Minimal-Dimensional Cocompact Models for Mapping Class Groups

Definitions and mathematical statement

The supplied record leaves the surface conventions and the range and quantification of the genus unspecified. Under the conventional interpretation, let $S_g$ be a closed, connected, orientable surface of integer genus $g$, with no boundary or marked points, and let \[ G_g=\operatorname{MCG}(S_g) =\pi_0\left(\operatorname{Homeo}^{+}(S_g)\right) \] be its group of orientation-preserving homeomorphisms modulo isotopy. These surface and group conventions are conditional interpretations requiring confirmation from the source.

Interpret a cell complex as a CW complex. A space $X$ is contractible if its identity map is homotopic to a constant map. For the discrete group $G_g$, interpret a continuous action on $X$ as proper if, for every compact subset $K\subseteq X$, the set \[ \{h\in G_g:hK\cap K\ne\varnothing\} \] is finite. The action is cocompact if $G_g\backslash X$ is compact. The dimension of $X$ is the supremum of its cell dimensions.

With these conventions, the existence question supplied by the record is: for an intended genus $g$, does there exist a contractible cell complex $X_g$, together with a proper cocompact action of $G_g$, such that \[ \dim X_g=4g-5? \] The dimension is stated as an equality. A nonempty CW complex has nonnegative dimension, so the conventional integer-genus formulation can only have an affirmative answer when $g\ge 2$. The record does not specify whether the question concerns every such genus, a fixed genus, or a narrower range. It also does not specify whether the action must preserve the cell structure.

The additional question, whether there are obstructions to the existence of such a complex and action, is retained as an open-ended request: no particular obstruction, invariant, or necessary condition is specified. The supplied dated assessment describes the extracted statement as only the fragment "Are there obstructions", whereas the supplied statement fields contain the fuller existence question above. This discrepancy prevents treating the fuller formulation as independently established source wording.

Short English statement

The supplied question asks whether the mapping class group of a genus-$g$ surface admits a proper cocompact action on a contractible cell complex of dimension exactly $4g-5$, and what might obstruct such an action. The genus quantifier, surface conventions, action requirements, and intended meaning of obstruction require review; the supplied open status is dated curation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001637$, source problem code catalogue entry, titled "Minimal-Dimensional Cocompact Models for Mapping Class Groups", in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2010$ and names no proposer.
• Supplied source citation: Teichmüller Theory (2011), Oberwolfach Reports, pp. 3085--3157, DOI 10.4171/owr/2010/53. Source URL and the supplied literature-source link labelled "original Oberwolfach problem report": https://doi.org/10.4171/owr/2010/53. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page is supplied.
• Supplied literature assessment, checked on 2026-08-21: status "open". It says that the extracted statement is only the fragment "Are there obstructions", does not specify a theorem, and that the underlying cell-complex existence question remains unresolved in the cited report. This is supplied dated curation derived from the supplied OWR material, not independently verified here; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification assertion is supplied metadata, not a verification performed for this draft.