{
  "id": "1103.6198",
  "version": "v1",
  "published": "2011-03-31T14:43:58.000Z",
  "updated": "2011-03-31T14:43:58.000Z",
  "title": "String topology and the based loop space",
  "authors": [
    "Eric J. Malm"
  ],
  "comment": "38 pages. Condensed version of the author's Stanford University Ph.D. thesis",
  "categories": [
    "math.AT"
  ],
  "abstract": "For M a closed, connected, oriented manifold, we obtain the Batalin-Vilkovisky (BV) algebra of its string topology through homotopy-theoretic constructions on its based loop space. In particular, we show that the Hochschild cohomology of the chain algebra C_*\\Omega M carries a BV algebra structure isomorphic to that of the loop homology $\\mathbb{H}_*(LM)$. Furthermore, this BV algebra structure is compatible with the usual cup product and Gerstenhaber bracket on Hochschild cohomology. To produce this isomorphism, we use a derived form of Poincar\\'e duality with C_*\\Omega M-modules as local coefficient systems, and a related version of Atiyah duality for parametrized spectra connects the algebraic constructions to the Chas-Sullivan loop product.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-31T14:43:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P50",
      "16E30",
      "55U30"
    ],
    "keywords": [
      "loop space",
      "string topology",
      "bv algebra structure isomorphic",
      "hochschild cohomology",
      "chas-sullivan loop product"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.6198M"
    }
  }
}