{
  "schema_version": 1,
  "problem_number": "OWR-14298580-008",
  "title": "A Negative Answer to the Strong Geography Question of Alfieri and Binns",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Alfieri and Binns say that an F[U]-module M satisfies the strong geography restriction if it has a direct summand F[U]/U^ℓ ⊕ F[U]/U^(ℓ−1) ⊕ ⋯ ⊕ F[U]/U, where ℓ is the least integer with U^ℓ M_red = 0. They showed that HF⁻(Y) satisfies it when Y is surgery on a knot in S³ or large surgery on a link, and asked whether it holds for every rational homology sphere Y. We show that the answer is no. For the Brieskorn sphere Y = Σ(30,47,83), with either orientation, the reduced Heegaard Floer homology is T_16 ⊕ T_14^10 ⊕ T_13^16 ⊕ ⋯ ⊕ T_1^92, where T_k = F[U]/U^k. So ℓ = 16, but F[U]/U^15 is not a direct summand of HF⁻(Y). The proof combines the Ozsváth–Szabó description of HF⁺ for plumbed manifolds, Némethi's reduction to the graded root of an explicit function τ, and an exact computer calculation. We give a certificate, the 1707 turning points of τ, from which the summand lengths can be rechecked by a short program. The manifold Y is an irreducible integral homology sphere, is not an L-space, and satisfies Lin's weaker restriction. It also shows that the word \"large\" cannot be removed from the link-surgery theorem of Alfieri and Binns. Computer searches with two independent programs show that among Seifert fibred integral homology spheres Σ(a_1,…,a_n), the counterexamples with the smallest product a_1⋯a_n are Σ(30,47,83) and three spheres with four singular fibres, all of product 117030. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GT"
  ],
  "keywords": [
    "OWR-14298580-008",
    "UnsolvedMath",
    "Oberwolfach Reports",
    "Heegaard Floer homology",
    "strong geography restriction",
    "Lin geography restriction",
    "Brieskorn sphere",
    "Seifert fibred homology sphere",
    "plumbed three-manifolds",
    "graded root",
    "counterexample",
    "math.GT",
    "open mathematics",
    "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-14298580-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-14298580-008/paper.pdf?v=246de76cb53d",
  "doi": "10.5281/zenodo.23063072",
  "zenodo_record_url": "https://zenodo.org/records/23063072",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the Alfieri–Binns strong geography question (OWR 34/2024; arXiv:2404.00490, Question 1.10) negatively with the Brieskorn sphere Σ(30,47,83), either orientation. The proof is computer-assisted with a printed certificate; the minimal-product statement rests on computer searches without per-sphere certificates. Lin's geography restriction and the L-space conjecture are not addressed.",
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    "source.zip": {
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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."
}