In the previous post, we recalled the statement of Taubes’s “SW=Gr” theorem, which equates the Seiberg-Witten invariants of a closed connected symplectic four-manifold with Taubes’s Gromov invariant which counts certain holomorphic curves. The goal of this post is to compute (some of) Taubes’s Gromov invariant for a certain family of examples, namely mapping tori cross . Later, this example will tell us a lot about how to define ECH.
Let be a closed connected symplectic two-manifold. That is, is a closed connected Riemann surface and is an area form on . Let be a symplectomorphism from to itself. That is is an (orientation-preserving) area-preserving diffeomorphism from to itself. The mapping torus of is the three-manifold
Note that fibers over with fiber , and defines a symplectic form on each fiber. We denote the coordinate on by . There is a canonical vector field on which increases the coordinate at unit speed. We will denote this vector field by . A fixed point of determines a periodic orbit of of period , and conversely a simple periodic orbit of of period determines fixed points of .
The fiberwise symplectic forms extend to a closed 2-form on which annihilates . We denote this closed 2-form on by the same letter . We can now define a symplectic form on by
where denotes the coordinate.
We now want to calculate Taubes’s Gromov invariant of the symplectic four-manifold . Remember that Taubes’s Gromov invariant also depends on a choice of homology class . For general this calculation may be difficult, and we are only going to consider the special case where for some . But this case is already interesting, and very useful in finding the definition of ECH.
Almost complex structure
Choose a fiberwise compatible almost complex structure on the fibers of . That is, for each , choose an almost complex structure on the fiber over , such that varies smoothly with . (Equivalently, choose an -compatible on for each such that for each , and depends smoothly on . Note that compatibility here just means that rotates positively with respect to the orientation on .) The fiberwise almost complex structure extends to a unique almost complex structure on such that . It is an exercise to check that is -compatible.
If is an embedded periodic orbit of , then by construction, is a -holomorphic torus. Conversely, I claim that every -holomorphic curve in in the class is a union of (possibly multiply covered) tori of the above form. To see this, note that because the class is -invariant while is pulled back via the projection to . On the other hand, by the construction of , the restriction of to is pointwise nonnegative, with equality only where is either tangent to the span of and or singular. And this implies the claim.
Transversality and nondegeneracy
The above argument shows that Taubes’s Gromov invariant for the class counts unions of (possibly multiply covered) periodic orbits of with total homology class . The next question is, what is the contribution to the count from each such union of periodic orbits? Also, under what circumstances are these holomorphic tori cut out transversely, so that the count is even defined?
Let be a periodic orbit of period , and let be one of the corresponding fixed points of . Recall that the fixed point of is called nondegenerate if the differential does not have as an eigenvalue. In this case, the Lefschetz sign is the sign of . Also, since the linear map is symplectic, its eigenvalues are either real with product , or on the unit circle (again with product ). We say that the fixed point is elliptic if the eigenvalues are on the unit circle, positive hyperbolic if the eigenvalues are real and positive, and negative hyperbolic if the eigenvalues are real and negative. In particular, the Lefschetz sign is in the fixed point is elliptic or negative hyperbolic, and if the fixed point is positive hyperbolic. All of the above conditions depend only on and not on the choice of corresponding fixed point .
I claim that the -holomorphic torus is cut out transversely if and only if is nondegenerate in the above sense. In this case, the sign defined in the previous installment agrees with the Lefschetz sign. I will now prove (most of) this, although I probably won’t be going into this much detail in the lectures or else I will never get very far.
The linearized operator
To prove the above claim, we first need to explain the linearized Cauchy-Riemann operator more carefully. And before that, we need some general formalism.
Suppose is a smooth vector bundle and is a smooth section. Let be a zero of . Then the derivative of the section at defines a canonical map
One way to say this is that you define using a connection on , and the value at does not depend on the choice of connection because the difference between any two connections is a tensor. Another way to say this is that the derivative of , regarded as a smooth map , has a differential , and you compose this with the projection .
Now I want to put the holomorphic curve equation into the above formalism. Let be an closed embedded surface in a closed four-manifold with an almost complex structure . Then is -holomorphic if and only if for every . Another way to say this is that the map
vanishes. Here is the normal bundle of , which we can define as the quotient to avoid making any choices. Now let be the space of all closed smooth embedded surfaces in ; this is an infinite dimensional manifold, and the tangent space at is the space of smooth sections of . There is an infinite dimensional smooth vector bundle over such that the fiber over is the space of smooth sections of . Now as defined above defines a section of this vector bundle whose zero set is exactly the space of embedded holomorphic curves. So by the previous formalism, for each embedded holomorphic curve there is a well-defined operator
Furthermore, since is holomorphic, the values of this operator anticommute with , so it is actually an operator
This is what we call the linearized operator .
One can write the operator in local coordinates as follows. Choose a local complex coordinate on . Extend and to local coordinates on . Use to locally trivialize , and use to define a local trivialization of . (I may be off by some factors of 2 here.) Then in these local coordinates,
Here is a real matrix determined by the derivative of in the directions. In particular, is a first-order elliptic differential operator. The earlier formalities show that the above formula gives a well-defined operator which does not depend on the local coordinates.
