{
  "id": "1506.06300",
  "version": "v1",
  "published": "2015-06-20T22:27:18.000Z",
  "updated": "2015-06-20T22:27:18.000Z",
  "title": "The co-rank of the fundamental group: the direct product, the first Betti number, and the topology of foliations",
  "authors": [
    "Irina Gelbukh"
  ],
  "comment": "Accepted to Mathematica Slovaca, 2015",
  "categories": [
    "math.AT"
  ],
  "abstract": "We study $b'_1(M)$, the co-rank of the fundamental group of a smooth closed connected manifold $M$. We calculate this value for the direct product of manifolds. We characterize the set of all possible combinations of $b'_1(M)$ and the first Betti number $b_1(M)$ by explicitly constructing manifolds with any possible combination of $b'_1(M)$ and $b_1(M)$ in any given dimension. Finally, we apply our results to the topology of Morse form foliations. In particular, we construct a manifold $M$ and a Morse form $\\omega$ on it for any possible combination of $b'_1(M)$, $b_1(M)$, $m(\\omega)$, and $c(\\omega)$, where $m(\\omega)$ is the number of minimal components and $c(\\omega)$ is the maximum number of homologically independent compact leaves of $\\omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-20T22:27:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F35",
      "57N65",
      "57R30"
    ],
    "keywords": [
      "first betti number",
      "direct product",
      "fundamental group",
      "combination",
      "morse form foliations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606300G"
    }
  }
}