{
  "id": "2003.07852",
  "version": "v1",
  "published": "2020-03-17T17:59:50.000Z",
  "updated": "2020-03-17T17:59:50.000Z",
  "title": "String topology of finite groups of Lie type",
  "authors": [
    "Jesper Grodal",
    "Anssi Lahtinen"
  ],
  "comment": "58 pages",
  "categories": [
    "math.AT",
    "math.GR",
    "math.RT"
  ],
  "abstract": "We show that the mod $\\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\\ell$ admits the structure of a module over the mod $\\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration. We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\\ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $\\ell$-compact fixed point group depending on the order of $q$ mod $\\ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-17T17:59:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "20D06",
      "55R35",
      "55P50"
    ],
    "keywords": [
      "finite group",
      "lie type",
      "string topology",
      "fundamental class",
      "connected untwisted classical groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}