{
  "id": "2011.03784",
  "version": "v1",
  "published": "2020-11-07T14:33:23.000Z",
  "updated": "2020-11-07T14:33:23.000Z",
  "title": "Distinguishing 4-dimensional geometries via profinite completions",
  "authors": [
    "Jiming Ma",
    "Zixi Wang"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a closed orientable 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except when the geometry is $ \\mathbb{H}^{4}$, $\\mathbb{H}^{2}_{\\mathbb{C}}$ or $\\mathbb{H}^2 \\times \\mathbb{H}^2$. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some 4-dimensional manifolds with Seifert fibred structures are indeed geometric. For a closed orientable Seifert fibred 4-manifold $M$, we show that whether $M$ is geometric could be detected by the profinite completion of its fundamental group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-07T14:33:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "57N16",
      "57M05"
    ],
    "keywords": [
      "profinite completion",
      "fundamental group",
      "geometric information",
      "distinguishing",
      "seifert fibred structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}