{
  "id": "1711.04762",
  "version": "v1",
  "published": "2017-11-13T18:58:42.000Z",
  "updated": "2017-11-13T18:58:42.000Z",
  "title": "Calculating the homology and intersection form of a 4-manifold from a trisection diagram",
  "authors": [
    "Peter Feller",
    "Michael Klug",
    "Trent Schirmer",
    "Drew Zemke"
  ],
  "comment": "14 pages, 2 figures. Submitted to appear as part of the Proceedings of the National Academy of Sciences collection on trisections of 4-manifolds",
  "categories": [
    "math.GT"
  ],
  "abstract": "Given a diagram for a trisection of a 4-manifold $X$, we describe the homology and the intersection form of $X$ in terms of the three subgroups of $H_1(F;\\mathbb{Z})$ generated by the three sets of curves and the intersection pairing on $H_1(F;\\mathbb{Z})$. This includes explicit formulas for the second and third homology groups of $X$ as well an algorithm to compute the intersection form. Moreover, we show that all $(g;k,0,0)$-trisections admit \"algebraically trivial\" diagrams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-13T18:58:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intersection form",
      "trisection diagram",
      "third homology groups",
      "explicit formulas",
      "calculating"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}