{
  "id": "1512.09345",
  "version": "v1",
  "published": "2015-12-31T19:13:17.000Z",
  "updated": "2015-12-31T19:13:17.000Z",
  "title": "On the traceless SU(2) character variety of the 6-punctured 2-sphere",
  "authors": [
    "Paul Kirk"
  ],
  "comment": "17 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.SG"
  ],
  "abstract": "We exhibit the traceless $SU(2)$ character variety of a 6-punctured 2-sphere as a 2-fold branched cover of ${\\mathbb{C}}P^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on $SU(2)$ character varieties by the 2-fold branched cover $F_{n-1}\\to S^2$ branched over $2n$ points, combined with the theorem of Narasimhan-Ramanan which identifies $R(F_2)$ with ${\\mathbb{C}} P^3$. The singular points of $R(S^2,6)$ correspond to abelian representations, and we prove that each has a neighborhood in $R(S^2,6)$ homeomorphic to a cone on $S^2\\times S^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-31T19:13:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "53D30"
    ],
    "keywords": [
      "character variety",
      "traceless su",
      "branched cover",
      "singular kummer surface",
      "binary dihedral representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151209345K"
    }
  }
}