# Verification report — OWR-4798-013 (Klee–Novik, Question 4: is B(i,d) a combinatorial S^i × B^(d−i−1)?)

Verification date: 2026-09-30 (revised the same day after a second independent verification run).

**Verdict.** The answer is yes. For all d ≥ 2 and 0 ≤ i ≤ d − 2, the Klee–Novik complex B(i,d) is PL homeomorphic
to S^i × B^(d−i−1), so it is a combinatorial triangulation of S^i × B^(d−i−1). Klee and Novik had already observed that
B(i,d) collapses onto the boundary of the (i+1)-dimensional cross-polytope and is a disc bundle over S^i
(Remark 3.7 of their paper). The new step is that |B(i,d)| is in fact a product, because that sphere is a join factor
of the boundary of the d-dimensional cross-polytope. Two independent verification runs (AI-assisted) found the
argument correct. The second reviewed the final text of the earlier draft and asked only for editorial and precision
fixes; they are applied in this version. The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report No. 08/2011, "Topological and Geometric Combinatorics" (organised by
  A. Björner, G. Kalai, I. Novik and G. M. Ziegler), Oberwolfach Rep. 8 (2011), no. 1, pp. 349–423,
  doi:10.4171/OWR/2011/08.
  - The report PDF was downloaded anonymously from EMS Press (via the DOI) and re-read during the revision.
  - The abstract in question is "Centrally symmetric manifolds with few vertices" by Steven Klee (joint work with
    Isabella Novik), pp. 370–373. The paper cites it by this speaker and title.
  - On p. 372 the abstract calls the question "very natural (yet strangely elusive)".
  - Question 4 asks whether B(i,d) is a combinatorial triangulation of S^i × B^(d−i−1). The surrounding theorem is
    stated for 0 ≤ i < d − 1.
- **Construction.** S. Klee and I. Novik, "Centrally symmetric manifolds with few vertices", Adv. Math. 229 (2012)
  487–500, doi:10.1016/j.aim.2011.07.024, arXiv:1102.0542.
  - arXiv v1 (the only arXiv version) was read in full. So was the authors' revised version dated July 27, 2011,
    from Novik's web page; it differs from v1 only in small edits and one added reference. Remark 3.7 and the lemma
    and theorem numbers are the same in both. The paper states that it uses this numbering.
  - The published version could not be accessed: ScienceDirect refused the anonymous request, and the Elsevier API
    needs a key.
  - Definition 3.1: B(i,d) is generated by the facets of the cross-polytope boundary C*_d whose xy-words have at
    most i switches.
  - Theorem 1.2: B(i,d) is a combinatorial manifold with the homology of S^i (d). Its complement C(i,d) is isomorphic
    to B(d−i−2,d) (c). ∂B(i,d) is homeomorphic to S^i × S^(d−i−2) (e).
  - Remark 3.7 says, without proof, that B(i,d) collapses onto B(i,d−1), …, B(i,i+1) = C*_(i+1). It adds that the
    results of Rourke–Sanderson, Ch. 3, imply that B(i,d) is a disc bundle over S^i.
- **Corpus record.** ulamai/UnsolvedMath, OWR-4798-013 (status `open`). Its statement is the question above, with
  the definition of B(i,d).

## Readings
| Reading | Answer | Where |
|---|---|---|
| B(i,d) is PL homeomorphic to S^i × B^(d−i−1) ("combinatorial triangulation"), 0 ≤ i ≤ d−2 | yes, for all such i, d | Theorem 1.2 |
| the same up to homeomorphism only | yes (weaker) | Theorem 1.2 |
| the boundary: ∂B(i,d) is a combinatorial triangulation of S^i × S^(d−i−2) | yes (PL; Klee–Novik proved homeomorphism, via Kreck's theorem when min{i, d−i−2} ≥ 2) | Corollary 1.3(a) |
| Cohen–Klee–Pannell conjecture: \|B(k,d)\| ≅ \|ST(k+1,d−k−1)\| | yes | Corollary 1.3(b) |
| Cohen–Klee–Pannell Question 1.1: can B(k,d) be obtained from ST(k+1,d−k−1) by bistellar moves, stellar exchanges, elementary shellings and their inverses? | yes, but only existentially: the theorem of Newman and Pachner (Lickorish, Thm 5.10) gives elementary shellings, inverse shellings and a simplicial isomorphism; no explicit sequence is constructed | Corollary 1.3(b) and the text after it |

