{
  "id": "1404.4231",
  "version": "v3",
  "published": "2014-04-16T13:07:54.000Z",
  "updated": "2014-05-07T13:22:06.000Z",
  "title": "Geometric structures, Gromov norm and Kodaira dimensions",
  "authors": [
    "Weiyi Zhang"
  ],
  "comment": "33 pages. v3, more references added",
  "categories": [
    "math.GT",
    "math.SG"
  ],
  "abstract": "We define the Kodaira dimension for $3$-dimensional manifolds through Thurston's eight geometries, along with a classification in terms of this Kodaira dimension. We show this is compatible with other existing Kodaira dimensions and the partial order defined by non-zero degree maps. For higher dimensions, we explore the relations of geometric structures and mapping orders with various Kodaira dimensions and other invariants. Especially, we show that a closed geometric $4$-manifold has nonvanishing Gromov norm if and only if it has geometry $\\mathbb H^2\\times \\mathbb H^2$, $\\mathbb H^2(\\mathbb C)$ or $\\mathbb H^4$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-07T13:22:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric structures",
      "non-zero degree maps",
      "nonvanishing gromov norm",
      "dimensional manifolds",
      "higher dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4231Z"
    }
  }
}