{
  "id": "2010.04579",
  "version": "v1",
  "published": "2020-10-09T13:50:58.000Z",
  "updated": "2020-10-09T13:50:58.000Z",
  "title": "Rational homotopy type of mapping spaces via cohomology algebras",
  "authors": [
    "Sang Xie",
    "Jian Liu",
    "Xiugui Liu"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\\map(X, Y)$ is determined by the cohomology algebra $H^*(X; \\Q)$ and the rational homotopy type of $Y$. From this, we deduce the existence of H-structures on a component of the mapping space $\\map(X, Y)$, assuming the cohomology algebras of $X$ and $Y$ are isomorphism. Finally, we will show that $\\map(X, Y; f)\\simeq\\map(X, Y; f')$ if the corresponding \\emph{Maurer-Cartan elements} are connected by an algebra automorphism of $H^\\ast(X, \\Q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-09T13:50:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P62",
      "54C35"
    ],
    "keywords": [
      "rational homotopy type",
      "cohomology algebra",
      "mapping space",
      "algebra automorphism",
      "two-stage space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}