## Results in the paper
- **Lemma 3.1** (Klee–Novik, Remark 3.7, with a proof).
  - For 0 ≤ i ≤ k−2, the faces of B(i,k) avoiding x_k, y_k form B(i,k−1), and B(i,k) ↘ B(i,k−1).
  - Proof: Klee–Novik's shelling of st(x_k) (Lemma 3.5, restrictions Sel(τ)) gives a shelling of lk(x_k). For j ≥ 2
    the restriction face R(G_j) = Sel(τ_j) is nonempty and proper. Removing the Boolean intervals of new faces through a
    vertex a_j shows that lk(x_k) is collapsible. The collapse lifts to the cone, and the same holds for y_k by the
    symmetry x_j ↔ y_j.
  - Hence B(i,d) ↘ C*_(i+1).
- **Lemma 3.2** (standard neighbourhood of a join factor).
  - C*_d = Σ * Σ', with Σ = C*_(i+1) and Σ' the cross-polytope boundary on the last d−i−1 pairs.
  - Writing z = (z', z'') ∈ R^(i+1) × R^(d−i−1), N_0 = {z ∈ |C*_d| : ‖z''‖_1 ≤ 1/2} is PL homeomorphic to |Σ| × D,
    D the ℓ1-ball of radius 1/2. The proof matches the face posets of two polyhedral complexes (cells Q, Q' and P, P')
    and uses derived subdivisions.
  - N_0 collapses onto |Σ| (cell by cell) and is a regular neighbourhood of |Σ|.
- **Lemma 3.3.**
  - C(i,d) is a combinatorial manifold (Klee–Novik Thm 1.2(c),(d)).
  - ∂B = ∂C, |B| ∪ |C| = |C*_d| and |B| ∩ |C| = |∂B|. The key step: for a nonempty face F of B ∩ C, |F| ≤ d−1, and the
    facet-adjacency graph of lk F is a cube graph, hence connected.
- **Remark 3.4.** Every vertex of C*_d lies in ∂B(i,d) (use the constant word and an alternating word, then Lemma 3.3).
  So |C*_(i+1)| meets ∂|B(i,d)|, |B(i,d)| is not itself a regular neighbourhood of it, and the outer collar is needed.
- **Theorem 1.2.**
  - N = |B| ∪ (outer collar in |C|) is PL homeomorphic to |B|, contains |B| in its interior and collapses onto |Σ|.
  - By the collapsing criterion it is a regular neighbourhood of |Σ| in the sphere |C*_d|.
  - By uniqueness of regular neighbourhoods, |B(i,d)| ≅ N ≅ N_0 ≅ S^i × B^(d−i−1).
- **Corollary 1.3.**
  - (a) ∂B(i,d) is a combinatorial triangulation of S^i × S^(d−i−2).
  - (b) Comparison with the staircase triangulations ST(m,n) of Cohen–Klee–Pannell, via the theorem of Newman and
    Pachner in the form of Lickorish, Thm 5.10.
- **PL facts used**, each stated explicitly in the paper as Facts 2.1–2.2 and cited to Rourke–Sanderson by chapter:
  - collars (Ch. 2); the paper now explains why the open collar is open, including near ∂M;
  - derived subdivisions of polyhedral complexes (Ch. 2);
  - triangulations of PL manifolds are combinatorial manifolds (Ch. 2);
  - the collapsing criterion for regular neighbourhoods, for X in the interior of N (Ch. 3);
  - uniqueness of regular neighbourhoods (Ch. 3).

  The book itself could not be consulted: the Internet Archive copy is lending-only, and its search refused the
  anonymous request (HTTP 403). The paper therefore says that statement numbers are not given, cites chapters, and
  states each fact in full.
- **Remark in the paper.** For i ≤ d−4 the sphere has codimension ≥ 3, and Zeeman's unknotting theorem together with
  Remark 3.7 would also give the result. The join argument is uniform and also covers codimensions one and two.

