# taut foliations and the existence of total transversals

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).

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

• 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. – Ian Agol Aug 9 '13 at 23:15