# Verification report — OWR-1275-010 (Fock's baby Teichmüller space)

Verification date: 2026-10-01 (first independent verification run: 2026-09-30; second independent verification run:
2026-10-01; both AI-assisted).

**Verdict.** Fock's question is answered completely, for every n ≥ 3. The baby Teichmüller space
B_n = {f: Z/n → RP^1 : f(i) ≠ f(i+1), at least three values}/SL(2,R) is a second countable real-analytic
(n−3)-manifold with exactly n − 1 connected components B_{n,k}, k = 1, …, n − 1 (the winding number).
- Every component with k ≠ n/2 is Hausdorff and contractible; for k = 1 and k = n − 1 it is the Teichmüller space of
  the ideal n-gon, homeomorphic to R^{n−3}.
- For even n the middle component B_{n,n/2} is not Hausdorff. Two distinct points have no disjoint neighbourhoods
  exactly when one is represented by a map constant on the even positions (the set E_n) and the other by a map constant
  on the odd positions (the set O_n); E_n and O_n are homeomorphic to S^{n/2−2}. The middle component is weakly homotopy
  equivalent to S^{n−3}.
- Singular cohomology: H^0 = Z^{n−1}, H^{n−3} = Z for even n, all other groups 0 (any coefficients likewise). The
  cohomology of the constant sheaf is the same. Its Čech cohomology agrees on the Hausdorff components and, on the
  middle component, in degrees ≤ 2; in all degrees it surjects onto sheaf cohomology. The de Rham cohomology of smooth
  forms differs (H^1_dR(B_{4,2}) = 0), and B_{4,2} is weakly homotopy equivalent to S^1 but not homotopy equivalent to it.
- Modulo PGL(2,R) there are ⌊n/2⌋ components; for even n the middle one is weakly homotopy equivalent to RP^{n−3}.
  For odd n the real variety of tame frieze patterns {c ∈ R^n : M(c) = −Id} has (n − 1)/2 contractible components.

The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report 26/2006, *Teichmüller Space (Classical and Quantum)*, organised by S. Morita,
  A. Papadopoulos and R. C. Penner, Oberwolfach Rep. 3 (2006), no. 2, 1537–1614, doi:10.4171/OWR/2006/26.
  - Read in the publisher's PDF (EMS Press). The problem session of June 1, 2006 is on pp. 1605–1609; the notes are by
    the three organisers, and the session was chaired by W. Goldman.
  - Problem 6, "Baby Teichmüller space; posed by Volodya Fock", p. 1607. Paraphrase: consider all functions
    f: Z/nZ → RP^1 with f(i) ≠ f(i+1) for all i (indices mod n) whose image has at least three points; their classes
    modulo SL(2,R) form the "baby Teichmüller space". The question begins "What is the cohomology of this space?" and
    asks, for instance, for the number of components.
  - The report records the problem without an answer.
- **Corpus record.** ulamai/UnsolvedMath, version 1.6.0, record OWR-1275-010 (status `open`). Its statement is the same
  as the source's.

## Readings
| Reading | Answered? | Where in the paper |
|---|---|---|
| B_n as defined in the source: labelled maps, at least three values, orbits of SL(2,R) (acting through PSL(2,R)); number of components | yes: n − 1 | Theorem 1.1(i) |
| "cohomology" = singular cohomology, any coefficients (and singular homology) | yes: A^{n−1} in degree 0, A in degree n − 3 for even n, 0 otherwise | Theorem 1.1(iv) |
| cohomology of the constant sheaf (derived-functor sheaf cohomology) | yes: the same groups | Proposition 5.3 |
| Čech cohomology of the constant sheaf (colimit over open covers) | on the Hausdorff components: the same; on B_{n,n/2}: the same in degrees ≤ 2, and a surjection onto sheaf cohomology in all degrees; injectivity in degrees ≥ 3 left open | Proposition 5.4, open question (2) |
| de Rham cohomology of smooth forms | differs for n = 4 (degree 1 vanishes); not determined for n ≥ 6 | Proposition 5.5(a), open question (3) |
| homotopy type | weakly a point (k ≠ n/2) or S^{n−3} (k = n/2); the Hausdorff components are contractible; B_{4,2} is not homotopy equivalent to S^1; the strong homotopy type of B_{n,n/2} for n ≥ 6 is left open | Propositions 3.4, 4.2, 5.5(b) |
| orbits of PGL(2,R) (unoriented configurations) | yes: ⌊n/2⌋ components; for even n the middle one is weakly RP^{n−3} | Corollary 1.2(a) |
| real friezes, odd n: {M(c) = −Id} | yes: (n − 1)/2 components, all contractible | Corollary 1.2(b) |
| maps up to cyclic relabelling, or including two-valued maps | not treated; the source defines neither | — |

