{
  "id": "0907.4412",
  "version": "v1",
  "published": "2009-07-25T11:53:06.000Z",
  "updated": "2009-07-25T11:53:06.000Z",
  "title": "The Cohomology Ring of the Space of Rational Functions",
  "authors": [
    "Dinesh Deshpande"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let Rat_k be the space of based holomorphic maps from S^2 to itself of degree k. Let beta_k denote the Artin's braid group on k strings and let Bbeta_k be the classifying space of beta_k. Let C_k denote the space of configurations of length less than or equal to k of distinct points in R^2 with labels in S^1. The three spaces Rat_k, Bbeta_{2k}, C_k are all stably homotopy equivalent to each other. For an odd prime p, the F_p-cohomology ring of the three spaces are isomorphic to each other. The F_2-cohomology ring of Bbeta_{2k} is isomorphic to that of C_k. We show that for all values of k except 1 and 3, the F_2-cohomology ring of Rat_k is not isomorphic to that of Bbeta_{2k} or C_k. This in particular implies that the HF_2-localization of Rat_k is not homotopy equivalent to HF_2-localization of Bbeta_{2k} or C_k. We also show that for k >= 1, Bbeta_{2k} and Bbeta_{2k+1} have homotopy equivalent HF_2-localizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-25T11:53:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P10",
      "55S12",
      "55P48"
    ],
    "keywords": [
      "rational functions",
      "cohomology ring",
      "isomorphic",
      "artins braid group",
      "distinct points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.4412D"
    }
  }
}