{
  "id": "2209.14535",
  "version": "v1",
  "published": "2022-09-29T03:44:37.000Z",
  "updated": "2022-09-29T03:44:37.000Z",
  "title": "Integral homology groups of double coverings and rank one $\\mathbb{Z}$-local system for minimal CW complex",
  "authors": [
    "Ye Liu",
    "Yongqiang Liu"
  ],
  "comment": "5 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $X$ be a connected finite CW complex. A connected double covering of $X$ is classified by a non-zero cohomology class $\\omega \\in H^1(X,\\mathbb{Z}_2)$. Denote the double covering space by $X^\\omega$. There exists a corresponding non-trivial rank one $\\mathbb{Z}$-local system $\\mathcal{L}_\\omega$ on $X$. What is the relation between the integral homology groups of $X^\\omega$ and the homology groups of the local system $\\mathcal{L}_\\omega$? When $X$ is homotopy equivalent to a minimal CW complex, we give a complete answer to this question. In particular, this settles a conjecture recently proposed by Ishibashi, Sugawara and Yoshinaga for hyperplane arrangement complement. As an application, when $X$ is a hyperplane arrangement complement and $\\mathcal{L}_\\omega$ satisfies certain conditions, we show that $H_*(X^\\omega,\\mathbb{Z})$ is combinatorially determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-29T03:44:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral homology groups",
      "minimal cw complex",
      "local system",
      "double covering",
      "hyperplane arrangement complement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}