## Results in the paper
- **Lemma 2.1 (coordinates).** Φ(f) = (ω(f(0)), θ(f)) identifies X_n with (R/2πZ) × ⊔_k Δ_k, where θ_i ∈ (0, 2π) is
  the arc from f(i) to f(i+1) and Δ_k = {θ ∈ (0,2π)^n : Σθ_i = 2πk}. The rotation group K acts on the first factor
  only. Two-valued maps correspond to an open segment L ⊂ Δ_{n/2}. The winding number k is G-invariant, and the
  B_{n,k} are the n − 1 components.
- **Lemma 3.1 (slices).** G acts freely with local slices; B_n is a second countable real-analytic (n−3)-manifold.
- **Lemma 3.2.** A fibre bundle with contractible fibre over an arbitrary base (no separation axiom) is a weak homotopy
  equivalence (Hatcher, Prop. 4.48, Thm. 4.41, Prop. 4.21).
- **Proposition 3.3 (two bundles).** (a) X_n^*/K → B_n is an R^2-bundle with total space ⊔_k Δ'_k. (b) The slice
  C_n = {f(0) = ∞, f(1) = 0} gives B_n ≅ C_n/D with D = {x ↦ λx, λ > 0} ≅ R, and C_n ≅ ⊔_k Y'_k, where
  Y_k = Δ_k ∩ {θ_0 = π} and Y'_{n/2} = Y_{n/2} minus the centre (π, …, π).
- **Proposition 3.4.** Δ'_k and Y'_k are convex for k ≠ n/2; Δ'_{n/2} and Y'_{n/2} are homotopy equivalent to S^{n−3}
  (radial retractions). Hence the weak homotopy types, by two independent routes.
- **Lemma 4.1 (inseparable pairs).** Exactly the (E_n, O_n) pairs, by a Cartan-decomposition argument (⇒) and explicit
  sequences, with b_j = ∞ allowed (⇐).
- **Proposition 4.2.** The components with k ≠ n/2 are Hausdorff and contractible (Milnor 1959, Cor. 1, and
  Whitehead); B_{n,1} ≅ R^{n−3}; E_n ≅ O_n ≅ S^{n/2−2}.
- **Example 4.3.** The explicit picture for n = 4: B_{4,2} consists of four points and four open arcs, each arc
  converging at one end to an E-point and an O-point; the incidences form an 8-cycle.
- **Lemma 5.1, Remark 5.2.** Combinatorial types and open stars; the preimages of open stars are convex. McCord's
  model and the Borel–Moore cellular model (used only for the computations).
- **Propositions 5.3–5.5.** Sheaf cohomology = singular cohomology (Hartshorne, Prop. III.8.1 and Ex. III.8.1);
  Čech cohomology (Godement, II §5.9 and §5.10; Leray's theorem, Hartshorne Ex. III.4.11); n = 4: de Rham cohomology of
  smooth forms, and failure of homotopy equivalence with S^1.
- **Corollary 1.2.** (a) PGL(2,R). (b) Friezes, via the normalized lift and ε = (−1)^k.
- **Lemma 6.1, Proposition 6.2.** Triangulations whose edges join distinct values (complete ear argument);
  Fock–Goncharov cross-ratio charts U_T/PGL(2,R) ≅ (R^*)^{n−3}; B_4^PGL is R^* with the point −1 doubled.

