{
  "id": "1502.03031",
  "version": "v1",
  "published": "2015-02-10T18:34:53.000Z",
  "updated": "2015-02-10T18:34:53.000Z",
  "title": "The Topological Complexity of a Surface",
  "authors": [
    "Aldo-Hilario Cruz-Cota"
  ],
  "comment": "17 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $p$ be a branched covering of a Riemann surface to the Riemann sphere $\\mathbb{P}^1$, with branching set $B \\subset \\mathbb{P}^1$. We define the complexity of $p$ as infinity, if $\\mathbb{P}^1 \\setminus B$ does not admit a hyperbolic structure, or the product of its degree and the hyperbolic area of $\\mathbb{P}^1 \\setminus B$, otherwise. The topological complexity of a surface $S$ is defined as the infimum of the set of all complexities of branched coverings $M \\to \\mathbb{P}^1$, where $M$ is a Riemann surface homeomorphic to $S$. We prove that if $S$ is a connected, closed, orientable surface of genus $g$, then its topological complexity, $C_{\\text{top}}(S)$, is given by: \\[C_{\\text{top}}(S)= \\left\\{ \\begin{array}{cl} 2\\pi(2g+1) & \\mbox{if } g \\geq 1, 6 \\pi & \\mbox{if } g=0. \\end{array} \\right.\\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-10T18:34:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M12",
      "30F99"
    ],
    "keywords": [
      "topological complexity",
      "riemann surface homeomorphic",
      "riemann sphere",
      "hyperbolic structure",
      "hyperbolic area"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150203031C"
    }
  }
}