{
  "id": "1611.04738",
  "version": "v1",
  "published": "2016-11-15T08:36:24.000Z",
  "updated": "2016-11-15T08:36:24.000Z",
  "title": "Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots",
  "authors": [
    "D. Crowley",
    "A. Skopenkov"
  ],
  "comment": "50 pages",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a complete readily calculable classification of embeddings $N\\to R^7$, up to the equivalence relation generated by isotopy and embedded connected sum with embeddings $S^4\\to R^7$. Such a classification was already known when $H_1=0$ by the work of Bo\\'echat, Haefliger and Hudson from 1970. Our results for $H_1\\ne0$ are new. The classification involves the Bo\\'echat-Haefliger invariant $\\varkappa(f)\\in H_2$, and two new invariants: a Seifert bilinear form $\\lambda(f):H_3\\times H_3\\to Z$ and $\\beta$-invariant $\\beta(f)$ which assumes values in a quotient of $H_1$ depending on the values of $\\varkappa(f)$ and $\\lambda(f)$. For $N=S^1\\times S^3$ we give a geometrically defined 1-1 correspondence between the set of equivalence classes of embeddings and an explicit quotient of the set $Z\\oplus Z$. Our proof is based on Kreck's modified surgery approach to the classification of embeddings, and also uses parametric connected sum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-15T08:36:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R40",
      "57R52",
      "57R67",
      "57Q35",
      "55R15"
    ],
    "keywords": [
      "classification modulo knots",
      "embeddings",
      "seifert bilinear form",
      "krecks modified surgery approach",
      "parametric connected sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}