## Computations (exact; scripts and outputs in reproducibility/)
- **Lead** (`lead/verify_owr4798013.py`, written with AI assistance, standard library only, about three minutes).
  - It treats all 66 pairs with 2 ≤ d ≤ 12 and 0 ≤ i ≤ d−2.
  - It checks Klee–Novik's filling criterion (Lemma 3.2) on all faces of C*_d.
  - It checks the isomorphism C(i,d) = A(B(d−i−2,d)), the pseudomanifold property, ∂B = ∂C (boundary ridges), and
    B ∩ C = ∂B.
  - It checks the shelling order and the restriction faces Sel(τ) of every star used.
  - It executes the explicit collapse of Lemma 3.1, checking every elementary collapse. The final complex is
    C*_(i+1) in all 66 cases; for (i,d) = (5,10), for example, there are 28180 elementary collapses.
  - For d ≤ 8 the mod-2 Betti numbers of B(i,d) and ∂B(i,d) are those of S^i and S^i × S^(d−i−2).
  - Two deliberately corrupted variants (wrong collapse order, wrong shelling order) fail at the first bad step, as
    they should.
  - In the revision only two comments and one docstring reference of this program were updated to the new notation
    (a_j instead of w, Lemma 3.3); its code and recorded output are unchanged.
- **Exploratory program** (`claimant/`, written while the proof was found). A single greedy collapse protecting
  C*_(i+1) succeeded for (i,d) = (1,5), (2,5), (3,5), (2,6), (3,6), (2,7), (3,7), (4,7). It is not used in the paper.
- **First independent verification run** (`independent_verification/`, AI-assisted, separate code).
  - Structured check for 3 ≤ d ≤ 10, 1 ≤ i ≤ d−2: at every stage the links of x_k and y_k collapse to a point, and
    the faces avoiding x_k, y_k form B(i,k−1). All 36 cases are OK.
  - A plain greedy collapse of the whole complex (d ≤ 9) gets stuck at (i,d) = (5,9) with 9779 faces left. A greedy
    collapse can get stuck on a collapsible complex, so this does not contradict Lemma 3.1. It shows that a single
    greedy collapse is not a reliable certificate, which is why the checks follow the structured collapse of the proof.
- **Second independent verification run** (`independent_run_2/`, AI-assisted). The run checked its results in memory
  and did not save its code. `check_owr4798013_run2.py` reproduces those checks; it was written during the revision
  from the text of the paper, with its own encoding of faces, and does not reuse the lead program or the first run's
  code (standard library only, about five seconds).
  - (A) The collapse of Lemma 3.1, built as in the proof and executed pair by pair with the elementary-collapse test,
    for all 45 pairs 2 ≤ d ≤ 10, 0 ≤ i ≤ d−2: every step is an elementary collapse, the complex after each stage is
    B(i,k−1), and the final complex is C*_(i+1). The numbers of steps agree with the lead program.
  - (B) Klee–Novik Lemma 3.5 (the paper's Lemma 2.3(a)) for the stars of x_d and y_d, 2 ≤ d ≤ 8: confirmed.
  - (C) Lemma 3.3 for 2 ≤ d ≤ 10 (every face of C*_d in B or C, every ridge of C*_d in exactly two facets, equal
    boundary ridges, B ∩ C = ∂B as complexes, and C(i,d) = A(B(d−i−2,d))): confirmed.
  - (D) Every vertex of C*_d lies in the boundary of B(i,d), 2 ≤ d ≤ 9 (Remark 3.4): confirmed.
  - The script was also run on deliberately wrong inputs (reversed order of switch sets, shifted restriction faces,
    complement without the exchange of letters); each made the corresponding check fail.

