Lecture notes on ECH 3: the mapping torus example

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 S^1. Later, this example will tell us a lot about how to define ECH.

Mapping tori

Let (\Sigma,\omega) be a closed connected symplectic two-manifold. That is, \Sigma is a closed connected Riemann surface and \omega is an area form on \Sigma. Let \phi be a symplectomorphism from (\Sigma,\omega) to itself. That is \phi is an (orientation-preserving) area-preserving diffeomorphism from \Sigma to itself. The mapping torus of \phi is the three-manifold

Y_\phi = [0,1]\times \Sigma/\sim,

(x,1) \sim (0,\phi(x)).

Note that Y_\phi fibers over S^1={\mathbb R}/{\mathbb Z} with fiber \Sigma, and \omega defines a symplectic form on each fiber. We denote the [0,1] coordinate on Y_\Sigma by t. There is a canonical vector field on Y_\phi which increases the [0,1] coordinate at unit speed. We will denote this vector field by \partial_t. A fixed point of \phi^d determines a periodic orbit of \partial_t of period d, and conversely a simple periodic orbit of \partial_t of period d determines d fixed points of \phi^d.

The fiberwise symplectic forms extend to a closed 2-form on Y_\Sigma which annihilates \partial_t. We denote this closed 2-form on Y_\Sigma by the same letter \omega. We can now define a symplectic form \Omega on S^1\times Y_\phi by

\Omega = ds\wedge dt + \omega

where s denotes the S^1 coordinate.

We now want to calculate Taubes’s Gromov invariant of the symplectic four-manifold (S^1\times Y_\phi,\Omega). Remember that Taubes’s Gromov invariant also depends on a choice of homology class A\in H_2(S^1\times Y_\phi). For general A this calculation may be difficult, and we are only going to consider the special case where A = [S^1]\times \Gamma for some \Gamma\in H_1(Y_\phi). 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 J on the fibers of Y_\phi\to S^1.  That is, for each t\in S^1={\mathbb R}/{\mathbb Z}, choose an almost complex structure J_t on the fiber over t, such that J_t varies smoothly with t. (Equivalently, choose an \omega-compatible  J_t on \Sigma for each t\in{\mathbb R} such that J_{t+1}\circ(\phi_t)_* = (\phi_t)_*\circ J_t for each t\in{\mathbb R}, and J_t depends smoothly on t. Note that compatibility here just means that J_t rotates positively with respect to the orientation on \Sigma.) The fiberwise almost complex structure extends to a unique almost complex structure J on S^1\times Y_\phi such that J\partial_s=\partial_t. It is an exercise to check that J is \Omega-compatible.

Holomorphic curves

If \gamma\subset Y_\phi is an embedded periodic orbit of \partial_t, then by construction, S^1\times\gamma\subset S^1\times Y is a J-holomorphic torus. Conversely, I claim that every J-holomorphic curve C in S^1\times Y_\phi in the class A=[S^1]\times\Gamma is a union of (possibly multiply covered) tori of the above form. To  see this, note that \langle[A],[\omega]\rangle=0 because the class A is S^1-invariant while \omega is pulled back via the projection to Y_\phi. On the other hand, by the construction of J, the restriction of \omega to C is pointwise nonnegative, with equality only where C is either tangent to the span of \partial_s and \partial_t or singular. And this implies the claim.

Transversality and nondegeneracy

The above argument shows that Taubes’s Gromov invariant for the class A=[S^1]\times\Gamma counts unions of (possibly multiply covered) periodic orbits of \partial_t with total homology class A. 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 \gamma be a periodic orbit of period d, and let x\in\Sigma be one of the corresponding fixed points of \phi^d. Recall that the fixed point x of \phi^d is called nondegenerate if the differential d\phi^d_x:T_x\Sigma\to T_x\Sigma does not have 1 as an eigenvalue. In this case, the Lefschetz sign is the sign of det(1-d\phi^d_x). Also, since the linear map d\phi^d_x is symplectic, its eigenvalues are either real with product 1, or on the unit circle (again with product 1). We say that the fixed point x 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 +1 in the fixed point is elliptic or negative hyperbolic, and -1 if the fixed point is positive hyperbolic. All of the above conditions depend only on \gamma and not on the choice of corresponding fixed point x.

I claim that the J-holomorphic torus C=S^1\times\gamma is cut out transversely if and only if \gamma is nondegenerate in the above sense. In this case, the sign \epsilon(C) 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 E\to B is a smooth vector bundle and \psi:B\to E is a smooth section. Let x\in B be a zero of \psi. Then the derivative of the section \psi at x defines a canonical map

\nabla\psi:T_xB\to E_x.

One way to say this is that you define \nabla\psi using a connection on E, and the value at x\in \psi^{-1}(0) 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 \psi, regarded as a smooth map B\to E, has a differential d\psi_{(x,0)}:T_xB\to T_xE, and you compose this with the projection T_{(x,0)}E=T_xB\oplus E_x\to E_x.

Now I want to put the holomorphic curve equation into the above formalism. Let C be an closed embedded surface in a closed four-manifold X with an almost complex structure J. Then C is J-holomorphic if and only if Jv\in TC for every v\in TC. Another way to say this is that the map

\overline{\partial}(C):TC\to N_C,

v\mapsto \pi_{N_C}(Jv)

vanishes. Here NC is the normal bundle of C, which we can define as the quotient TX|_{C}/TC to avoid making any choices. Now let B be the space of all closed smooth embedded surfaces in X; this is an infinite dimensional manifold, and the tangent space at C is the space of smooth sections of N_C. There is an infinite dimensional smooth vector bundle E over B such that the fiber over C is the space of smooth sections of Hom(TC,N_C). Now \overline{\partial} 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 C there is a well-defined operator

\nabla(\overline{\partial}):\Gamma(TC)\to\Gamma(Hom(TC,N_C)).

