{
  "id": "2109.00515",
  "version": "v1",
  "published": "2021-09-01T17:59:53.000Z",
  "updated": "2021-09-01T17:59:53.000Z",
  "title": "Heisenberg homology on surface configurations",
  "authors": [
    "Christian Blanchet",
    "Martin Palmer",
    "Awais Shaukat"
  ],
  "comment": "61 pages, 13 figures",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "We study the action of the mapping class group of $\\Sigma = \\Sigma_{g,1}$ on the homology of configuration spaces with coefficients twisted by the discrete Heisenberg group $\\mathcal{H} = \\mathcal{H}(\\Sigma)$, or more generally by any representation $V$ of $\\mathcal{H}$. In general, this is a twisted representation of the mapping class group $\\mathfrak{M}(\\Sigma)$ and restricts to an untwisted representation on the Chillingworth subgroup $\\mathrm{Chill}(\\Sigma) \\subseteq \\mathfrak{M}(\\Sigma)$. Moreover, it may be untwisted on the Torelli group $\\mathfrak{T}(\\Sigma)$ by passing to a $\\mathbb{Z}$-central extension, and, in the special case where we take coefficients in the Schr\\\"odinger representation of $\\mathcal{H}$, it may be untwisted on the full mapping class group $\\mathfrak{M}(\\Sigma)$ by passing to a double covering. We illustrate our construction with several calculations for $2$-point configurations, in particular for genus-$1$ separating twists.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-01T17:59:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K20",
      "55R80",
      "55N25",
      "20C12"
    ],
    "keywords": [
      "surface configurations",
      "heisenberg homology",
      "representation",
      "discrete heisenberg group",
      "full mapping class group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}