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.
To be a bit more precise, let me insist in what follows that is closed and oriented, and that is oriented and co-oriented. The smoothness of is an issue in some of the arguments I will give, but I will not make a big deal of this. Then one has the following:
Theorem: Let be a foliation of a 3-manifold as above. A cohomology class is represented by a smooth closed 2-form positive on if and only if for every nontrivial transverse invariant measure for .
This requires a bit of explanation. A transverse measure assigns a non-negative number to any segment transverse to , which is countable additive on unions. Such a measure is invariant if it takes the same value on two transverse segments , related to each other by holonomy transport; thus, a transverse measure is really a measure defined on the local leaf space of the foliation, which is compatible on the overlap of leaf space charts (one would like to think of it as a measure on the global leaf space space, but since this space is typically non-Hausdorff, one tends not to express things in such terms). If the foliation is orientable and co-orientable, we can define the measure on oriented transversals in such a way that changing the orientation changes the sign: , where denotes with the opposite orientation. We still insist in this case that is non-negative whenever is positively oriented (with respect to the co-orientation).
Such a measure pairs with 1-chains, and the invariance property implies that it vanishes on 1-boundaries. Thus as above defines a 2-dimensional homology class , by how it pairs with 1-cycles, and appealing to Poincaré duality.
Here is another interpretation of an invariant transverse measure. Any compact subsurface contained in a union of leaves of determines a transverse measure by defining to be the number of intersections of with , when is a positively-oriented transversal. Now, let’s suppose that is a sequence of compact subsurfaces in unions of leaves of , and suppose further that (i.e. the form a “Følner sequence” for ). If we denote the area of by , we can define a sequence of measures , and then some subsequence will converge to a limiting transverse measure which is invariant. This is because most of the intersections of any transversal with (for big ) are contained deep in the interior, so that any nearby intersects in almost the same number of points, and is very close to (here we really need to restrict attention to whose boundary is not too complicated; it is enough for it to have bounded geodesic curvature, for instance). We can think of each weighted surface as a de Rham 2-chain by how it pairs with smooth 2-forms; the limit converges to a well-defined de Rham 2-cycle, representing the 2-dimensional homology class . All invariant transverse measures are of this form. When expressed in this language, we refer to such invariant transverse measures as foliation cycles.
Thus one immediately sees one direction of the Theorem: if is a closed 2-form strictly positive on every leaf of , it pairs (uniformly) positively with each , and therefore also with .
The converse direction is also easy to see, modulo some functional analysis. A sketch of the idea is as follows. In the space of de Rham 2-chains, the weighted surfaces carried by the foliation as above are dense in a closed convex cone . An element of the dual space is positive on if it is positive on . It is closed if it vanishes on all de Rham 2-boundaries . Since is closed, by the Hahn-Banach theorem such a exists if and only if ; equivalently, if and only if there is no foliation cycle representing 0 in (de Rham) homology. Such a can be approximated by a smooth 2-form (since such forms are dense in ) which is also positive on . A foliation with no null-homologous foliation cycle is said to be homologically taut, so we deduce that any homologically taut foliation admits a smooth closed form positive on the leaves. But by the same reasoning, we can find in a particular cohomology class if and only if does not intersect the subspace of de Rham 2-cycles pairing to zero with . This concludes the sketch of the proof of the theorem; for details consult Sullivan’s paper, Thm.II.3.
Note that the theorem is very interesting even in the case that the foliation admits no invariant transverse measure. In some sense, this is the generic situation for a taut foliation of a 3-manifold; the existence of a nontrivial invariant transverse measure imposes strong (polynomial!) growth conditions on leaves in the support of the measure. In this case, every cohomology class is represented by a form positive on the leaves of the foliation.
It is worth pointing out an important application. A foliation is said to be geometrically taut if there exists a Riemannian metric for which all the leaves are minimal surfaces. A necessary and sufficient condition for this is the existence of a form as above which is closed and positive on , and furthermore is pure: i.e. the kernel of is a complementary subspace to at each point. In codimension one this condition is vacuous, but in higher codimension Sullivan shows how to derive a pure (closed) form from an arbitrary one by an algebraic operation called purification. Anyway, from this one (i.e. Sullivan) deduces Sullivan’s theorem, to wit: a foliation is homologically taut if and only if it is geometrically taut. Note that this theorem is interesting even for 1-dimensional foliations — i.e. flows, since geometrically taut is equivalent to geodesibility of the flow.
The proof above is short, but the appeal to Hahn-Banach and the analytic details in Sullivan’s paper is unsatisfying. Here is the sketch of a topological argument which gets to the point. First consider a special case: suppose some homologically trivial loop is transverse to and intersects every leaf. Then we can find representatives of that contain any tiny transverse segment, and by swapping a negative tiny segment of for the (positive) rest of it, we can replace any loop with a homologically equivalent loop which is positive; in this case every class is representable. In the general case, the support of the nontrivial invariant transverse measures is some closed union of leaves, and we focus attention on a complementary open pocket. Because this pocket has no invariant transverse measure, lots of directions in many leaves have contracting holonomy; thus we can find small intervals in the leaf space so that for every subinterval there is a pair of elements and a point so that takes to not equal to , and takes the interval properly inside . Thus the commutator (which is homologically trivial) represents a transverse loop intersecting any given collection of leaves in the pocket. So by the argument above we can take any transverse loop which intersects the invariantly measured leaves positively, and replace it by a positively oriented transverse loop in the same homology class.
OK, back to 3-manifolds and cohomology classes. What about Taubes’ question: when can the Euler class of be represented by a form positive on the leaves of ? To answer this we need to talk about the Euler characteristic of a transverse measure, and the foliated Gauss-Bonnet theorem. Recall that the ordinary Gauss-Bonnet theorem says that for a closed oriented surface we have an equality
where is the curvature of any Riemannian metric on . For a surface with boundary there is a correction term, which involves the integral of the geodesic curvature over the boundary. If we apply this theorem to each of our surfaces in turn we see that we can define
On the other hand, if denotes the Euler class of , thought of as an element of , then . Taking limits as above, we deduce the formula for any foliation cycle .
A theorem of Ghys says that for any Riemann surface lamination and any invariant transverse measure with , some positive measure of leaves must be 2-spheres. For a foliation of a 3-manifold, the Reeb stability theorem says that the existence of one spherical leaf implies that (up to taking double covers) the manifold is with the product foliation by spheres. So we can ignore this possibility by fiat.
If there is a transverse measure with then a theorem of Candel implies that some positive measure of leaves must be conformally parabolic. If we assume that the foliation is taut, then this implies that contains an essential torus. So if we restrict attention to the “generic” case that is a hyperbolic 3-manifold, then for any taut foliation , every leaf is conformally hyperbolic. In this case, Candel shows that leafwise uniformization is continuous, so that admits a metric in which every leaf has constant curvature . In particular, every invariant transverse measure has . Thus for a (coorientable, orientable) taut foliation of a hyperbolic 3-manifold, the negative of the Euler class is always represented by a closed 2-form , positive on every leaf.
Feels like old times . . .