{
  "id": "1406.6269",
  "version": "v1",
  "published": "2014-06-24T15:00:15.000Z",
  "updated": "2014-06-24T15:00:15.000Z",
  "title": "Rational homotopy theory of function spaces and Hochschild cohomology",
  "authors": [
    "Ilias Amrani"
  ],
  "categories": [
    "math.AT",
    "math.AC",
    "math.KT",
    "math.RA"
  ],
  "abstract": "Given a map $f: X\\rightarrow Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by $\\Omega_{f} Map(X,Y)$. For $n> 0$, we give a model categorical interpretation of the existence (in functorial way) of an injective map of $\\mathbb{Q}$-vector spaces $\\pi_{n} \\Omega_{f}Map(X,Y_{\\mathbb{Q}}) \\rightarrow HH^{-n}(C^{\\ast}(Y),C^{\\ast}(X)_{f})$, where $HH^{\\ast}$ is the (negative) Hochschild cohomology and $C^{\\ast}(X)_{f}$ is the rational cochain complex associated to $X$ equipped with a structure of $C^{\\ast}(Y)$-differential graded bimodule via the induced map of differential graded algebras $f^{\\ast}: C^{\\ast}(Y)\\rightarrow C^{\\ast}(X)$. Moreover, we identifiy the image in presice way by using the Hodge filtration on Hochschild cohomology. In particular, when $X=Y$, we describe the fundamental group of the identity component of the monoid of self equivalence of a (rationalization of) space $X$ i.e., $\\pi_{1} Aut(X_{\\mathbb{Q}})_{id}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-24T15:00:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational homotopy theory",
      "hochschild cohomology",
      "function spaces",
      "fundamental group",
      "finite type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.6269A"
    }
  }
}