{
  "id": "1407.2148",
  "version": "v1",
  "published": "2014-07-08T16:06:41.000Z",
  "updated": "2014-07-08T16:06:41.000Z",
  "title": "Open book decompositions versus prime factorizations of closed, oriented 3-manifolds",
  "authors": [
    "Paolo Ghiggini",
    "Paolo Lisca"
  ],
  "comment": "8 pages, 1 figure. Submitted to the proceedings of the conference \"Interactions between low dimensional topology and mapping class groups\", July 1-5, 2013, Max Planck Institute for Mathematics, Bonn",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $M$ be a closed, oriented, connected 3--manifold and $(B,\\pi)$ an open book decomposition on $M$ with page $\\Sigma$ and monodromy $\\varphi$. It is easy to see that the first Betti number of $\\Sigma$ is bounded below by the number of $S^2\\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is realized if and only if $\\varphi$ is trivial and $M$ is a connected sum of $S^2\\times S^1$'s. We also give some applications of our main result, such as a new proof of the result by Birman and Menasco that if the closure of a braid with $n$ strands is the unlink with $n$ components then the braid is trivial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-08T16:06:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10",
      "57M25"
    ],
    "keywords": [
      "open book decomposition",
      "prime factorization",
      "main result",
      "first betti number"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2148G"
    }
  }
}