{
  "id": "1904.08714",
  "version": "v1",
  "published": "2019-04-18T12:03:57.000Z",
  "updated": "2019-04-18T12:03:57.000Z",
  "title": "Aspherical completions and rationally inert elements",
  "authors": [
    "Yves Felix",
    "Steve Halperin"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $X$ be a connected space. An element $[f]\\in \\pi_n(X)$ is called rationally inert if $\\pi_*(X)\\otimes \\mathbb Q \\to \\pi_*(X\\cup_fD^{n+1})\\otimes \\mathbb Q$ is surjective. We extend the results obtained in the simply connected case, and prove in particular that if $X\\cup_fD^{n+1}$ is a Poincar\\'e duality complex and the algebra $H(X)$ requires at least two generators then $[f]\\in \\pi_n(X)$ is rationally inert. On the other hand, if $X$ is rationally a wedge of at least two spheres and $f$ is rationally non trivial, then $f$ is rationally inert. Finally if $f$ is rationally inert then the rational homotopy of the homotopy fibre of the injection $X \\to X\\cup_fD^{n+1}$ is the completion of a free Lie algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-18T12:03:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P62"
    ],
    "keywords": [
      "rationally inert elements",
      "aspherical completions",
      "free lie algebra",
      "poincare duality complex",
      "homotopy fibre"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}