{
  "abstract": "We construct an explicit, globally continuous domain exchange involution F : S 2 → S 2 that maps a closed Jordan domain into its complementary region on the two-dimensional sphere. The transformation is uniquely anchored via a projected spherical center of mass computed from the boundary embedding, guaranteeing global existence and uniqueness for any non-origin-symmetric closed Jordan curve. Governed by this reference pole and its antipode, the transformation operates locally as a homeomorphism between the punctured complementary domains and globally as an intrinsic inversion of the entire manifold S 2 . This concurrent deformation is formulated as a dual family of synchronized gradient homotopies under a canonical Morse height function. By exploiting the structural cut loci of the complementary domains, we prove that the global mapping extends continuously across the singular subsets to form a global bijection. The resulting operator preserves the topological invariants of the sphere with a stable topological degree of deg(F ) = +1. Finally, the global deformation is formalized as an inversion homotopy restricted entirely to the intrinsic geometry of the two-dimensional manifold, establishing a structural reciprocity between the contraction of a 1-manifold loop and its dual 1-dimensional cut locus complexes. Finally, the framework is extended to higher-dimensional spheres S n (n ≥ 3), formalizing a generalized Point-Hyperloop Duality where the topological degree deg(F ) = +1 remains stable due to orientation consistency under opposite driving elds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Abdellatif Aitelhad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cak6hzat18ckf.bib",
  "cited_by": 0,
  "doi": "10.13140/rg.2.2.19734.51529",
  "fulltext_url": null,
  "id": "cak6hzat18ckf",
  "issns": [],
  "landing_url": "https://hal.science/hal-05840149v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221780398",
  "pdf_url": "https://hal.science/hal-05840149/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cak6hzat18ckf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05840149",
      "source": "hal",
      "url": "https://hal.science/hal-05840149v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Geometric and Algebraic Topology",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An Intrinsic Domain Exchange Involution via Dual Morse Flows and Point-Loop Duality",
  "title_text": "An Intrinsic Domain Exchange Involution via Dual Morse Flows and Point-Loop Duality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:01Z",
  "url": "https://research.draconitic.com/paper/cak6hzat18ckf",
  "venue": "HAL",
  "versions": []
}