{
  "id": "1811.03708",
  "version": "v1",
  "published": "2018-11-08T23:01:57.000Z",
  "updated": "2018-11-08T23:01:57.000Z",
  "title": "Geography of Genus 2 Lefschetz fibrations",
  "authors": [
    "Kai Nakamura"
  ],
  "categories": [
    "math.GT",
    "math.SG"
  ],
  "abstract": "Questions of geography of various classes of $4$-manifolds have been a central motivating question in $4$-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal $4$-manifolds admit a genus $2$ Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize all but one with the exception of a genus $2$ Lefschetz fibration on a symplectic $4$-manifold homeomorphic, but not diffeomorphic to $3 \\mathbb{CP}^2 \\# 11\\overline{\\mathbb{CP}}^2$. We give a positive factorization of type $(10,10)$ that corresponds to such a genus $2$ Lefschetz fibration. Furthermore, we observe two restrictions on the geography of genus $2$ Lefschetz fibrations, we find that they satisfy the Noether inequality and a BMY like inequality. We then find positive factorizations that describe genus $2$ Lefschetz fibrations on simply connected, minimal symplectic $4$-manifolds for many of these points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-08T23:01:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lefschetz fibration",
      "positive factorization",
      "manifold topology",
      "central motivating question",
      "homeomorphism types"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}