{
  "id": "math/0603429",
  "version": "v5",
  "published": "2006-03-17T14:11:55.000Z",
  "updated": "2008-01-15T17:38:23.000Z",
  "title": "A classification of smooth embeddings of 3-manifolds in 6-space",
  "authors": [
    "A. Skopenkov"
  ],
  "comment": "32 pages, a link to http://www.springerlink.com added, to appear in Math. Zeit",
  "journal": "Math. Zeitschrift, 260:3 (2008) 647-672",
  "doi": "10.1007/s00209-007-0294-1",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "We work in the smooth category. If there are knotted embeddings S^n\\to R^m, which often happens for 2m<3n+4, then no concrete complete description of embeddings of n-manifolds into R^m up to isotopy was known, except for disjoint unions of spheres. Let N be a closed connected orientable 3-manifold. Our main result is the following description of the set Emb^6(N) of embeddings N\\to R^6 up to isotopy. The Whitney invariant W : Emb^6(N) \\to H_1(N;Z) is surjective. For each u \\in H_1(N;Z) the Kreck invariant \\eta_u : W^{-1}u \\to Z_{d(u)} is bijective, where d(u) is the divisibility of the projection of u to the free part of H_1(N;Z). The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on Emb^6(N) by embedded connected sum. It was proved that the orbit space of this action maps under W bijectively to H_1(N;Z) (by Vrabec and Haefliger's smoothing theory). The new part of our classification result is determination of the orbits of the action. E. g. for N=RP^3 the action is free, while for N=S^1\\times S^2 we construct explicitly an embedding f : N \\to R^6 such that for each knot l:S^3\\to R^6 the embedding f#l is isotopic to f. Our proof uses new approaches involving the Kreck modified surgery theory or the Boechat-Haefliger formula for smoothing obstruction.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2008-01-15T17:38:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R40",
      "57R52"
    ],
    "keywords": [
      "smooth embeddings",
      "classification",
      "concrete complete description",
      "kreck modified surgery theory",
      "disjoint unions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3429S"
    }
  }
}