{
  "id": "math/0601750",
  "version": "v4",
  "published": "2006-01-31T04:55:15.000Z",
  "updated": "2008-09-04T19:01:35.000Z",
  "title": "Surfaces in a background space and the homology of mapping class groups",
  "authors": [
    "Ralph L. Cohen",
    "Ib Madsen"
  ],
  "comment": "final version, to be published in the Proceedings of the Seattle Summer Institute in Algebraic Geometry, Proc. Symp. Pure Math., AMS",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, \\gamma). This is the space consisting of triples, (F_{g,n}, \\phi, f), where F_{g,n} is a Riemann surface of genus g and n-boundary components, \\phi is a parameterization of the boundary, and f : F_{g,n} \\to X is a continuous map that satisfies a boundary condition \\gamma. We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S_{\\infty, n}(X; \\gamma), defined to be the limit as the genus g gets large, of the spaces S_{g,n} (X; \\gamma). Our result about this stable topology is a parameterized version of the theorem of Madsen and Weiss proving a generalization of the Mumford conjecture on the stable cohomology of mapping class groups. Our second result describes a stable range in which the homology of S_{g,n} (X; \\gamma) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are generalizations of stability theorems of Harer and Ivanov.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2008-09-04T19:01:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R50",
      "30F99",
      "57M07"
    ],
    "keywords": [
      "mapping class groups",
      "background space",
      "stability theorem",
      "riemann surface",
      "stable homology type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1750C"
    }
  }
}