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.)

  • $\begingroup$ 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.) $\endgroup$ – ThiKu Sep 3 '15 at 23:27

The claim is true and you find it on page 319 of Farb-Margalit's book.

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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