{
  "id": "math/0502311",
  "version": "v1",
  "published": "2005-02-15T15:59:05.000Z",
  "updated": "2005-02-15T15:59:05.000Z",
  "title": "Rank of the fundamental group of a component of a function space",
  "authors": [
    "Gregory Lupton",
    "Samuel Bruce Smith"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "We compute the rank of the fundamental group of an arbitrary connected component of the space map(X, Y) for X and Y nilpotent CW complexes with X finite. For the general component corresponding to a homotopy class f : X --> Y, we give a formula directly computable from the Sullivan model for f. For the component of the constant map, our formula expresses the rank in terms of classical invariants of X and Y. Among other applications and calculations, we obtain the following: Let G be a compact simple Lie group with maximal torus T^n. Then the fundamental group of map(S^2, G/T^n; f) is a finite group if and only if f: S^2 --> G/T^n is essential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-02-15T15:59:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q52",
      "55P15"
    ],
    "keywords": [
      "fundamental group",
      "function space",
      "compact simple lie group",
      "nilpotent cw complexes",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......2311L"
    }
  }
}