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

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 3¹⁺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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-16415-018 (D. Gekhtman, Problem 12 of the problem session in Oberwolfach Reports 15 (2018), Report No. 40/2018, doi:10.4171/OWR/2018/40, p. 2528).