{
  "schema_version": 1,
  "problem_number": "OWR-4798-013",
  "title": "The Klee–Novik Complexes B(i,d) Triangulate S^i × B^(d−i−1)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For 0 ≤ i ≤ d − 2, Klee and Novik defined B(i,d) as the subcomplex of the boundary of the d-dimensional cross-polytope generated by the facets whose xy-words have at most i switches, and asked whether B(i,d) is a combinatorial triangulation of S^i × B^(d−i−1). We show that it is, for all 0 ≤ i ≤ d − 2. Klee and Novik observed that B(i,d) collapses onto the boundary of the (i+1)-dimensional cross-polytope and noted that it is therefore a disc bundle over S^i. We show that |B(i,d)| is in fact a product. That sphere is a join factor of the boundary of the d-dimensional cross-polytope, so it has a regular neighbourhood that is a product, and |B(i,d)| together with an outer collar is another regular neighbourhood of it; uniqueness of regular neighbourhoods gives the result. Consequently ∂B(i,d) is PL homeomorphic to S^i × S^(d−i−2), and a conjecture of Cohen, Klee and Pannell holds. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "keywords": [
    "combinatorial manifold",
    "cross-polytope",
    "centrally symmetric triangulation",
    "sphere products",
    "collapsing",
    "regular neighbourhood",
    "join",
    "staircase triangulation",
    "Oberwolfach Reports",
    "OWR-4798-013",
    "math.GT",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-4798-013/",
  "pdf_url": "https://eulersolve.org/papers/owr-4798-013/paper.pdf?v=258be8794773",
  "doi": "10.5281/zenodo.23063421",
  "zenodo_record_url": "https://zenodo.org/records/23063421",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Klee and Novik's Question 4 (OWR 8/2011, p. 372) affirmatively for all 0 ≤ i ≤ d − 2. The collapse onto the boundary of the (i+1)-cross-polytope and the disc-bundle observation are due to Klee and Novik; what is new is the product structure via the join factor. Cohen–Klee–Pannell's Question 1.1 is answered only existentially.",
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    "source.zip": {
      "sha256": "27174fcf4c55b3ad06ea114d679dd44119dcd324cdb81248fcb1a8618b9044d0"
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    "verification_report.md": {
      "sha256": "f5acd168d6da5fd6b7f04d50ef219b271e11b8df288232b23525b573feb51302"
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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."
}