## Computations (scripts and outputs in reproducibility/)
- **Author** (`lead/`, standard library Python, 70–80 s):
  - `sanity_checks.py`: invariance of the winding number (3200 configurations, n ≤ 10, exact); the sequences of
    Lemma 4.1 (exact); Lemma 6.1 on all 22,077 coincidence patterns with n ≤ 10; the frieze relations ε = (−1)^k and
    M(c) = ε·Id (floating point, 1200 configurations).
  - `types_poset.py`: type counts for n ≤ 8; order-complex homology of the McCord model over three fields: a point
    for all k ≠ n/2, S^1 and S^3 for (4,2) and (6,3).
  - `bm_cellular.py`: integral Borel–Moore homology for all 35 components with n ≤ 9 (up to 206,023 cells), no
    torsion, results as in Theorem 1.1; the join computation for n = 4, 6, 8.
  - `paper_checks.py` (written for this release): the reflection involution on types; GF(2) cohomology of RP^{n−3}
    for the middle PGL-component (n = 4, 6, 8); GF(2) cohomology of S^{n/2−2} for E_n (n ≤ 14).
- **First independent verification run** (`verifier/`, AI-assisted, written from scratch without importing the author's
  code): type counts, sign-free GF(2) Borel–Moore homology for the 27 components with n ≤ 8, full order complexes for
  n ≤ 6, Euler characteristics of the E/O strata, and 53,700 convexity tests of open stars; no discrepancy. Its re-runs
  of the author's scripts reproduced the recorded outputs up to timings.
- **Second independent verification run** (`independent_run_2/`, AI-assisted, 2026-10-01, written from scratch without
  importing the author's code or the first run's code; about 140 s):
  - types and windings for n ≤ 9 (502,748 types for n = 9);
  - Borel–Moore homology of all 35 components with n ≤ 9 over GF(2), GF(3) and GF(10007), with incidence numbers
    obtained geometrically from integer determinants of tangent frames (random orientation flips), every facet
    relation verified geometrically, and d∘d = 0 over Z;
  - π^{-1}(E_n) and π^{-1}(O_n) (n ≤ 8; E_n also for n = 10, 12): the cohomology of S^{n/2−2};
  - the integral cohomology of Δ'_{n/2}/ρ from the orientation-twisted subcomplex: H^*(RP^{n−3}; Z) for n = 4, 6, 8,
    with the 2-torsion;
  - Lemma 6.1 by interval dynamic programming for all coincidence patterns with n ≤ 11;
  - friezes in exact arithmetic (odd n ≤ 11): M(c) = ε·Id, ε = (−1)^k, SL(2,Z)-invariance, the inverse construction,
    and k = 1 for all Conway–Coxeter quiddities;
  - Example 4.3, the sequences of Lemma 4.1 (b_j = ∞ included), the half-space description of the open stars
    (Lemma 5.1) and Proposition 6.2, all exact;
  - a rerun of `lead/run_all.sh` and `verifier/indep_check.py` from an extracted copy of the source archive, with 0
    differing lines after removing timings.
  All checks passed.
- All outputs were regenerated for this release with the scripts in `reproducibility/`, with identical results up to
  timings.
- The finite models reach B_n only through the bundle π, so they check the combinatorics of Δ'_k, not the bundle step;
  the bundle step is proved, in two ways (Proposition 3.3).

## Independent verification runs

### First run (AI-assisted, 2026-09-30)
The first independent verification run checked the claim and its computations.

| Item | Verdict |
|---|---|
| Source fidelity | PASS (definition, group, labelled maps, at least three values) |
| Proofs | PASS (every step checked; the main homotopy statement re-derived by the slice argument, now Proposition 3.3(b)) |
| Computations | PASS (re-runs identical up to timings; independent code agrees) |
| Answer as posed | complete answer to both parts of the question |
| Novelty | no earlier answer found; ingredients to be credited |
| Presentation | fixes required (below) |

