{
  "schema_version": 1,
  "problem_number": "OWR-16415-018",
  "title": "A Finite Cover of the Genus-Two Surface in Which Every Four-Holed Sphere Has Connected Preimage: A Partial Answer to a Question of D. Gekhtman",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In the problem session of the Oberwolfach workshop \"New Trends in Teichmüller Theory and Mapping Class Groups\" (2018), D. Gekhtman asked: given a finite cover f : S_h → S_g of closed orientable surfaces with h > g ≥ 2, is there always a four-holed sphere H ⊂ S_g with essential boundary components such that f^{-1}(H) is disconnected, and is there always such a three-holed sphere? We give a partial answer. For the four-holed question the answer is no in genus 2: there is a cover S_10 → S_2 of degree 9, with monodromy group 3^2:Q_8 of order 72, such that the preimage of every four-holed sphere with essential boundary components is connected. The mechanism is a lifting obstruction: the monodromy homomorphism does not lift to the central extension 3^{1+2}:Q_8, whereas it would lift if some four-holed sphere had disconnected preimage. This settles the four-holed question in genus 2 only, where every such four-holed sphere is the complement of two disjoint curves, so that its boundary curves are isotopic in pairs; if the four boundary curves are required to be pairwise non-isotopic, there is no such subsurface in genus 2, the question begins in genus 3, and there it remains open. The example is not a regular cover, and its Galois closure is not an example. It belongs to a family of covers of S_2 of degree p^2 with group p^2:Q_8, for every prime p ≡ 3 (mod 4); the same groups give no example in genus at least 3. In the positive direction we find, in every genus, three-holed and four-holed spheres with disconnected preimage for covers which factor through a non-trivial abelian cover, for abelian-by-cyclic monodromy groups, for regular covers with characteristic kernel, and when the genus is at least three times the order of the monodromy group. Exact computations in genus 2 show that 9 is the smallest degree of such an example, that in degrees 9 and 10 the only examples are the covers with group 3^2:Q_8 whose monodromy does not lift, and that there is none among the regular covers with deck group of order less than 12180. The three-holed question remains open (our covers do have pairs of pants with disconnected preimage), and so do the four-holed question in genus at least 3 and both questions for regular covers. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "keywords": [
    "finite covers of surfaces",
    "four-holed sphere",
    "pair of pants",
    "monodromy",
    "lifting obstruction",
    "central extension",
    "Schur multiplier",
    "mapping class group",
    "genus two",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-16415-018",
    "math.GT",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-09",
  "publication_date": "2026-10-09",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-09",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-16415-018/",
  "pdf_url": "https://eulersolve.org/papers/owr-16415-018/paper.pdf?v=04335fe54ddf",
  "doi": "10.5281/zenodo.23272252",
  "zenodo_record_url": "https://zenodo.org/records/23272252",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Problem 12 of D. Gekhtman (Oberwolfach Reports 15 (2018)): for the four-holed question the answer is no in genus 2 only, where every four-holed sphere with essential boundary components is the complement of two disjoint curves, so that its boundary curves are isotopic in pairs. If the four boundary curves are required to be pairwise non-isotopic, the question begins in genus 3 and is open there. The example is not a regular cover, and its Galois closure is not an example. The three-holed question and both questions for regular covers remain open. The lifting obstruction used is the classical invariant in the second homology of the group; no novelty is claimed for it, and no priority is claimed.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}