{
  "abstract": "The critical dimension for free maps on m-manifolds is $q_m=m(m+3)/2$. We show that $Free^\\infty(\\mathbb{T}^m,\\mathbb{R}^{q_m})\\neq\\emptyset$ for every $m\\ge 1$. The main tool is a product construction: given free maps in critical dimension on $\\mathbb{T}^a$ and on $\\mathbb{T}^b$, together with a cross-free map on $\\mathbb{T}^a\\times \\mathbb{T}^b$, whose mixed Hessian is invertible everywhere, we obtain a free map in critical dimension on $\\mathbb{T}^{a+b}$, the osculating matrix of the product being block-triangular. Cross-free maps are additive in each argument, and in the torus setting none exists when one factor is one-dimensional. We construct explicit cross-free maps on $\\mathbb{T}^2\\times \\mathbb{T}^2$ and on $\\mathbb{T}^3\\times\\mathbb{T}^4$ using quaternionic multiplication. Combined with the low-dimensional cases $m\\leq 5$, constructed explicitly by the author in a previous publication, these yield the result by induction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23172",
  "author_total": 1,
  "authors": [
    {
      "name": "Roberto De Leo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bj7tqdv8yc8n7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23172",
  "fulltext_url": null,
  "id": "bj7tqdv8yc8n7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23172",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7213999511",
  "pdf_url": "https://arxiv.org/pdf/2609.23172",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bj7tqdv8yc8n7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23172",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23172"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Free maps in critical dimension on $\\mathbb{T}^m$",
  "title_text": "Free maps in critical dimension on 𝕋^m",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bj7tqdv8yc8n7",
  "venue": "arXiv",
  "versions": []
}