{
  "id": "2104.01419",
  "version": "v1",
  "published": "2021-04-03T14:35:13.000Z",
  "updated": "2021-04-03T14:35:13.000Z",
  "title": "Small Lefschetz Fibrations on Simply-Connected $4$-Manifolds",
  "authors": [
    "Tulin Altunoz"
  ],
  "comment": "20 pages, 7 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We consider simply-connected $4$-manifolds admitting Lefschetz fibrations over the $2$-sphere. We explicitly construct Lefschetz fibrations of genus $3$ and $4$ on simply-connected $4$-manifolds which are exotic symplectic $4$-manifolds in the homeomorphism classes of $\\mathbb{C} P^{2}\\#7\\overline{\\mathbb{C} P^{2}}$ and $\\mathbb{C} P^{2}\\#8\\overline{\\mathbb{C} P^{2}}$, respectively. From these, we provide upper bounds for the minimal number of singular fibers of such fibrations. In addition, we prove that this number is equal to $14$ for $g=2$. When such fibrations are hyperelliptic, we prove that it is $18$ for $g=3$. Moreover, we discuss the number for higher genera.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-03T14:35:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "small lefschetz fibrations",
      "manifolds admitting lefschetz fibrations",
      "explicitly construct lefschetz fibrations",
      "singular fibers",
      "exotic symplectic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}