{
  "id": "0804.1999",
  "version": "v1",
  "published": "2008-04-12T07:36:35.000Z",
  "updated": "2008-04-12T07:36:35.000Z",
  "title": "Intersection of subgroups in free groups and homotopy groups",
  "authors": [
    "Hans-Joachim Baues",
    "Roman Mikhailov"
  ],
  "journal": "Internat. J. Algebra Comput, 18 (2008), 803-823",
  "categories": [
    "math.GR",
    "math.AT"
  ],
  "abstract": "We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group $\\pi_3$. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the computation of $\\pi_2$. Let $K$ be a two-dimensional CW-complex with subcomplexes $K_1,K_2,K_3$ such that $K=K_1\\cup K_2\\cup K_3$ and $K_1\\cap K_2\\cap K_3$ is the 1-skeleton $K^1$ of $K$. We construct a natural homomorphism of $\\pi_1(K)$-modules $$ \\pi_3(K)\\to \\frac{R_1\\cap R_2\\cap R_3}{[R_1,R_2\\cap R_3][R_2,R_3\\cap R_1][R_3,R_1\\cap R_2]}, $$ where $R_i=ker\\{\\pi_1(K^1)\\to \\pi_1(K_i)\\}, i=1,2,3$ and the action of $\\pi_1(K)=F/R_1R_2R_3$ on the right hand abelian group is defined via conjugation in $F$. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-12T07:36:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free group",
      "intersection",
      "right hand abelian group",
      "third homotopy group",
      "group homology"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.1999B"
    }
  }
}