{
  "id": "1510.03770",
  "version": "v1",
  "published": "2015-10-13T16:53:46.000Z",
  "updated": "2015-10-13T16:53:46.000Z",
  "title": "On Betti Numbers of Milnor Fiber of Hyperplane Arrangements",
  "authors": [
    "KaiHo Tommy Wong",
    "Yun Su"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\mathcal{A}$ be a central hyperplane arrangement in $\\mathbb{C}^{n+1}$ and $H_i,i=1,2,...,d$ be the defining equations of the hyperplanes of $\\mathcal{A}$. Let $f=\\prod_i H_i$. There is a global Milnor fibration $F\\hookrightarrow \\mathbb{C}^{n+1} \\setminus \\mathcal{A} \\xrightarrow{f} \\mathbb{C}^*,$ where $F$ is called the Milnor fiber and can be identified as the affine hypersurface $f=1$ in $\\mathbb{C}^{n+1}$. Many open questions have been raised subject to $F$. In particular, it has been conjectured that the integral homology, or the characteristic polynomial, hence the Betti numbers, of $F$ are also determined by the intersection lattice $L(\\mathcal{A})$. In this paper, we find a combinatorial upper bound for the first the characteristic polynomial of the Milnor fiber for central hyperplane arrangements, which improves existing results in the study. As a corollary, we obtain a combinatorial obstruction for trivial algebraic monodromy of the first homology of Milnor fiber. Calculations and comparisons to known examples computed using Fox calculus will be provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-13T16:53:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "milnor fiber",
      "betti numbers",
      "central hyperplane arrangement",
      "characteristic polynomial",
      "global milnor fibration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}