{
  "abstract": "We show that there exist exotic spheres with metrics of strictly positive sectional curvature in infinitely many dimensions: in every dimension $n\\equiv3\\pmod4$ with $n\\geq11$ and $n\\not\\equiv255\\pmod{256}$. Previously, every known closed simply connected manifold of positive sectional curvature in a dimension above $24$ was diffeomorphic to a compact rank-one symmetric space. The spheres we consider are certain homotopy spheres arising as quotients of special $S^3$-$S^3$ bundles introduced by Sperança. These homotopy spheres are determined by an orthogonal representation $ρ$ of $S^3$ and an $S^3$-conjugation-equivariant map $β$, and are denoted $Σ_{ρ,β}$. By recent constructions, such spheres admit Riemannian metrics of strictly positive sectional curvature. In this paper we consider the spheres $Σ_{ρ_b,β}$ of dimension $4b+3$, $b\\geq2$, where $ρ_b$ is the representation of $S^3$ on $\\mathbb{R}^3\\oplus\\mathbb{H}^b$ by left multiplication on the second summand. For these spheres, we show that the Eells-Kuiper invariant is the product of an integer $k(β)$ depending only on $β$ and a rational number $r_b$ given by one coefficient of a power series built from the $\\hat A$- and $L$-genera. We then show that, for every $b\\geq2$ with $b\\not\\equiv63\\pmod{64}$, there is a map $β$ for which the Eells-Kuiper invariant does not vanish, and thus the corresponding sphere is exotic. A computer calculation also gives the dimensions $n\\equiv255\\pmod{256}$ below $2^{16}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31528",
  "author_total": 1,
  "authors": [
    {
      "name": "Fernando Galaz-García"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6vem6rap2wzxy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31528",
  "fulltext_url": null,
  "id": "6vem6rap2wzxy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31528",
  "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.DG"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7223279790",
  "pdf_url": "https://arxiv.org/pdf/2610.31528",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6vem6rap2wzxy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31528",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31528"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Positive sectional curvature on exotic spheres in infinitely many dimensions",
  "title_text": "Positive sectional curvature on exotic spheres in infinitely many dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/6vem6rap2wzxy",
  "venue": "arXiv",
  "versions": []
}