## Independent verification runs
Both runs were AI-assisted verification runs; the note has not been peer reviewed.

**First independent verification run** (2026-09-30, on the argument before the paper was written):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (OWR 8/2011 p. 372 re-read; corpus record matches) |
| Proofs | CONFIRMED (every step checked) |
| Computations | CONFIRMED (structured collapse, d ≤ 10) |
| Answer as posed | CONFIRMED (positive) |
| Novelty | CONFIRMED as far as can be checked; the new content is only the product structure (the triviality of the bundle) |
| Presentation | CONFIRMED_WITH_FIXES |

All four required fixes were applied:
1. Credit Klee–Novik Remark 3.7 (the collapse and the disc bundle), and present the new content as the product
   structure via the join factor.
2. State the PL results used explicitly: the collapsing characterisation of regular neighbourhoods with X in the
   interior of N, uniqueness, and the product structure of the neighbourhood of a join factor (the paper proves the
   last one, Lemma 3.2). Also state the existence of the outer collar via C(i,d) ≅ B(d−i−2,d).
3. Check the published version for changes to Remark 3.7. The published version was inaccessible; the authors'
   revised version of July 2011 has the same Remark 3.7.
4. Use the structured collapse, not a single greedy collapse, in the computer check.

**Second independent verification run** (2026-09-30, on the final text of the earlier draft). Verdict: correct with
fixes; the mathematics is correct and complete, and the fixes are editorial or about precision. It checked Lemmas 3.1–3.3,
the proof of Theorem 1.2 (not circular; Klee–Novik Thm 1.2(e) is not used), Corollary 1.3(b) (including the wording of
Lickorish, Thm 5.10, and the separate case i = 0) and the attributions to Klee–Novik, Cohen–Klee–Pannell and Machacek.
Its fixes, and how they were applied:

| Fix | Applied as |
|---|---|
| 1. The outer collar: \|B(i,d)\| together with an outer collar is the regular neighbourhood, not B(i,d) itself | Introduction and abstract reworded; new Remark 3.4 proves that every vertex of C*_d lies in ∂B(i,d), so the collar is needed |
| 2. "This bundle is trivial" → "\|B(i,d)\| is in fact a product" | abstract, introduction and scope paragraph |
| 3. The answer to CKP Question 1.1 is existential and non-constructive, and includes a simplicial isomorphism | text after Corollary 1.3; Question 1.1 re-read in the 2011 preprint |
| 4. Add i = d−2 [CKP14] to the known cases | scope paragraph |
| 5. "Novik's web page" instead of "the second author's web page" | scope paragraph |
| 6. Verification paragraph: remove the self-contradictory description of the AI-assisted runs; explain the exploratory greedy collapse; a greedy collapse stuck at (5,9) does not contradict collapsibility; harmonise "a program by the author" with the AI disclosure; say where the programs are | Verification paragraph rewritten |
| 7. Notation: M(σ,τ) clashed with M = \|C*_d\|; w was both a vertex and a coordinate | M(σ,τ) renamed Q'(σ,τ); the vertex is now a_j and the coordinates are z = (z', z''); the derived-subdivision points are now written P̂ |
| 8. "the second and third claims"; the facet-adjacency graph of lk F is a cube graph, hence connected | Lemma 3.3 |
| 9. Statement numbers: Rourke–Sanderson (or say which are unavailable), KN numbering from arXiv:1102.0542, CKP numbering from the 2011 preprint | new "Sources" paragraph at the end of the introduction; Rourke–Sanderson was unavailable, so no statement numbers are given |
| 10. (Optional) Fact 2.1(a): openness at t = 0 comes from the neighbourhood property | Fact 2.1(a) |
| Also: R(G_j) ≠ ∅ for j ≥ 2 in Lemma 3.1 | Lemma 3.1 |
| Also: \|F\| ≤ d−1 in Lemma 3.3 | Lemma 3.3 |
| Also: Machacek never mentions Klee–Novik | scope paragraph. Machacek's arXiv:1909.04640v2 indeed does not mention them. The identification of his space with the antipodal quotient of \|B(i,d)\| is given in the paper via the filling criterion, but it is not claimed as new: Karp–Machacek (arXiv:2104.02786; Combinatorial Theory 3 (2023)) note that his face poset is a quotient of the face poset of B(l,n), and the paper cites them |

The run could not check, from the files it had, the quotation on p. 372 of the report and the speaker of the abstract,
or what Wang–Zheng's Lemma 2.2 says. Both were checked during the revision from anonymous downloads (see "Statement
checked" and "Closest works"): the abstract is by Steven Klee, joint work with Isabella Novik, and the paper now cites
it by speaker and title.

