{
  "id": "1401.7726",
  "version": "v2",
  "published": "2014-01-30T03:32:09.000Z",
  "updated": "2014-10-21T18:17:38.000Z",
  "title": "Foliations, orders, representations, L-spaces and graph manifolds",
  "authors": [
    "Steven Boyer",
    "Adam Clay"
  ],
  "comment": "50 pages. A gap in the proof of Corollary 1.2 has been filled. The corollary, which now appears as Theorem 1.2, is a consequence of the smoothness results to appear in \"Slope detection and foliations in graph manifolds.\"",
  "categories": [
    "math.GT"
  ],
  "abstract": "We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology $3$-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions - using representations, using left-orders, using foliations, and using Heegaard-Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next we provide necessary and sufficient conditions for the existence of a co-oriented taut foliation on a graph manifold rational homology $3$-sphere, respectively a left-order on its fundamental group, which depend solely on families of detected slopes on the boundaries of its pieces. The fact that Heegaard-Floer methods can be used to detect families of slopes on the boundary of a Seifert fibred manifold combines with certain conjectures in the literature to suggest an L-space gluing theorem for rational homology $3$-sphere graph manifolds as well as other interesting problems in Heegaard-Floer theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-30T03:32:09.000Z",
      "comment": "50 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-21T18:17:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M50",
      "57M99"
    ],
    "keywords": [
      "sphere graph manifolds",
      "co-oriented taut foliation",
      "representations",
      "seifert fibred manifold",
      "fundamental group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.7726B"
    }
  }
}