{
  "id": "math/0507514",
  "version": "v2",
  "published": "2005-07-25T14:55:29.000Z",
  "updated": "2007-05-16T15:28:28.000Z",
  "title": "The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points",
  "authors": [
    "Pavel Etingof",
    "Andre Henriques",
    "Joel Kamnitzer",
    "Eric Rains"
  ],
  "comment": "43 pages, 4 figures; This is a revised version, containing more detailed explanations",
  "categories": [
    "math.AT",
    "math.QA"
  ],
  "abstract": "We compute the Poincare polynomial and the cohomology algebra with rational coefficeints of the manifold M_n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M_n. As was shown by E. Rains in arXiv:math/0610743 the cohomology of M_n does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of M_n is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of 2-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld's theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra L_n of H^*(M_n,Q) (associated to such quasibialgebras) factors through the the natural projection of L_n to the associated graded Lie algebra of the prounipotent completion of the fundamental group of M_n. This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces M_n are not formal starting from n=6.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-05-16T15:28:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "moduli space",
      "real locus",
      "stable curves",
      "marked points",
      "cohomology ring"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7514E"
    }
  }
}