{
  "abstract": "Let $Z$ be any fixed closed connected PL $4$-manifold. We give a polynomial-time many-to-one reduction from the compressed word problem in Thompson's group $F$ to fixed-target recognition of $Z\\#_{2}(S^2\\times S^2)$. Once a finite triangulation of $Z$ has been fixed, the construction sends a straight-line program $A$ to a closed triangulated PL $4$-manifold $X_{A,Z}$ such that $\\operatorname{val}(A)=1$ in $F$ if and only if $X_{A,Z}$ is PL-homeomorphic to $Z\\#_2(S^2\\times S^2)$. Since the compressed word problem in $F$ is PSPACE-complete, every such recognition problem is PSPACE-hard. In particular, fixed-target recognition of $\\#_n(S^2\\times S^2)$ is PSPACE-hard for every fixed $n\\geq2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15890",
  "author_total": 1,
  "authors": [
    {
      "name": "Rhuaidi Antonio Burke"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mjegdrahzdyj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15890",
  "fulltext_url": null,
  "id": "9mjegdrahzdyj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15890",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7213233136",
  "pdf_url": "https://arxiv.org/pdf/2609.15890",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mjegdrahzdyj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15890",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15890"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "PL Recognition After Two $S^2\\times S^2$ Stabilisations is PSPACE-Hard",
  "title_text": "PL Recognition After Two S^2× S^2 Stabilisations is PSPACE-Hard",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9mjegdrahzdyj",
  "venue": "arXiv",
  "versions": []
}