This week I visited Washington University in St. Louis to give a colloquium, and caught up with a couple of my old foliations friends, namely Rachel Roberts and Larry Conlon. Actually, I had caught up with Rachel (to some extent) the weekend before, when we both spoke at the Texas Geometry and Topology Conference in Austin, where Rachel gave a talk about her recent proof (joint with Will Kazez) that every taut foliation on a 3-manifold (other than ) can be approximated by both positive and negative contact structures; it follows that admits a symplectic structure with pseudoconvex boundary, and one deduces nontriviality of various invariants associated to (Seiberg-Witten, Heegaard Floer Homology, etc.). This theorem was known for sufficiently smooth (at least ) foliations by Eliashberg-Thurston, as exposed in their *confoliations* monograph, and it is one of the cornerstones of -dimensional symplectic geometry; unfortunately (fortunately?) many natural constructions of foliations on 3-manifolds can be done only in the or world. So the theorem of Rachel and Will is a big deal.

If we denote the foliation by which is the kernel of a 1-form and suppose the approximating positive and negative contact structures are given by the kernels of 1-forms where and pointwise, the symplectic form on is given by the formula

for some small , where is any closed 2-form on which is (strictly) positive on (and therefore also positive on the kernel of if the contact structures approximate the foliation sufficiently closely). The existence of such a closed 2-form is one of the well-known characterizations of tautness for foliations of 3-manifolds. I know several proofs, and at one point considered myself an expert in the theory of taut foliations. But when Cliff Taubes happened to ask me a few months ago which *cohomology classes* in are represented by such forms , and in particular whether the Euler class of could be represented by such a form, I was embarrassed to discover that I had never considered the question before.

The answer is actually well-known and quite easy to state, and is one of the applications of Sullivan’s theory of foliation cycles. One can also give a more hands-on topological proof which is special to codimension 1 foliations of 3-manifolds. Since the theory of taut foliations of 3-manifolds is a somewhat lost art, I thought it would be worthwhile to write a blog post giving the answer, and explaining the proofs.