{
  "id": "math/0401081",
  "version": "v1",
  "published": "2004-01-08T22:42:41.000Z",
  "updated": "2004-01-08T22:42:41.000Z",
  "title": "Higher string topology on general spaces",
  "authors": [
    "P. Hu"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex $X$ and $k \\geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, and show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the $(k+1)$-dimensional unframed little disk operad $\\mathcal{C}_{k+1}$. I also prove Kontsevich's conjecture that the Quillen cohomology of a based $\\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad $C_{\\ast}\\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual $C_{\\ast}\\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and the chain complex of $\\Omega^k X$ are Koszul dual to each other as $C_{\\ast}\\mathcal{C}_k$-algebras, and that the chain complex of $Maps(S^k, X)^{S(X)}$ is naturally equivalent to their (equivalent) Hochschild cohomology in the category of $C_{\\ast}\\mathcal{C}_k$-algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-08T22:42:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P48",
      "16E40",
      "55N45",
      "18D50"
    ],
    "keywords": [
      "higher string topology",
      "general spaces",
      "chain complex",
      "dimensional unframed little disk operad",
      "koszul dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1081H"
    }
  }
}