3
$\begingroup$

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).

$\endgroup$
1
  • 1
    $\begingroup$ Ahem, you probably intend $M$ to be connected? :) $\endgroup$
    – Ian Agol
    Aug 9, 2013 at 23:13

1 Answer 1

6
$\begingroup$

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

$\endgroup$
3
  • $\begingroup$ This is great, thanks! I wonder if the same results holds for non-compact M. $\endgroup$ Aug 9, 2013 at 21:28
  • 3
    $\begingroup$ 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, 2013 at 23:15
  • $\begingroup$ This completely answers my question. Thanks, I appreciate it. $\endgroup$ Aug 11, 2013 at 14:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.