## Relation to the literature, novelty and scope
- **Searches (September 30, 2026; anonymous requests only; logged).**
  - arXiv: metadata of 1102.0542 (one version only) and 1811.08505 through the API; full texts of 1102.0542v1,
    1811.08505v2, 1909.04640v2, 2104.02786v2 and of the citing works marked as read.
  - Crossref: DOIs of all references.
  - zbMATH Open: four documents citing Zbl 1236.52010.
  - Semantic Scholar: 24 citing works.
  - OpenAlex: the list of citing works could not be retrieved (rate limit).
  - Novik's publication page, and one web search.
  - EMS Press: the full Oberwolfach Report 08/2011.
  - The list of citing works, with the ones read marked, is in `reproducibility/lead/citing_works_KN12_2026-09-30.txt`.
- **Closest works.**
  - Cohen–Klee–Pannell, "Bistellar equivalences of two families of simplicial complexes", J. Combin. Math. Combin.
    Comput. 89 (2014) 65–85 (Zbl 1304.05162). We read the preprint dated August 18, 2011 (an archived copy of Klee's
    former web page) and the zbMATH review; the paper uses the preprint's numbering. The preprint conjectures that
    |B(k,d)| and |ST(k+1,d−k−1)| are homeomorphic, asks in Question 1.1 whether B(k,d) can be obtained from
    ST(k+1,d−k−1) by bistellar moves, stellar exchanges, elementary shellings and their inverses, answers it for k = 0
    (Section 2, an isomorphism) and k = d−2 (Section 3), and treats B(1,4) and B(1,5) (Section 4).
  - Wang–Zheng, "Centrally symmetric and balanced triangulations of S^2 × S^(d−3) with few vertices", Eur. J. Combin.
    85 (2020) 103043, arXiv:1811.08505. In v2 (p. 4), Lemma 2.2, which the authors describe as essentially
    Theorem 1.2 of Klee–Novik, records in item 3 that ‖B(0,d)‖ and ‖B(1,d)‖ are homeomorphic to D^(d−1) × S^0 and
    D^(d−2) × S^1. It says nothing of this kind for i ≥ 2. The paper cites it for exactly this, with the numbering of v2.
  - Machacek (Ann. Inst. Henri Poincaré D 9 (2022) 543–565; arXiv:1909.04640) proves that the space of points of
    RP^(n−1) with sign variation at most m (Theorem 3.4: a PL manifold; Theorem 3.6: it collapses onto RP^m, in the
    numbering of arXiv v2). By the filling criterion this space is the antipodal quotient of |B(m,n)|; this is the
    projective analogue of Remark 3.7, and he determines the homotopy type only. He does not mention Klee and Novik.
    Karp–Machacek (Combinatorial Theory 3 (2023); arXiv:2104.02786) note that the face poset of his space is a quotient
    of the face poset of B(l,n).
  - Other works read that do not treat the question: Bagchi–Datta (2011, 2013) on Klee–Novik manifolds, Murai–Nevo,
    Murai–Novik, Datta–Murai, Spreer, Novik–Zheng, Kühnel–Spreer, Spreer–Tobin. Also the surveys Klee–Novik (2016),
    Novik (2019) and Novik (ICM 2022).
- **Caveats.**
  - The published versions of Klee–Novik and of Cohen–Klee–Pannell were not seen.
  - Rourke–Sanderson was not accessible, so statement numbers are not quoted.
  - The argument is short and uses only standard PL topology. For i ≤ d−4 the result also follows from the collapse of
    Remark 3.7, the outer collar and Zeeman's unknotting theorem, so it may be known to experts.

  This negative search is not a proof of priority.
- **Scope.** The note answers Question 4 of OWR 8/2011 as posed, for all 0 ≤ i ≤ d−2. It also gives the PL form of
  Klee–Novik Theorem 1.2(e), confirms the conjecture of Cohen–Klee–Pannell, and gives an existential answer to their
  Question 1.1.

## 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/