{
  "id": "1701.01711",
  "version": "v1",
  "published": "2017-01-06T18:00:40.000Z",
  "updated": "2017-01-06T18:00:40.000Z",
  "title": "Functions on surfaces and constructions of manifolds in dimensions three, four and five",
  "authors": [
    "David T Gay"
  ],
  "comment": "19 pages, 5 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We offer a new proof that two closed oriented 4-manifolds are cobordant if their signatures agree, in the spirit of Lickorish's proof that all closed oriented 3-manifolds bound 4-manifolds. Where Lickorish uses Heegaard splittings we use trisections. In fact we begin with a subtle recasting of Lickorish's argument: Instead of factoring the gluing map for a Heegaard splitting as a product of Dehn twists, we encode each handlebody in a Heegaard splitting in terms of a Morse function on the surface and build the 4-manifold from a generic homotopy between the two functions. This extends up a dimension by encoding a trisection of a closed 4-manifold as a triangle (circle) of functions and constructing an associated 5-manifold from an extension to a 2-simplex (disk) of functions. This borrows ideas from Hatcher and Thurston's proof that the mapping class group of a surface is finitely presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-06T18:00:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M99",
      "57R90",
      "57R45"
    ],
    "keywords": [
      "constructions",
      "heegaard splitting",
      "dehn twists",
      "lickorishs proof",
      "lickorishs argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}