{
  "abstract": "Hambleton and Kreck described the group of based homotopy self-equivalences of a closed oriented $4$-manifold by a commutative braid of exact sequences built from bordism groups and the Postnikov $2$-type. We construct the analogous braid for closed oriented $BO\\langle n+1\\rangle$-manifolds $M$ of dimension $2n$ or $2n-1$ whose fundamental group is finite or infinite cyclic and which satisfy one condition on the action of $π_1$ on the homotopy of the fibre of $M\\to P_n(M)$. We compute $\\operatorname{Aut}_\\bullet(S^1\\times S^{m-1})\\cong(\\mathbb{Z}/2)^2$ for $m\\ge4$, and $\\operatorname{Aut}_\\bullet(S^3\\times S^3)\\cong(\\mathbb{Z}/12)^2\\rtimes SL_2(\\mathbb{Z})$ with the standard action modulo $12$. Further, $\\operatorname{Aut}_\\bullet(S^3\\times S^4)\\cong(\\mathbb{Z}/2\\oplus\\mathbb{Z}/24)\\rtimes(\\mathbb{Z}/2)^2$ has order $192$, that $\\operatorname{Aut}_\\bullet(S^4\\times S^4)\\cong D_4\\times\\mathbb{Z}/2$ has order $16$, and that for connected sums of $r\\ge2$ copies of $S^3\\times S^3$ the kernel contains a subgroup $(\\mathbb{Z}/12)^{2r}$ on which the permutations of the summands act non-trivially. For $S^p\\times S^p$ the kernel is a sum of two copies of $π_{2p}(S^p)$ modulo Whitehead products with $ι_p$, and the image is determined by an obstruction on the Whitehead square. In particular $\\operatorname{Aut}_\\bullet(S^5\\times S^5)$ is isomorphic to the theta subgroup of $SL_2(\\mathbb{Z})$ and $\\operatorname{Aut}_\\bullet(S^7\\times S^7)$ is an extension of $SL_2(\\mathbb{Z})$ by $(\\mathbb{Z}/120)^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02244",
  "author_total": 1,
  "authors": [
    {
      "name": "Mehmetcik Pamuk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fy8ax0ggx80fd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02244",
  "fulltext_url": null,
  "id": "fy8ax0ggx80fd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02244",
  "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",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7220426016",
  "pdf_url": "https://arxiv.org/pdf/2610.02244",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fy8ax0ggx80fd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.02244",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02244"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Homotopy Self-Equivalences of Manifolds",
  "title_text": "Homotopy Self-Equivalences of Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/fy8ax0ggx80fd",
  "venue": "arXiv",
  "versions": []
}