{
  "id": "math/9805058",
  "version": "v2",
  "published": "1998-05-12T15:45:18.000Z",
  "updated": "1998-07-02T17:04:42.000Z",
  "title": "The homology of abelian coverings of knotted graphs",
  "authors": [
    "R. A. Litherland"
  ],
  "comment": "Plain TeX, 40 pages, 1 figure; corrected the proof of Theorem 8.8, and some typos",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2 subgroup H of G, M_H = N/H is a double branched cover of M. Sakuma has proved that the first homology of N is isomorphic, modulo 2-torsion, to the direct sum of the first homology groups of the M_H, and has shown that H_1(N) is determined up to isomorphism by the direct sum of the H_1(M_H) in certain cases; specifically, when d=2 and the coloring is such that the branch set of each cover M_H -> M is connected, and when d=3 and Gamma is the complete graph K_4. We prove this for a larger class of coverings: when d=2, for any coloring of a connected graph; when d=3 or 4, for an infinite class of colored graphs; and when d=5, for a single coloring of the Petersen graph.",
  "revisions": [
    {
      "version": "v2",
      "updated": "1998-07-02T17:04:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M12",
      "57M25",
      "57M15"
    ],
    "keywords": [
      "abelian coverings",
      "knotted graphs",
      "direct sum",
      "branch set",
      "trivalent graph gamma"
    ],
    "note": {
      "typesetting": "Plain TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......5058L"
    }
  }
}