of nowhere dense subsets of $X$.

Source questions. Does $\operatorname{Homeo}_0(S^2)$ have the generic point property? Independently, does $\operatorname{Homeo}_0(\mathbb{RP}^2)$ have the generic point property?

Unresolved scope of the property. The supplied record does not define the generic point property or specify the class of flows over which it quantifies. To make the missing convention explicit, let $\mathcal C$ be a specified class of nonempty compact Hausdorff spaces. The corresponding precise property is \[ \begin{gathered} \text{for every }X\in\mathcal C\text{ and every continuous minimal action }G\curvearrowright X,\\ \text{there exists }x\in X\text{ such that }G\cdot x\text{ is comeager in }X. \end{gathered} \] The two questions ask whether this property holds for each of the two indicated groups, with the source's intended class $\mathcal C$. In particular, whether the intended quantification includes all compact Hausdorff flows or only compact metrizable flows must be checked in the cited source before this expansion is treated as definitive; no choice between these formulations is asserted here.

The supplied extract also states Corollary 4: if $X$ is a closed surface other than the sphere or the real projective plane, then $\operatorname{Homeo}_0(X)$ does not have the generic point property. This is retained as supplied source context, without independent verification or extension to the two exceptional surfaces.

Short English statement

The problem asks separately whether the identity components of the homeomorphism groups of the sphere and the real projective plane have the generic point property. In terms of generic orbits, the intended question concerns the existence of a comeager orbit in every minimal flow in the source's specified class. That class is absent from the supplied extract and requires verification. The supplied curation labels both questions open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006162$, source problem code catalogue entry, title "Generic Point Property for Exceptional Surface Homeomorphisms." The record assigns the problem to 2025 and supplies no individual proposer.
• Set Theory (2025), Oberwolfach Reports, DOI 10.4171/owr/2025/2, pp. 75--140. Supplied source URL: https://doi.org/10.4171/owr/2025/2. The supplied extract identifies the problem as Question 5, with the internal reference "[1], Question 1.3," and includes Corollary 4 concerning the other closed surfaces. The extract does not supply the bibliography entry corresponding to [1].
• Supplied literature source: "Oberwolfach Report 2/2025," https://content.ems.press/assets/public/full-texts/serials/owr/22/1/14299082/online/10.4171-owr-2025-2.pdf.
• Supplied literature source: Basso--Zucker, Topological groups with tractable minimal dynamics, https://arxiv.org/abs/2412.05659. The supplied record does not explicitly identify this reference with the extract's bibliography entry [1].
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. Its status label is "open," and its assessment says that the report proves failure of the generic point property for all other closed surfaces while leaving the sphere and real projective plane unresolved. The background describes this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims; this draft does not independently confirm them.