# Are all transversely oriented, transversely measured foliations given by closed forms?

This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the book "Differential Geometry of Foliations: The Fundamental Integrability Problem" by Reinhart places a restriction on the transversely oriented foliation to have trivial holonomy. On the other hand, the latter result pertains to $$C^k$$-foliations, while the first one refers to the smooth case.

Is the claim in the paper false?

EDIT: "Smooth" is a bit of misnomer. What I had in mind is foliations suited to train tracks (so the singular leaves only have cusps as singularities.)

• FWIW the paper seems to be talking about foliations with singularities. Actually a compact surface of genus $g\ge 2$ never admits a foliation without singularities. (And the foliations of the torus can in some sense be classified, see the book by Hector-Hirsch.) Sep 3 '15 at 23:27

The point is that the transition maps between foliation charts (with the leaves as y-level sets) are of the form $(x,y)\to (f(x,y),\pm y+c)$. (This is because the transverse measure has to be preserved. So the transverse measure seems essential for the claim to be true.)
Clearly $dy$ or $-dy$ define the singular foliation in a chart, and by coorientability of the foliation you can then consistently choose one of the two for each chart.