{
  "id": "1612.04776",
  "version": "v1",
  "published": "2016-12-14T19:22:32.000Z",
  "updated": "2016-12-14T19:22:32.000Z",
  "title": "Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification",
  "authors": [
    "D. Crowley",
    "A. Skopenkov"
  ],
  "comment": "16 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 readily calculable classification of embeddings $N\\to R^7$ up to isotopy, with an indeterminancy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9. The group of knots $S^4\\to R^7$ acts on the set of embeddings $N\\to R^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1\\ne0$, with an indeterminancy. Besides the invariants of Part I, the classification involves a refinement of the Kreck invariant from our work of 2008 which detects the action of knots. For $N=S^1\\times S^3$ we give a geometrically defined 1--1 correspondence between the set of isotopy classes of embeddings and a quotient of the set $Z\\oplus Z\\oplus Z_{12}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-14T19:22:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R52",
      "57R67",
      "55R15"
    ],
    "keywords": [
      "smooth classification",
      "embeddings",
      "isotopy classes",
      "indeterminancy",
      "smooth category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}