While is defined on smooth sections, often one needs to extend this operator to appropriate Banach space completions, for example to regard it as a map from sections to sections. However since it is elliptic, its kernel will still consist of smooth sections, and its cokernel can be identified with the kernel of the formal adjoint , which in local coordinates as above is given by
Deformations of tori
Returning to the mapping torus example, let’s prove (most of) the claim from before. I am going to make this into a series of exercises, since my first minicourse is supposed to have homework, and these exercises may help you understand what is going on.
Let be the embedded -holomorphic torus determined by an embedded periodic orbit of . We can identify the normal bundle to in with , where denotes the vertical tangent bundle of the fiber bundle . The linearization of the flow along defines a connection on this bundle. Also, we can use to trivialize .
Exercise 1: With these identifications (possibly up to factors of 2),
Exercise 2: Every element of is -invariant, so is identified with the kernel of the operator sending sections of to itself. Hint: use a linear analogue of the proof that every holomorphic curve in an -invariant homology class is -invariant.
Exercise 3: If has period and is a fixed point of corresponding to , then there is a canonical identification .
The above three exercises imply that is transverse if and only if is nondegenerate. To prove that agrees with the Lefschetz sign, one deforms the order zero term in the operator to zero (in a -invariant manner) and sees when during the deformation there is a kernel. I’ll skip the detailed calculation for now because this will be clearer after we talk about the Conley-Zehnder index later.
How to count the multiple covers
Recall from the previous installment that for each embedded holomorphic torus as above, there is a generating function encoding how its multiple covers are counted. Specifically, the coefficient of is the number of times you count the (current given by the) -fold cover of . Since corresponds to an embedded periodic orbit of , we can denote this generating function by .
I claim that is if is elliptic, if is positive hyperbolic, and if is negative hyperbolic. Proof: to compute the generating function , we need to compute the sign of (which we have already done) as well as the signs of the three connected double covers of . Let denote the double cover obtained by doubling in the direction, let denote the double cover obtained by doubling in the direction, and let denote the third connected double cover. It is not hard to see that , because you can compute the kernels of the relevant operators in the same way. After a change of coordinates, one can similarly show that . Finally is the sign corresponding to the double cover of ; so this is positive if is elliptic, and negative if is positive or negative hyperbolic. If you believe these sign calculations, then the claim follows directly from the formulas for the generating functions in the previous installment.
The above calculation shows the following: Suppose that all periodic orbits of are nondegenerate (which you can arrange by a perturbation of the map ). Then counts unions of possibly multiply covered periodic orbits of whose total homology class is . Hyperbolic orbits are not allowed to be multiply covered (because the generating function for hyperbolic has no higher powers of ). The sign with which a union of periodic orbits is counted is to the number of negative hyperbolic orbits.
We obtained the generating functions by quoting Taubes’s work, but without understanding the latter in detail, these generating functions may seem rather unmotivated. So let me now say a bit more about why the above generating functions make sense.
First, what if we try to define an invariant of the isotopy class of using other generating funtions? Suppose we count elliptic orbits with a generating function , positive hyperbolic orbits with a generating funtion , and negative hyperbolic orbits with a generating function . These generating functions must satisfy certain relations in order to be invariat under isotopy of . First, it is possible for a bifurcation to occur in which an elliptic orbit cancels a positive hyperbolic orbit of the same period. To obtain invariance under this bifurcation, we must have
Second, a “period-doubling” bifurcation can occur in which an elliptic orbit turns into a negative hyperbolic orbit of the same period and an elliptic orbit of twice the period. For invariance under this bifurcation we need
The generating functions we are using above are and , which satisfy the above relations. If we allowed multiply covered hyperbolic orbits also and counted them with their signs, then the generating functions would be , , and , which do not satisfy the above relations. So that would not give you an isotopy invariant. Throwing out all multiple covers and defining and would not work either.
There are of course other triples of generating functions which satisfy the above relations. For example, if we were to calculate the Euler characteristic of the mapping torus analogue of symplectic field theory (for those of you who know about this), then the generating functions would be
where the omission of even powers of in corresponds to the forbidding of “bad” orbits (without which we would not have invariance of period doubling).
Given that there are different triples of generating functions that satisfy the relations, why is the triple that we found above the right one for determining the Seiberg-Witten invariant of ? Here is one answer: let us put together all of the -invariant Gromov invariants into a single generating function
Here denotes the intersection number of the homology class with a fiber of . Then a fun calculation which I will omit for now shows, using our above calculation of , that
where denotes the Lefschetz number (i.e. algebraic count of fixed points) of the symmetric product of acting on the symmetric product of . But this is what we expect as a result of work of Salamon relating the corresponding Seiberg-Witten invariants to vortex moduli spaces. (That last sentence will be cryptic to the uninitiated, but I am not going to try to explain it now because I have already written so much and I am hardly at the beginning of my planned lectures.)
In the next installment, I will briefly review three-dimensional Seiberg-Witten theory and describe what ECH is supposed to accomplish. In the following installment maybe I can actually start talking about ECH!