{
  "id": "math/0107169",
  "version": "v3",
  "published": "2001-07-24T01:36:26.000Z",
  "updated": "2003-12-30T02:02:56.000Z",
  "title": "Harmonic maps M^3 --> S^1 and 2-cycles, realizing the Thurston norm",
  "authors": [
    "Gabriel Katz"
  ],
  "comment": "13 figures",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a \\emph{harmonic} map $f: M^3 \\to S^1$ with Morse-type singularities delivers the Thurston norm $\\chi_-([F_{best}])$ of its homology class $[F_{best}] \\in H_2(M^3; \\Z)$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $\\Sigma \\subset M$ in the homology class of a fiber, we show that the Thurston number $\\chi_-(\\Sigma)$ satisfies an inequality $$\\chi_-(\\Sigma) \\geq \\chi_-(F_{best}) - \\rho^\\circ(\\Sigma, f)\\cdot Var_{\\chi_-}(f).$$ Here the variation $Var_{\\chi_-}(f)$ is can be expressed in terms of the $\\chi_-$-invariants of the fiber components, and the twist $\\rho^\\circ(\\Sigma, f)$ measures the complexity of the intersection of $\\Sigma$ with a particular set $F_R$ of \"bad\" fiber components. This complexity is tightly linked with the optimal \"$\\tilde f$-height\" of $\\Sigma$, being lifted to the $f$-induced cyclic cover $\\tilde M^3 \\to M^3$. Based on these invariants, for any Morse map $f$, we introduce the notion of its \\emph{twist} $\\rho_{\\chi_-}(f)$. We prove that, for a harmonic $f$, $\\chi_-([F_{best}]) = \\chi_-(F_{best})$, if and only if, $\\rho_{\\chi_-}(f) = 0$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2003-12-30T02:02:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M99",
      "57R30"
    ],
    "keywords": [
      "thurston norm",
      "harmonic maps",
      "fiber components",
      "homology class",
      "morse-type singularities delivers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......7169K"
    }
  }
}