Required fixes and how they were applied:
1. **The citing work.** Identified as H. Bai, *A uniqueness property for the quantization of Teichmüller spaces*,
   Geom. Dedicata 128 (2007), 1–16, doi:10.1007/s10711-007-9176-2; it cites the report as a whole and does not address
   Problem 6. Stated in the paragraph on scope and priority.
2. **Credits and novelty wording.** Morier-Genoud–Ovsienko–Schwartz–Tabachnikov, Thm. 3.4.1(ii) (friezes versus n-gons
   for odd n); Ovsienko 2018, §5 (the rotation index as the homotopy class of the n-gon, and the sign of the
   monodromy); Coxeter and Conway–Coxeter; Fock–Goncharov (the positive part as Teichmüller space; non-separated
   schemes from positive atlases). All credited in the introduction ("Context and credits"). The novelty wording is
   modest: the component count is called elementary and possibly known to experts.
3. **Triangulation lemma.** The paper gives a complete proof by ears (Lemma 6.1), including the case of exactly three
   values; it is also checked exhaustively for n ≤ 10.
4. **Lemma 4.1 (⇐).** The case b_j = ∞ is allowed explicitly, with the convention λ·∞ = ∞.
5. **De Rham cohomology.** The paper speaks of the de Rham cohomology of smooth forms (the naive de Rham complex) and
   cites O'Connell, arXiv:2310.17151. On reading that paper we describe it as follows: O'Connell proves a de Rham
   theorem for non-Hausdorff manifolds glued from Hausdorff ones under assumptions on the gluing sets, among them
   regular openness (Thm. 4.4), and shows with the line with two origins, where these assumptions fail, that the two
   cohomologies can differ (§4). Proposition 5.5(a) is an instance of this phenomenon.
6. **Finite models.** The paper states that both finite models reach B_n only through π and check the combinatorics of
   Δ'_k, not the bundle step (Remark 5.2 and the Verification paragraph). The second proof via the slice C_n modulo
   R_{>0} is included (Proposition 3.3(b) and Proposition 3.4).
7. **Precise references.** Godement, Chap. II, §5.9 (Čech = sheaf cohomology in degrees ≤ 1, injective in degree 2) and
   §5.10 (paracompact spaces); Leray's theorem on acyclic covers (Hartshorne, Ex. III.4.11); Hartshorne, Prop. III.8.1
   (stalks of higher direct images) and Ex. III.8.1; McCord 1966, Thm. 6; Milnor 1959, Cor. 1; Hatcher, Prop. 4.48,
   Thm. 4.41, Prop. 4.21 and Thm. 4.5.
8. **Caveats kept as open questions.** Whether the contractible components with 2 ≤ k ≤ n − 2, n ≥ 6, are homeomorphic
   to R^{n−3}; Čech cohomology of the middle component in degrees ≥ 3; the de Rham cohomology and strong homotopy type
   of B_{n,n/2} for n ≥ 6.

Further changes in writing the note:
- The agreement of Čech and sheaf cohomology on B_{n,n/2} is stated in degrees ≤ 2: the finite cover by open stars is
  a Leray cover, so the natural map is surjective in every degree, and it is injective in degree 2 on every space
  (Godement).
- A new computation (`paper_checks.py`) checks Corollary 1.2(a) and Proposition 4.2(c) mod 2.
- The join decomposition of the middle component and a barycentre model, which were in the working notes, are not part
  of the note; the join computation remains in `lead/outputs/bm_join_out.txt` as a consistency check.

### Second run (AI-assisted, 2026-10-01)
A second independent verification run checked the revised note line by line, the source (the publisher's PDF of the
report, fetched anonymously; p. 1607 rendered and read), every cited result that is openly available, the computations
(its own code and a rerun of the package) and the literature.

| Item | Verdict |
|---|---|
| Source fidelity | PASS (the definition and the question as on p. 1607; the corpus statement is the same) |
| Proofs | PASS (no error; see the points below) |
| Strengthened Čech statement (Proposition 5.4) | justified and kept; precision fix: Čech cohomology of the constant sheaf |
| Computations | PASS (own code: 0 failed checks; package rerun: identical up to timings) |
| Answer as posed | complete answer to both parts of the question |
| Novelty | no earlier answer found; ingredients credited |
| Presentation | minor fixes (below); all pages rendered and inspected |

