{
  "id": "2203.02462",
  "version": "v1",
  "published": "2022-03-04T17:47:57.000Z",
  "updated": "2022-03-04T17:47:57.000Z",
  "title": "Algebraic models for classifying spaces of fibrations",
  "authors": [
    "Alexander Berglund",
    "Tomáš Zeman"
  ],
  "comment": "54 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "We construct an algebraic model for the rational homotopy type of the classifying space $B\\operatorname{aut}(X)$ for fibrations with fiber a simply connected finite CW-complex $X$. The ingredients are a nilpotent dg Lie algebra $\\mathfrak{g}(X)$ in the category of algebraic representations of a certain reductive algebraic group $R(X)$ together with an arithmetic subgroup $\\Gamma(X)$ of $R(X)$. The algebraic model reduces the computation of the rational cohomology ring of $B\\operatorname{aut}(X)$ to the computation of Chevalley-Eilenberg cohomology of dg Lie algebras and cohomology of arithmetic groups with coefficients in algebraic representations. This has strong consequences for the structure of this ring.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-04T17:47:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R40",
      "55P62",
      "55R15",
      "11F75"
    ],
    "keywords": [
      "classifying space",
      "fibrations",
      "nilpotent dg lie algebra",
      "algebraic representations",
      "rational homotopy type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}