{
  "id": "1408.3918",
  "version": "v2",
  "published": "2014-08-18T08:04:39.000Z",
  "updated": "2015-09-22T09:24:42.000Z",
  "title": "The classification of certain linked $3$-manifolds in $6$-space",
  "authors": [
    "Sergey Avvakumov"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "We work entirely in the smooth category. An embedding $f:(S^2\\times S^1)\\sqcup S^3\\rightarrow {\\mathbb R}^6$ is {\\it Brunnian}, if the restriction of $f$ to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$ such that $m\\equiv n \\pmod{2}$, we explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\\times S^1)\\sqcup S^3 \\rightarrow {\\mathbb R}^6$ such that the following theorem holds. Theorem: Any Brunnian embedding $f:(S^2\\times S^1)\\sqcup S^3\\rightarrow {\\mathbb R}^6$ is isotopic to $f_{k,m,n}$ for some integers $k,m,n$ such that $m\\equiv n \\pmod{2}$. Two embeddings $f_{k,m,n}$ and $f_{k',m',n'}$ are isotopic if and only if $k=k'$, $m\\equiv m' \\pmod{2k}$ and $n\\equiv n' \\pmod{2k}$. We use Haefliger's classification of embeddings $S^3\\sqcup S^3\\rightarrow {\\mathbb R}^6$ in our proof. The following corollary shows that the relation between the embeddings $(S^2\\times S^1)\\sqcup S^3\\rightarrow {\\mathbb R}^6$ and $S^3\\sqcup S^3\\rightarrow {\\mathbb R}^6$ is not trivial. Corollary: There exist embeddings $f:(S^2\\times S^1)\\sqcup S^3\\rightarrow {\\mathbb R}^6$ and $g,g':S^3\\sqcup S^3\\rightarrow {\\mathbb R}^6$ such that the componentwise embedded connected sum $f\\#g$ is isotopic to $f\\#g'$ but $g$ is not isotopic to $g'$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-18T08:04:39.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-09-22T09:24:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth category",
      "brunnian embedding",
      "theorem holds",
      "haefligers classification",
      "restriction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3918A"
    }
  }
}