Points examined in detail:
- **Čech cohomology (Proposition 5.4).**
  - The open-star cover of B_{n,n/2} is a finite Leray cover (Lemma 5.1, Proposition 5.3), so by Leray's theorem
    (Hartshorne, Ex. III.4.11, valid on any topological space) Ȟ^j(U; A) ≅ H^j(B_{n,n/2}; A) for every j.
  - This isomorphism factors through Ȟ^j(B_{n,n/2}; A), so the natural map is surjective in every degree.
  - On every space Ȟ^j → H^j is bijective for j ≤ 1 and injective for j = 2 (Godement, II §5.9). The run re-derived the
    degree-2 injectivity without the book: embed F in a flasque sheaf, use the vanishing of Čech cohomology of flasque
    sheaves, the vanishing of Ȟ^0 of a locally zero presheaf, and Ȟ^1 = H^1 (Hartshorne, Ex. III.4.4).
  - Hence the map is bijective for j ≤ 2. The statement concerns Čech cohomology with coefficients in the constant
    sheaf A; the note now says so. Injectivity in degrees ≥ 3 remains open question (2).
- **Lemma 6.1.** The rewritten ear argument is correct; the false parenthetical of an earlier sketch is not used. Checked
  independently by dynamic programming for n ≤ 11.
- **Bundles over the non-Hausdorff base.** Local triviality of both bundles (Proposition 3.3); Hatcher's proof of
  Prop. 4.48 uses only compactness of I^n × I and local trivializations (read in the online edition); Thm. 4.41 and
  Prop. 4.21 hold for arbitrary spaces. Lemma 4.1 (both directions, b_j = ∞ included), E_n ≅ O_n ≅ S^{n/2−2} and the two
  proofs of the weak S^{n−3} type (via Δ'_{n/2} and via the slice C_n) were checked.
- **n = 4 and O'Connell.** The naive de Rham argument and the Hausdorff quotient argument are correct. O'Connell,
  arXiv:2310.17151v3, was read: Thm. 4.4 is a de Rham theorem under the criteria of Remark 1.7 and Thm. 3.2 and
  regular openness of the unions of the gluing sets; the line with two origins (Section 4) has H^1_dR = 0 but H^1 = R.
  The note's description is accurate.
- **Corollary 1.2.** PGL(2,R): the equivariant retraction and the non-Hausdorffness argument are correct. Friezes:
  MGOST, Thm. 3.4.1(i),(ii) (read in arXiv:1309.3880; ground field R) and Ovsienko, §5 and §5.4 (Definition 5.6: index
  1/2 means non-osculating, that is, totally positive; read in arXiv:1710.02996) are cited correctly.
- **Citations.** Hatcher, Prop. 4.48, Thm. 4.41, Prop. 4.21 and Thm. 4.5 (online edition); Milnor 1959, Cor. 1 (AMS
  PDF: "every separable manifold belongs to the class W_0"); McCord 1966, Thm. 6, the basis-like cover theorem, as used
  in Barmak–Minian, arXiv:math/0611158; Sella, arXiv:1602.06674 (abstract); Fock–Goncharov 2016, §1.2 (cross-ratio,
  possibly non-separated scheme). All 15 DOIs in the bibliography resolve in Crossref with matching metadata, including
  Ovsienko's corrected DOI 10.1007/s40687-018-0139-z. Godement and Hartshorne were not accessible; the numbering used
  is the standard one.

Required fixes and how they were applied:
1. **Record the second run.** Done: the Verification paragraph of the note (item (3)), this section, and the folder
   `reproducibility/independent_run_2/` with its README.
2. **Čech precision.** The abstract, Theorem 1.1(iv), Proposition 5.4 and open question (2) now speak of the Čech
   cohomology of the constant sheaf A. The proof of Proposition 5.4 notes that on the Leray cover the nonempty finite
   intersections are connected, so Ȟ(U; A) is the simplicial cohomology of its nerve.
3. **Abstract, PGL sentence.** Now "for even n the middle one is weakly homotopy equivalent to RP^{n−3}".
4. **Lemma 5.1.** The proof now notes that only finitely many hyperplanes of the arrangement meet the closed polytope,
   so the open star is open.
5. **Package.** Rebuilt: source archive, the three Zenodo files and their checksums, and the deposit description.

