{
  "id": "2005.13893",
  "version": "v1",
  "published": "2020-05-28T10:34:46.000Z",
  "updated": "2020-05-28T10:34:46.000Z",
  "title": "A pro-algebraic fundamental group for topological spaces",
  "authors": [
    "C. Deninger"
  ],
  "categories": [
    "math.AT",
    "math.GN"
  ],
  "abstract": "Consider a connected topological space $X$ with a point $x \\in X$ and let $K$ be a field with the discrete topology. We study the Tannakian category of finite dimensional (flat) vector bundles on $X$ and its Tannakian dual $\\pi_K (X,x)$ with respect to the fibre functor in $x$. The maximal pro-\\'etale quotient of $\\pi_K (X,x)$ is the \\'etale fundamental group of $X$ studied by Kucharczyk and Scholze. For well behaved topological spaces, $\\pi_K (X,x)$ is the pro-algebraic completion of the ordinary fundamental group $\\pi_1 (X,x)$. We obtain some structural results on $\\pi_K (X,x)$ by studying (pseudo-)torsors attached to its quotients. This approach uses ideas of Nori in algebraic geometry and a result of Deligne on Tannakian categories. We also calculate $\\pi_K (X,x)$ for some generalized solenoids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-28T10:34:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q99",
      "55R15"
    ],
    "keywords": [
      "pro-algebraic fundamental group",
      "topological space",
      "tannakian category",
      "ordinary fundamental group",
      "maximal pro-etale quotient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}