Here is the overdue part 2 post on symplectic fillability. In part 1 , we gave several equivalent definitions of weak symplectic fillings, strong symplectic fillings, and Stein fillings. From these definitions we had the following inclusions:
Now I want to summarize the results that show that each of these notions are distinct from each other, namely that each of these inclusions is strict, and also mention some conditions that can ensure some of these notions do coincide.
Historically, it was first shown that there are weakly symplectically fillable contact manifolds which are not strongly symplectically fillable by Eliashberg (the paper is called “Unique holomorphically fillable contact structure on the 3-torus”). The examples were contact structures on . Identify , with coordinates . Then one can verify that for , is a contact form (). Then is a contact structure on for each , and it was shown by Giroux and independently Kanda that these contact structures are not contactomorphic for different values of , and all tight contact structures on are contactomorphic to one of these. The claim is that if , is not strongly symplectically fillable, but all are weakly symplectically fillable.
First we will show that is weakly symplectically fillable. View as the boundary of . Put coordinates on the factor, and let parametrize the boundary of . We can put a symplectic structure on by taking the product of area forms on and . We want to show that this is a weak symplectic filling of . This is not immediately obvious, so instead we look at a family of contact structures, which are contactomorphic to , and show that is a weak symplectic filling of for some . It is clear that the submanifolds for are symplectic submanifolds of , so is positive on the tangent planes to these submanifolds. These tangent planes are described by on . Clearly these planes do not form a contact structure on since they are integrable, but will evaluate positively on a nearby contact structure. With this in mind, set
so is a contact structure for any . When , so we have a homotopy through contact structures from to any for . By Gray’s theorem there are diffeomorphisms such that , therefore each is contactomorphic to for . When is close to zero, is close to . Therefore, since is an open condition, there exists some such that . Therefore is a weak symplectic filling of which is contactomorphic to . This shows that all are weakly symplectically fillable.
Eliashberg shows that for , is not strongly symplectically fillable by contradiction. Recall that if it did have a strong symplectic filling, this filling could be glued symplectically to any symplectic manifold whose concave boundary is . The idea is to find a symplectic manifold with concave boundary, such that if we could close off this boundary symplectically, we would obtain a symplectic manifold that cannot exist.
Consider . Take the product of circles . This is a Lagrangian 2-torus inside . Any Lagrangian submanifold, has a neighborhood symplectomorphic to a neighborhood of the zero section of its cotangent bundle. One can explicitly show that a disk bundle inside has convex boundary inducing the contact structure , using the radially outward pointing Liouville vector field. Therefore the complement of a neighborhood of this Lagrangian torus in has concave boundary inducing the contact structure . We can take an n-fold cover of the complement of this neighborhood in so that on the boundary, the factor parameterized by is covered n times. This cover has a symplectic form given by the pull-back for which the boundary is still concave and the induced contact structure on the resulting boundary is . This n-fold cover has ends, each symplectomorphic to the complement of a large compact set in . If it were possible to cap off the concave boundary component with a convex symplectic filling of , we would obtain a symplectic manifold without boundary, but with n standard ends. A theorem of Gromov states that this is impossible (if someone has a good explanation of the idea of this proof, that might make a good new post that I would appreciate). Thus we have reached a contradiction to the assumption that is strongly symplectically fillable for .
While this example establishes the inequivalence of weak and strong symplectic fillability, the proof by contradiction and the reliance on a difficult theorem of Gromov which requires holomorphic curve techniques makes it difficult to see what the difference between weak and strong fillability would be in general. A generalization of this example was established by Ding and Geiges. They proved that a more general class of 2-torus bundles over the circle have contact structures which are weakly but not strongly symplectically fillable. Their proof that there are no strong fillings reduces the more general case to the original examples on using contact surgery. While this doesn’t make the underlying cause of non-fillability more clear, it does illuminate the fact that certain surgery operations on contact manifolds preserve different types of fillability and non-fillability.
In particular, Legendrian surgery (surgery along a Legendrian knot, with framing given by the contact framing with one additional negative twist) preserves weak symplectic fillability, strong symplectic fillability, and Stein fillability. The proofs that it preserves strong symplectic fillability and Stein fillability are due to Weinstein and Eliashberg respectively, who show that the convex symplectic or Stein structures can be extended over the corresponding handle attachments which provide cobordisms between the original manifold and the result of Legendrian surgery. The fact that Legendrian surgery preserves weak symplectic fillability is proven here in Theorem 2.3 by Etnyre and Honda, by showing that in a neighborhood of the Legendrian knot, the contact structure can be slightly perturbed so that the weak symplectic filling is a convex filling in a neighborhood of the knot. Then the surgery can be performed to preserve the strong fillability in a neighborhood of the knot, thus preserving the weak fillability of the entire contact manifold.
