{
  "id": "2110.02431",
  "version": "v2",
  "published": "2021-10-06T00:43:12.000Z",
  "updated": "2022-07-08T06:13:49.000Z",
  "title": "Presentation of the fundamental groups of complements of shadows",
  "authors": [
    "Masaharu Ishikawa",
    "Yuya Koda",
    "Hironobu Naoe"
  ],
  "comment": "25 pages, 20 figures",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "A shadowed polyhedron is a simple polyhedron equipped with half integers on regions, called gleams, which represents a compact, oriented, smooth 4-manifold. The polyhedron is embedded in the 4-manifold and it is called a shadow of that manifold. A subpolyhedron of a shadow represents a possibly singular subsurface in the 4-manifold. In this paper, we focus on contractible shadows obtained from the unit disk by attaching annuli along generically immersed closed curves on the disk. In this case, the 4-manifold is always a 4-ball. Milnor fibers of plane curve singularities and complexified real line arrangements can be represented in this way. We give a presentation of the fundamental group of the complement of a subpolyhedron of such a shadow in the 4-ball. The method is very similar to the Wirtinger presentation of links in knot theory.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-07-08T06:13:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S55",
      "32S30",
      "57M05",
      "52C35"
    ],
    "keywords": [
      "fundamental group",
      "presentation",
      "complement",
      "plane curve singularities",
      "complexified real line arrangements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}