A codimension one foliation $\cal F$ on a smooth manifold $M$ is taut if every leaf of $\cal F$ meets a closed transversal (i.e., a simple closed curve that is everywhere transversal to the leaves of the foliation). Is it true that a taut foliation admits a closed transversal $\gamma$ that meets all leafs of the foliation?

(I've seen such a curve $\gamma$ being referred to as a total transversal).

  • $\begingroup$ Ahem, you probably intend $M$ to be connected? :) $\endgroup$ – Ian Agol Aug 9 '13 at 23:13

Yes, this is true. You can find a proof in Calegari's book, "Foliations and the Geometry of 3–Manifolds", lemma 4.26.

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  • $\begingroup$ This is great, thanks! I wonder if the same results holds for non-compact M. $\endgroup$ – Manfredi Maggiore Aug 9 '13 at 21:28
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    $\begingroup$ This is false for non-compact $M$ (interpreting $\gamma$ in this case to be an embedded line), since the leaf space of foliations of $\mathbb{R}^3$ can be non-Hausdorff. Such examples might also be in Calegari's book. $\endgroup$ – Ian Agol Aug 9 '13 at 23:15
  • $\begingroup$ This completely answers my question. Thanks, I appreciate it. $\endgroup$ – Manfredi Maggiore Aug 11 '13 at 14:55

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