{
  "id": "1603.03931",
  "version": "v1",
  "published": "2016-03-12T15:27:03.000Z",
  "updated": "2016-03-12T15:27:03.000Z",
  "title": "Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting",
  "authors": [
    "Weiyan Chen"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GT",
    "math.AG",
    "math.AT",
    "math.CO"
  ],
  "abstract": "We consider two families of algebraic varieties $Y_n$ indexed by natural numbers $n$: the configuration space of unordered $n$-tuples of distinct points on $\\mathbb{C}$, and the space of unordered $n$-tuples of linearly independent lines in $\\mathbb{C}^n$. Let $W_n$ be any sequence of virtual $S_n$-representations given by a character polynomial, we compute $H^i(Y_n; W_n)$ for all $i$ and all $n$ in terms of double generating functions. One consequence of the computation is a new recurrence phenomenon: the stable twisted Betti numbers $\\lim_{n\\to\\infty}\\dim H^i(Y_n; W_n)$ are linearly recurrent in $i$. Our method is to compute twisted point-counts on the $F_q$-points of certain algebraic varieties, and then pass through the Grothendieck-Lefschetz fixed point formula to prove results in topology. We also generalize a result of Church-Ellenberg-Farb about the configuration spaces of the affine line to those of a general smooth variety.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-12T15:27:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "configuration space",
      "maximal tori",
      "twisted cohomology",
      "algebraic varieties",
      "general smooth variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160303931C"
    }
  }
}