{
  "abstract": "We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space $Π(X)$ of homotopy classes of paths (relative endpoints) to the path space $X^I$ that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if $X$ has the dpp, then its universal cover is contractible and, in particular, $X$ is aspherical. The space $Π(X)$ is a fiber bundle over $X$ whose fiber is the universal cover of $X$, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from $Π(X)$ to $X^I$ and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04440",
  "author_total": 2,
  "authors": [
    {
      "name": "Elias Gabriel Minian"
    },
    {
      "name": "Juan Martín Perez Garber"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kfq80vnhxzkg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04440",
  "fulltext_url": null,
  "id": "3kfq80vnhxzkg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04440",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7211894642",
  "pdf_url": "https://arxiv.org/pdf/2609.04440",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3kfq80vnhxzkg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04440",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04440"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A topological version of the Cartan-Hadamard Theorem and asphericity complexity",
  "title_text": "A topological version of the Cartan-Hadamard Theorem and asphericity complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/3kfq80vnhxzkg",
  "venue": "arXiv",
  "versions": []
}