{
  "id": "math/0203028",
  "version": "v2",
  "published": "2002-03-04T14:12:40.000Z",
  "updated": "2002-03-04T22:29:29.000Z",
  "title": "Combinatorics and Topology of partitions of spherical measures by 2 and 3 fans",
  "authors": [
    "Rade T. Zivaljevic"
  ],
  "comment": "16 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "abstract": "An arrangement of k-semilines in the Euclidean (projective) plane or on the 2-sphere is called a k-fan if all semilines start from the same point. A k-fan is an $\\alpha$-partition for a probability measure $\\mu$ if $\\mu(\\sigma_i)=\\alpha_i$ for each $i=1,...,k$ where $\\{\\sigma_i\\}_{i=1}^k$ are conical sectors associated with the k-fan and $\\alpha = (\\alpha_1,... ,\\alpha_k)$. The set of all $\\alpha = (\\alpha_1,... ,\\alpha_m)$ such that for any collection of probability measures $\\mu_1,... ,\\mu_m$ there exists a common $\\alpha$-partition by a k-fan is denoted by ${\\cal A}_{m,k}$. We prove, as a central result of this paper, that ${\\cal A}_{3,2} = \\{(s,t)\\in \\mathbb{R}^2\\mid s+t=1 {\\rm and} s,t>0\\}$. The result follows from the fact that under mild conditions there does not exist a $Q_{4n}$-equivariant map $f : S^3\\to V\\setminus {\\cal A}(\\alpha)$ where ${\\cal A}(\\alpha)$ is a $Q_{4n}$-invariant, linear subspace arrangement in a $Q_{4n}$-representation V, where $Q_{4n}$ is the generalized quaternion group. This fact is established by showing that an appropriate obstruction in the group $\\Omega_1(Q_{4n})$ of $Q_{4n}$-bordisms does not vanish.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-03-04T22:29:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F20",
      "52C35",
      "55N91",
      "57R91"
    ],
    "keywords": [
      "spherical measures",
      "combinatorics",
      "probability measure",
      "linear subspace arrangement",
      "generalized quaternion group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......3028Z"
    }
  }
}