{
  "id": "math/0601085",
  "version": "v8",
  "published": "2006-01-04T23:40:06.000Z",
  "updated": "2009-08-07T20:29:36.000Z",
  "title": "The bar complex of an E-infinity algebra",
  "authors": [
    "Benoit Fresse"
  ],
  "comment": "51 pages. Preprint put in Elsevier format. Minor additional writing corrections",
  "journal": "Adv. Math. 223 (2010), pp. 2049--2096",
  "categories": [
    "math.AT"
  ],
  "abstract": "The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction. We apply our construction to cochain complexes of topological spaces, which are instances of E-infinity algebras. We prove that the n-th iterated bar complexes of the cochain algebra of a space X is equivalent to the cochain complex of the n-fold iterated loop space of X, under reasonable connectedness, completeness and finiteness assumptions on X.",
  "revisions": [
    {
      "version": "v8",
      "updated": "2009-08-07T20:29:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57T30",
      "55P48",
      "18G55",
      "55P35"
    ],
    "keywords": [
      "e-infinity algebra",
      "standard reduced bar complex",
      "n-fold iterated loop space",
      "bar construction defines",
      "n-th iterated bar complexes"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1085F"
    }
  }
}