## Relation to the literature, novelty and scope
- **Sources read.** The report OWR 26/2006 (publisher's PDF); the later Oberwolfach reports on Teichmüller theory
  53/2010, 7/2014, 40/2018 and 33/2023 (searched for the space; no mention); several arXiv papers of Fock and
  Fock–Goncharov (math/0311149, math/0311245, math/0702397, math/0508408, 0811.3356, 1104.0407, 1812.11199); the papers of
  Morier-Genoud–Ovsienko–Schwartz–Tabachnikov (1309.3880), Ovsienko (1710.02996), Ovsienko–Tabachnikov (1312.3021),
  Conley–Ovsienko (2107.01234, 1812.04271), Arnold–Fuchs–Izmestiev–Tabachnikov (1812.05337), Short–Van Son–Zabolotskii
  (2312.12953), Bai (math/0509679, and the Crossref reference list of its journal version), O'Connell (2310.17151) and
  Barmak–Minian (math/0611158). The statements of Milnor 1959 and the numbering of the results cited from Hatcher's
  book were checked in the sources (AMS PDF; the author's online edition). The results cited from Hartshorne,
  Godement and Bredon are standard; their numbering was not re-checked in the books for this release.
- **Second verification run (2026-10-01).** Read again: the report OWR 26/2006 and OWR 53/2010 (publisher's PDFs; no
  mention of the space; Fock's items there concern other topics), O'Connell (2310.17151v3), Ovsienko (1710.02996v5),
  Morier-Genoud–Ovsienko–Schwartz–Tabachnikov (1309.3880), Fock–Goncharov (1104.0407v2), Barmak–Minian
  (math/0611158v2), Hatcher's book (online edition) and Milnor 1959 (AMS PDF). New searches: 12 arXiv API queries
  (the name of the space; friezes with connected components, homotopy, winding number, Sturm theory; rotation numbers of
  polygons on the projective line; discrete Hill equations; non-Hausdorff configuration spaces; real points of cluster
  varieties), zbMATH Open, Crossref, OpenAlex (the report has one citing work; the list query was rate-limited),
  Semantic Scholar (one citing work, Bai 2007), OpenCitations (none) and one web search. The only new hits were
  unrelated: arXiv:2609.20967 (integer friezes with wild entries), 2508.16287 (expository homotopy of closed polygonal
  lines) and 2603.19872 (continuous 2-friezes). All requests were anonymous.
- **Searches (September 2026).** arXiv API (well over 50 queries on the name of the space, polygons and configurations
  on the projective line, winding and rotation numbers, real friezes, non-Hausdorff moduli and cluster varieties),
  Crossref, OpenAlex (partly rate-limited), zbMATH Open, Semantic Scholar, OpenCitations, MathOverflow, and web
  searches. All requests were anonymous.
- **Citing works.** Crossref records no work citing the report; OpenAlex counts one; Semantic Scholar lists one, Bai
  (Geom. Dedicata 2007), which cites the report as a whole and concerns the quantization of Teichmüller spaces;
  OpenCitations lists none.
- **Assessment.** No earlier answer to Fock's question was found. The decomposition by winding number is elementary and
  may be known to experts; the rotation index and its relation with the sign of the monodromy (Ovsienko), the
  isomorphism between friezes and n-gons for odd n (Morier-Genoud–Ovsienko–Schwartz–Tabachnikov) and the positive
  part (Fock–Goncharov) are known. We did not find the description of the non-Hausdorff middle component, its weak
  homotopy type, the comparison of cohomology theories, or the component count for real friezes. This negative search
  is not a proof of priority.
- **Scope.** The note answers the question for the space defined in the source and for every n ≥ 3. Left open: the
  homeomorphism type of the contractible components with 2 ≤ k ≤ n − 2 for n ≥ 6; Čech cohomology of the middle
  component in degrees ≥ 3; the de Rham cohomology of smooth forms and the strong homotopy type of the middle component
  for n ≥ 6.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/