Disconnected Preimages of Subsurfaces Under Finite Covers

Definitions and mathematical statement

The notation for the surfaces is not explicitly defined in the supplied record. Under the conventional interpretation, let $S_j$ denote a closed, connected, orientable surface of genus $j$, and let a finite cover mean an unbranched covering map with finitely many sheets. The formulation below is conditional on these conventions; whether they match the source requires review.

For $k\in\{3,4\}$, a $k$-holed sphere in $S_g$ is a compact embedded subsurface $H\subset S_g$ homeomorphic to a sphere with the interiors of $k$ pairwise disjoint closed disks removed. Require every component of $\partial H$ to be essential in $S_g$, meaning that it is not null-homotopic in $S_g$. No requirement that distinct boundary components be nonisotopic is added. For a subset of a surface, $\operatorname{int}_{S_j}$ denotes its interior in that surface.

For each $k\in\{3,4\}$, consider the assertion \[ \begin{gathered} \text{For every pair of integers }h>g\geq 2 \text{ and every finite covering map }f:S_h\longrightarrow S_g,\\ \text{there exists a }k\text{-holed sphere }H\subset S_g \text{ whose boundary components are essential in }S_g\\ \text{such that }\operatorname{int}_{S_h}\!\left(f^{-1}(H)\right) \text{ is disconnected.} \end{gathered} \] The source asks whether this assertion holds for $k=4$, and asks the corresponding question for $k=3$. The subsurface $H$ may depend on the covering map $f$. These are questions, not asserted conclusions.

Short English statement

For every finite cover $S_h\to S_g$ with $h>g\geq2$, must there be an embedded four-holed sphere with essential boundary whose full preimage has disconnected interior? Ask the same question for a three-holed sphere. This formulation uses the usual closed, connected, orientable surface convention, which the supplied record does not explicitly specify.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003955$, titled Disconnected Preimages of Subsurfaces Under Finite Covers; exact source problem code catalogue entry. The collection is Oberwolfach Reports Open Problems.
• Supplied report citation: New Trends in Teichmüller Theory and Mapping Class Groups (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/40, pp. 2475--2534. Supplied link: https://doi.org/10.4171/owr/2018/40.
• Supplied literature source, labeled "Oberwolfach Report 40, Gekhtman problem": https://publications.mfo.de/bitstream/handle/mfo/3662/OWR_2018_40.pdf?isAllowed=y&sequence=1. The label associates the problem with Gekhtman; the registry's proposer field is empty, and no fuller attribution is supplied. The proposed year is recorded as 2018.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and reports that no general theorem answering the four-holed or three-holed sphere versions was found. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that notation, typography, and omitted definitions were repaired while preserving mathematical content. These are supplied editorial claims, not independent mathematical verification of this draft.