While we still do not seem to have a full understanding of when strong and weak fillability coincide and when they differ, there are certain situations where weak fillability implies strong fillability. Eliashberg proved the following proposition:
Proposition 4.1 in A few remarks about symplectic filling: Suppose that a symplectic manifold weakly fills a contact manifold . Then if the form is exact near then it can be modified into a symplectic form such that is a strong symplectic filling of .
The idea is essentially to consider the primitive for near the boundary and the contact form , and to interpolate between the primitives and , so that near the boundary and a little further in from the boundary so it glues up to the original on the interior of the weak filling. The condition that be a weak filling ensures one can do this interpolation while maintaining the symplectic (non-degeneracy) condition.
A consequence of this proposition is that in order to find a 3-manifold which supports weakly fillable contact structures that are not strongly fillable, the 3-manifold must carry some nonzero second homology. In particular, an integer homology sphere which is weakly fillable is also strongly fillable. This fact comes into play in the next example.
While Eliashberg’s examples distinguishing weak and strong fillings were published in 1996, it took much longer to find examples of contact 3-manifolds which were strongly fillable but not Stein fillable. The technology needed to obstruct Stein fillability was the Heegaard Floer contact invariant. Ghiggini found the first examples in this paper, which were the Brieskorn homology spheres (with reversed orientation) ( even and ). These manifolds can be understood as 0-surgery on the positive trefoil knot in , together with -surgery on its meridian (or alternatively as Seifert fibered spaces over with three singular fibers with coefficients 2,-3,).
The 3-manifold obtained from by 0-surgery on the positive trefoil is actually a bundle over , because the trefoil is a fibered knot of genus 1. Its monodromy is well understood, and in fact this is one of the 3-manifolds considered by Ding and Geiges, which have contact structures similar to those on (the contact structures twist n times in the direction of the monodromy). These contact structures are all weakly symplectically fillable (the filling is a Lefschetz fibration over a disk with tori as regular fibers, and the argument that this is a weak filling is similar to the argument that is a weak filling of above).
So we have a weak symplectic filling of 0-surgery on the positive trefoil with any of the contact structures , but we would like to also do surgery on a meridian of this trefoil to obtain the Brieskorn spheres of interest. Ghiggini shows that this meridian can be realized as a Legendrian knot whose its contact framing twists times around it, therefore performing surgery on this meridian corresponds to Legendrian surgery. Since the above result said that Legendrian surgery preserves weak fillability, the resulting Brieskorn sphere with its corresponding contact structure is weakly symplectically fillable.
Furthermore, because the result of the Legendrian surgery gives an integer homology sphere, a weak symplectic filling can be perturbed into a strong symplectic filling, as mentioned above (since the restriction of the symplectic structure to the boundary is an exact form). Therefore we have strongly symplectically fillable contact structures on these Brieskorn spheres.
On the other hand, these contact manifolds are not Stein fillable. Ghiggini proves this by showing that the contact invariant is in the fixed point set of an involution on Heegaard Floer homology, and then studying generators of this fixed point set. Properties of these generators imply that all of them are sent to zero by a map induced by a Stein cobordism between and . Therefore the contact invariant of , which is a linear combination of these generators is sent to zero by the map induced by a Stein cobordism, which is a contradiction. (One of the first theorems proved about the Heegaard Floer contact invariant is that it is sent to the generator of by any map induced by a Stein cobordism from , and thus the contact invariant of a Stein fillable contact structure is non-zero.)
More specifically, the generators of the fixed point set can be written as where is the contact structure obtained by Legendrian surgery on as in the picture below, where the Legendrian unknot has cusps on one side and cusps on the other side.
Note that each of these contact structures is Stein fillable (since it is obtained from Legendrian surgery), so their contact invariants are non-vanishing. By showing that , Ghiggini proved that when is even, the fixed point set of the involution on Heegaard Floer homology is generated by . Furthermore, any map induced by a 4-manifold cobordism to sends and to the same element, so in coefficients, the image of each generator of the fixed point set is zero.
In conclusion, these examples distinguish these three notions of fillability, but I think there is still a lot left to understand about when such examples can arise. Given an arbitrary 3-manifold, there is not usually much one can say about whether it supports contact structures that are strongly symplectically fillable but not Stein fillable, and unless it is a homology sphere it is hard to tell whether there could be weakly but not strongly symplectically fillable contact strucutres. The Heegaard Floer contact invariant has been a useful probe to obstruct Stein (and with twisted coefficients, symplectic) fillability, so a related question is to understand geometric conditions under which the contact invariant vanishes for tight contact structures (see Honda, Kazez, and Matic’s work for some situations when this occurs). Because there was a significant period of time where it was unknown whether weak, strong, and Stein fillability were equivalent notions, it seems that examples where these notions diverge are rare, but I don’t think we really have any idea how prevalent these examples can be, or what geometric properties can allow this phenomenon to occur.