{
  "abstract": "We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold $\\RP^2\\times S^2$. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover $S^2\\times S^2$ has degree $+1$ has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel $π_4(\\RP^2\\times S^2)\\cong(\\Z/2)^2$ generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton--Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to $\\RP^2\\times S^2$ and to $\\RP^4$, and for the Hambleton--Hillman manifold with fundamental group $(\\Z/2)^2$ we determine the normal $1$-type and show that self-equivalences act on $π_2$ through diagonal sign matrices. For $\\RP^2\\times S^2$ the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16606",
  "author_total": 1,
  "authors": [
    {
      "name": "Mehmetcik Pamuk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ye1vz513jy4f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16606",
  "fulltext_url": null,
  "id": "7ye1vz513jy4f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16606",
  "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.AT"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221778557",
  "pdf_url": "https://arxiv.org/pdf/2610.16606",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ye1vz513jy4f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16606",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16606"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Homotopy Self-Equivalences of $\\mathbb{R}P^2\\times S^2$ and a Braid for non-orientable $4$-Manifolds",
  "title_text": "Homotopy Self-Equivalences of ℝP^2× S^2 and a Braid for non-orientable $4$-Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7ye1vz513jy4f",
  "venue": "arXiv",
  "versions": []
}