{
  "schema_version": 1,
  "problem_number": "OWR-1275-010",
  "title": "Components and Cohomology of Fock's Baby Teichmüller Space",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "At an Oberwolfach problem session in 2006, V. Fock asked for the cohomology and the number of components of the space B_n of maps f: ℤ/n → ℝP¹ with f(i) ≠ f(i+1) and at least three values, modulo SL(2,ℝ). We answer both questions for all n ≥ 3. The space B_n is a real-analytic (n−3)-manifold with exactly n−1 components B_{n,k}, indexed by the winding number k. For k ≠ n/2 they are Hausdorff and contractible, and for k = 1 and k = n−1 they are the Teichmüller space of the ideal n-gon. For even n the middle component is not Hausdorff: two distinct points have no disjoint neighbourhoods exactly when one is represented by a map constant on the even positions and the other by a map constant on the odd positions, and both loci are spheres S^{n/2−2}. The middle component is weakly homotopy equivalent to S^{n−3}. Hence H⁰(B_n; ℤ) ≅ ℤ^{n−1}, H^{n−3}(B_n; ℤ) ≅ ℤ for even n, and all other singular cohomology vanishes; the cohomology of the constant sheaf agrees, and so does its Čech cohomology on the Hausdorff components and, on the middle component, in degrees at most 2. For n = 4 the de Rham cohomology of smooth forms differs, and B_{4,2} is weakly homotopy equivalent to a circle but not homotopy equivalent to it. Modulo PGL(2,ℝ) there are ⌊n/2⌋ components, and for even n the middle one is weakly homotopy equivalent to ℝP^{n−3}; for odd n the space of real tame frieze patterns of width n−3 has (n−1)/2 components, all contractible. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GT",
    "math.AT",
    "math.CO"
  ],
  "keywords": [
    "baby Teichmüller space",
    "configurations on the projective line",
    "PSL(2,R)",
    "winding number",
    "non-Hausdorff manifold",
    "weak homotopy equivalence",
    "sheaf cohomology",
    "Čech cohomology",
    "de Rham cohomology",
    "frieze patterns",
    "Fock–Goncharov coordinates",
    "Oberwolfach Reports",
    "OWR-1275-010",
    "math.GT",
    "math.AT",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-01",
  "publication_date": "2026-10-01",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-01",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1275-010/",
  "pdf_url": "https://eulersolve.org/papers/owr-1275-010/paper.pdf?v=6fc9ab9046e9",
  "doi": "10.5281/zenodo.23071801",
  "zenodo_record_url": "https://zenodo.org/records/23071801",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Fock's question (OWR 26/2006, p. 1607) for all n ≥ 3: n − 1 components; singular and constant-sheaf cohomology determined; Čech cohomology agrees on the Hausdorff components and, on the middle component, in degrees at most 2. Open: whether the contractible components with 2 ≤ k ≤ n − 2 (k ≠ n/2, n ≥ 6) are homeomorphic to R^{n−3}, higher Čech cohomology of the middle component, and its de Rham cohomology and strong homotopy type for n ≥ 6.",
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    "source.zip": {
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      "sha256": "1033ec09570f69febe4d26fb9a9cdbee4e421decff6a1194c9308fbb981057c9"
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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."
}