{
  "id": "1308.4397",
  "version": "v2",
  "published": "2013-08-20T19:53:01.000Z",
  "updated": "2014-12-17T18:42:52.000Z",
  "title": "Twisted homological stability for configuration spaces",
  "authors": [
    "Martin Palmer"
  ],
  "comment": "v2: 23 pages. Title changed; added and generalised corollaries of the main result; otherwise minor changes",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let M be an open, connected manifold. A classical theorem of McDuff and Segal states that the sequence of configuration spaces of n unordered, distinct points in M is homologically stable with coefficients in Z: in each degree, the integral homology is eventually independent of n. The purpose of this note is to prove that this phenomenon also holds for homology with twisted coefficients. We first define an appropriate notion of finite-degree twisted coefficient system for configuration spaces and then use a spectral sequence argument to deduce the result from the untwisted homological stability result of McDuff and Segal. The result and the methods are generalisations of those of Betley for the symmetric groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-20T19:53:01.000Z",
      "title": "Twisted homology of configuration spaces",
      "abstract": "Fix a connected open manifold M and a path-connected space X. Then the sequence C_n(M,X) of configuration spaces of n distinct unordered points in M equipped with labels from X is known to be homologically stable: in each degree, the integral homology is eventually independent of n. In this note we prove that this is also true for homology with twisted coefficients. Obviously one cannot choose local coefficients randomly for each space in the sequence and expect stability: what is needed is a so-called finite-degree twisted coefficient system for {C_n(M,X)}, which we begin by explaining in detail. We then use the untwisted homological stability result to deduce twisted homological stability in this setting. The result and the methods are generalisations of those of [Betley, 2002] in the case of the symmetric groups.",
      "comment": "17 pages; comments welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-17T18:42:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R80",
      "57N65"
    ],
    "keywords": [
      "configuration spaces",
      "twisted homology",
      "finite-degree twisted coefficient system",
      "choose local coefficients",
      "symmetric groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.4397P"
    }
  }
}