{
  "id": "1502.04470",
  "version": "v1",
  "published": "2015-02-16T09:19:12.000Z",
  "updated": "2015-02-16T09:19:12.000Z",
  "title": "Classification of knotted tori",
  "authors": [
    "A. Skopenkov"
  ],
  "comment": "18 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "For a smooth manifold N denote by E^m(N) the set of smooth isotopy classes of smooth embeddings N -> R^m. A description of the set E^m (S^p x S^q) was known only for p=0, m > q+2 or for 2m > 3p+3q+3 (in terms of homotopy groups of spheres and Stiefel manifolds). For m > 2p+q+2 we introduce an abelian group structure on E^m (S^p x S^q) and describe this group up to an extension problem: this group and E^m (D^{p+1} x S^q) + ker l + E^m (S^{p+q}) are associated to the same group for some filtrations of length four. Here l : E -> pi_q(S^{m-p-q-1}) is the linking coefficient defined on the subset E of E^m (S^q U S^{p+q}) formed by isotopy classes of embeddings whose restriction to each component is unknotted. This result and its proof have corollaries which, under stronger dimension restrictions, more explicitly describe E^m (S^p x S^q) in terms of homotopy groups of spheres and Stiefel manifolds. In the proof we use a recent exact sequence of M. Skopenkov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-16T09:19:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R40",
      "57R52",
      "55Q40"
    ],
    "keywords": [
      "knotted tori",
      "classification",
      "stiefel manifolds",
      "homotopy groups",
      "abelian group structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}