{
  "id": "math/0508294",
  "version": "v3",
  "published": "2005-08-16T15:54:02.000Z",
  "updated": "2005-10-27T16:34:53.000Z",
  "title": "The Growth Rate of the First Betti Number in Abelian Covers of 3-Manifolds",
  "authors": [
    "Tim D. Cochran",
    "Joseph D. Masters"
  ],
  "comment": "Minor changes. Final version, to appear in Math. Proc. Camb. Phil. Soc",
  "journal": "Math.Proc.Cambridge Phil.Soc., 141, no.3 , (2006), 465-476",
  "doi": "10.1017/S0305004106009479",
  "categories": [
    "math.GT"
  ],
  "abstract": "We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exist a family of $\\mathbb{Z}_n$ covering spaces, $M_n$, in which $\\beta_1(M_n)$ increases linearly with $n$. The latter generalizes work of M. Katz and C. Lescop [KL], by showing that the non-vanishing of any one of these invariants of $M$ is sufficient to guarantee certain optimal systolic inequalities for $M$ (by work of Ivanov and Katz [IK]).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-10-27T16:34:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first betti number",
      "abelian covers",
      "growth rate",
      "finite abelian covering spaces increases",
      "optimal systolic inequalities"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2006,
      "month": "Nov",
      "volume": 141,
      "number": 3,
      "pages": 465
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006MPCPS.141..465C"
    }
  }
}