{
  "id": "2004.00290",
  "version": "v1",
  "published": "2020-04-01T08:54:37.000Z",
  "updated": "2020-04-01T08:54:37.000Z",
  "title": "Phantom maps and fibrations",
  "authors": [
    "Hiroshi Kihara"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "Given pointed $CW$-complexes $X$ and $Y$, $\\rmph(X, Y)$ denotes the set of homotopy classes of phantom maps from $X$ to $Y$ and $\\rmsph(X, Y)$ denotes the subset of $\\rmph(X, Y)$ consisting of homotopy classes of special phantom maps. In a preceding paper, we gave a sufficient condition such that $\\rmph(X, Y)$ and $\\rmsph(X, Y)$ have natural group structures and established a formula for calculating the groups $\\rmph(X, Y)$ and $\\rmsph(X, Y)$ in many cases where the groups $[X,\\Omega \\widehat{Y}]$ are nontrivial. In this paper, we establish a dual version of the formula, in which the target is the total space of a fibration, to calculate the groups $\\rmph(X, Y)$ and $\\rmsph(X, Y)$ for pairs $(X,Y)$ to which the formula or existing methods do not apply. In particular, we calculate the groups $\\rmph(X,Y)$ and $\\rmsph(X,Y)$ for pairs $(X,Y)$ such that $X$ is the classifying space $BG$ of a compact Lie group $G$ and $Y$ is a highly connected cover $Y' \\langle n \\rangle$ of a nilpotent finite complex $Y'$ or the quotient $\\gbb / H$ of $\\gbb = U, O$ by a compact Lie group $H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-01T08:54:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q05",
      "55P60"
    ],
    "keywords": [
      "compact lie group",
      "homotopy classes",
      "special phantom maps",
      "natural group structures",
      "nilpotent finite complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}