{
  "id": "2411.00227",
  "version": "v1",
  "published": "2024-10-31T21:53:40.000Z",
  "updated": "2024-10-31T21:53:40.000Z",
  "title": "Veering triangulations and transverse foliations",
  "authors": [
    "Jonathan Zung"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "We present a combinatorial approach to the existence of foliations and contact structures transverse to a given pseudo-Anosov flow. Let $\\varphi$ be a transitive pseudo-Anosov flow on a closed oriented 3-manifold. Our main technical result is that every codimension 1 foliation transverse to $\\varphi$ is carried by a single branched surface coming from a veering triangulation. Combined with recent breakthrough work of Massoni, this reduces the existence problem for transverse foliations to something like the feasibility of a system of inequalities (rather than equations!) over $Homeo_+([0,1])$. As a proof of concept, we show that for the hyperbolic, fibered, non-L-space knot $10_{145}$, the natural pseudo-Anosov flow on the slope $s$ Dehn surgery admits a transverse foliation for $s\\in (-\\infty, 3)$, but does not admit such a foliation for $s\\in [5,\\infty)$. The negative result is part of a more general Milnor--Wood type phenomenon which puts limitations on some well known methods for constructing taut foliations on Dehn surgeries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-31T21:53:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R30"
    ],
    "keywords": [
      "transverse foliation",
      "veering triangulation",
      "general milnor-wood type phenomenon",
      "contact structures transverse",
      "natural pseudo-anosov flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}