{
  "id": "1111.7134",
  "version": "v1",
  "published": "2011-11-30T12:04:04.000Z",
  "updated": "2011-11-30T12:04:04.000Z",
  "title": "Operations on Spaces over Operads and Applications to Homotopy Groups",
  "authors": [
    "Wenbin Zhang"
  ],
  "comment": "29 pages, part of the author's Ph.D. thesis",
  "categories": [
    "math.AT"
  ],
  "abstract": "We establish certain smash operations on spaces over operads which are general analogues of the Samelson product on single loop spaces, and obtain a conceptual description of the structure of the homotopy groups of spaces $Y$ over a symmetric $K(\\pi,1)$ operad: $\\pi_*Y$ is a module over the free algebraic symmetric operad generated by operations on homotopy groups induced by these smash operations. In particular the homotopy groups of double loop spaces is a module over the free algebraic symmetric operad generated by the conjugacy classes of Brunnian braids modulo the conjugation action of pure braids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-30T12:04:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homotopy groups",
      "free algebraic symmetric operad",
      "smash operations",
      "applications",
      "brunnian braids modulo"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.7134Z"
    }
  }
}