Today I want to tell you about a magic trick for proving transversality and/or defining obstruction bundles in certain four-dimensional situations where the usual approach doesn’t work. This is helpful for defining cobordism maps between various kinds of contact homology in three dimensions (although to completely define cobordism maps you need much more than just this trick).
Here is the setup. Let be a four-dimensional strong symplectic cobordism between nondegenerate contact three-manifolds and . Let denote the usual symplectization completion of , and let be a generic almost complex structure on satisfying the usual conditions.
Let be an immersed -holomorphic curve in . Let be a degree branched cover of with branch points.
When the number of branch points , we would like a criterion which guarantees that is cut out transversely. When the number of branch points , it is more or less impossible for to be cut out transversely, but we would still like to be “good” in the following sense.
Recall that there is a deformation operator
where denotes the normal bundle to . This operator is surjective if and only if is cut out transversely. The kernel of this operator consists of deformations of which are -holomorphic to first order. The index of this operator is what we call the Fredholm index of , denoted by , which when is cut out transversely is the dimension of the moduli space of holomorphic curves near . Now there is an induced operator
To define this, choose a local complex trivialization of . Then locally, in this trivialization,
where is some -form on , determined by the derivative of the almost complex structure in directions normal to . Using the same local trivialization for , we define
Roughly speaking, considers deformations of in directions normal to , so that the branch points don’t move.
Definition. The branched cover is good if .
If there are no branch points and , then is good if and only if it is transverse. If there are branch points, then one needs all branched covers in the moduli space of branched covers of containing to be good, in order to define an “obstruction bundle” over the moduli space of such branched covers in order to do gluing as in my joint papers with Taubes.
So the question is now, under what circumstances is guaranteed to be good?
The usual approach
Recall that the Fredholm index of is given in my notation by
By the Riemann-Hurwitz formula, we can rewrite this as
Note that this is not the Fredholm index of the operator . Rather, we have
Intuitively, this is because the operator does not consider deformations of that move the branch points, so the dimension of its domain is too small.
Anyway, using the above notation, we can now state the result given by the usual approach. Let denote the genus of . Let denote the number of ends of that are at positive hyperbolic orbits (including even covers of negative hyperbolic orbits).
Proposition 1. Suppose that . Then is good.
Proof. Suppose is not identically zero. We know, e.g. from the Carleman similarity principle, that every zero of is isolated and has positive multiplicity, so the signed count of zeroes of is nonnegative. On the other hand, we can bound the number of zeroes of similarly to my previous blog post “automatic transversality for dummies”. I’ll leave the calculation as an exercise; the result is
If the right hand side is negative this is a contradiction. Thus if the right hand side is negative then is good. QED.
Where the usual approach doesn’t work
Here are two examples where Proposition 1 is not applicable but we would still like to be able to prove that is good.
Example 2. Suppose that is a cylinder with Fredholm index zero which has one positive end and one negative end, each at positive hyperbolic orbits. Suppose that is also a cylinder, so that there are no branch points. If we are trying to define cobordism maps on cylindrical contact homology, we would like to be able to prove that is cut out transversely, or equivalently good. Proposition 1 is not applicable here because the inequality that one needs is an equality in this case.
To motivate our next example, note that to define the cobordism maps needed to show that ECH does not depend on the almost complex structure, one needs to consider certain cases where and both have ECH and Fredholm index equal to zero, with all ends at elliptic orbits. One wants to show that is good so that one can do obstruction bundle gluing as in my paper with Taubes to prove the chain map equation. Here is a simple example of such a case where the inequality needed to apply Proposition 1 does not hold.
Example 3. Suppose that is embedded, has genus one, and has exactly one end, which is a positive end at a simple elliptic orbit with monodromy angle (with respect to some trivialization ) . Suppose further that the Fredholm index . Then the ECH index also (this is true for any embedded curve whose ends are at distinct simple orbits by the relative adjunction formula). More precisely, and . Now let be a double cover of with one branch point, and then necessarily exactly one end, which is a positive end at the double cover of . Then it follows from the above calculations that . But , so the inequality needed to apply Proposition 1 is an equality in this case.
The magic trick
Now I can show you the magic trick. The statement is a bit complicated and not the most general possible, but this will illustrate the technique and is sufficient to handle the above two examples.
To proceed with the statement and the proof, we need to consider the normal bundle . We can regard this as a kind of (completed) symplectic cobordism between the disjoint union over the ends of of the normal bundle of the corresponding (possibly multiply covered) Reeb orbit. Let me not try to define exactly what I mean by “symplectic cobordism” in this context. Suffice it to say that , regarded as the zero section, defines an embedded holomorphic curve in whose ends are all at distinct simple Reeb orbits, even if none of the above is true for the original curve .
There is now a well defined notion of the ECH index of in the normal bundle . This is defined by copying the usual formulas, in the normal bundle . We can think of this as the ECH index “relative to “, and we denote it by . If is embedded in and all its ends are at distinct simple Reeb orbits, then the ECH index of in agrees with the ECH index of in , i.e. . If does not have these properties, then it is possible that .
We also say that satisfies the ECH partition conditions “relative to ” if it satisfies the usual ECH partition conditions in the normal bundle . If all ends of are at distinct simple Reeb orbits then satisfies the ECH partition conditions if and only if satisfies the ECH partition conditions relative to .
We can now state:
Proposition 4. Assume that . Suppose that either , or and does not satisfy the ECH partition conditions relative to . Furthermore, if factors through a branched cover whose degree is between and , then assume that the above condition also holds with replaced by . Then is good.
Here is why Proposition 4 is applicable to Examples 2 and 3. In both of these examples, . However neither of these examples satisfies the ECH partition conditions relative to . (In Example 2, the partition conditions relative to would require that has positive ends and negative ends. In Example 3, the partition conditions relative to would require that has two positive ends.)
Proof of Proposition 4. First observe that there is a unique almost complex structure on the normal bundle (regarded here as four-manifold, not a bundle) whose restriction to the fibers agrees with the almost complex structure on , such that a local section is in the kernel of the operator if and only if is a holomorphic map from a neighborhood in to .
Now suppose is a nonzero element of . Let denote the image of under the projection . Then is a holomorphic curve in .
By the assumption that and the assumption about intermediate branched covers in Proposition 4, we can assume without loss of generality that is somewhere injective.
Now a version of the ECH index inequality tells us that , with equality only if satisfies the ECH partition conditions relative to . This is proved just like the usual ECH index inequality, except that in this case one does not need Siefring’s nonlinear analysis; instead one can use the much easier, linear analysis in my paper “An index inequality for pseudoholomorphic curves in symplectizations”.
Now in the first sentence of the above paragraph, one can replace everywhere by without changing anything. We then have a contradiction to the assumptions of Proposition 4, which means that couldn’t exist, so is good. QED.
Remark. There is a variant of Example 3 in which is the same as before, but now there are no branch points, so that has genus one and two ends. Here neither Proposition 1 nor Proposition 4 is applicable to show that is good, i.e. transverse. However I don’t think one needs to consider this case to define ECH cobordism maps. The reason is that the curves you would want to glue this to would have just one negative end (by the ECH partition conditions), so you would want to add a branch point so that has just one positive end as in Example 3.