Furthermore, since C is holomorphic, the values of this operator anticommute with J, so it is actually an operator

\Gamma(TC)\to\Gamma(Hom(T^{0,1}C,N_C)).

This is what we call the linearized operator D_C.

One can write the operator D_C in local coordinates as follows. Choose a local complex coordinate z=s+it on C. Extend s and t to local coordinates s,t,u,v on X. Use \frac{1}{2}d\overline{z} to locally trivialize T^{0,1}C, and use u,v to define a local trivialization of N_C. (I may be off by some factors of 2 here.) Then in these local coordinates,

D_C\psi = \partial_s\psi + J\partial_t\psi + M(s,t)\psi.

Here M(s,t) is a 2\times 2 real matrix determined by the derivative of J in the u,v directions. In particular, D_C 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 D_C 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 L^2_1 sections to L^2 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 D_C^*, which in local coordinates as above is given by

D_C^* = -\partial_s +J\partial_t + M(s,t)^*.

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 C be the embedded J-holomorphic torus S^1\times\gamma determined by an embedded periodic orbit \gamma of \partial_t. We can identify the normal bundle to \gamma in Y_\phi with T^{vert}Y_\phi|_\gamma, where T^{vert}Y_\phi denotes the vertical tangent bundle of the fiber bundle Y_\phi\to S^1. The linearization of the flow \partial_t along \gamma defines a connection \nabla on this bundle. Also, we can use \frac{1}{2}(ds-idt) to trivialize T^{0,1}C.

Exercise 1: With these identifications (possibly up to factors of 2),

D_C = \partial_s + J\nabla_t.

Exercise 2: Every element of Ker(D_C) is s-invariant, so Ker(D_C) is identified with the kernel of the operator J\nabla_t sending sections of T^{vert}Y_\gamma to itself. Hint: use a linear analogue of the proof that every holomorphic curve in an S^1-invariant homology class is S^1-invariant.

Exercise 3: If \gamma has period d and x is a fixed point of \phi^d corresponding to \gamma, then there is a canonical identification Ker(J\nabla_t) = Ker(1-d\phi^d_x).

The above three exercises imply that C is transverse if and only if \gamma is nondegenerate. To prove that \epsilon(C) agrees with the Lefschetz sign, one deforms the order zero term in the operator D_C to zero (in a s-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 C as above, there is a generating function f_C(t) encoding how its multiple covers are counted. Specifically, the coefficient of t^k is the number of times you count the (current given by the) k-fold cover of C. Since C corresponds to an embedded periodic orbit \gamma of \partial_t, we can denote this generating function by f_\gamma(t).

I claim that f_\gamma(t) is (1-t)^{-1}=1+t+t^2+\cdots if \gamma is elliptic, 1-t if \gamma is positive hyperbolic, and 1+t if \gamma is negative hyperbolic. Proof: to compute the generating function f_\gamma(t), we need to compute the sign of C (which we have already done) as well as the signs of the three connected double covers of C. Let C_s denote the double cover obtained by doubling in the s direction, let C_t denote the double cover obtained by doubling in the t direction, and let C_{s,t} denote the third connected double cover. It is not hard to see that \epsilon(C_s)=\epsilon(C), because you can compute the kernels of the relevant operators in the same way. After a change of coordinates, one can similarly show that \epsilon(C_{s,t})=\epsilon(C). Finally \epsilon(C_t) is the sign corresponding to the double cover of \gamma; so this is positive if \gamma is elliptic, and negative if \gamma 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.

Conclusion

The above calculation shows the following: Suppose that all periodic orbits of \partial_t are nondegenerate (which you can arrange by a perturbation of the map \phi). Then Gr(S^1\times Y_\phi,\Omega,[S^1]\times\Gamma) counts unions of possibly multiply covered periodic orbits of \gamma whose total homology class is \Gamma. Hyperbolic orbits are not allowed to be multiply covered (because the generating function f_\gamma for \gamma hyperbolic has no higher powers of t). The sign with which a union of periodic orbits is counted is -1 to the number of negative hyperbolic orbits.

Addendum

We obtained the generating functions f_\gamma 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 \phi using other generating funtions? Suppose we count elliptic orbits with a generating function e(t), positive hyperbolic orbits with a generating funtion h_+(t), and negative hyperbolic orbits with a generating function h_-(t). These generating functions must satisfy certain relations in order to be invariat under isotopy of \phi. 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

e(t)h_+(t)=1.

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

e(t)=h_-(t)e(t^2).

The generating functions we are using above are e(t)=(1-t)^{-1} and h_\pm(t)=1\pm t, 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 e(t)=(1-t)^{-1}, h_+(t)=1-t-t^2-\cdots, and h_-(t)=1+t-t^2+\cdots, which do not satisfy the above relations. So that would not give you an isotopy invariant. Throwing out all multiple covers and defining e(t)=h_-(t)=1+t and h_+(t)=1-t 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

e(t)=(1-t)^{-1}(1-t^2)^{-1}\cdots,

h_+(t) = (1-t)(1-t^2)\cdots,

h_-(t) = (1-t)^{-1}(1-t)^{-3}\cdots,

where the omission of even powers of (1-t)^{-1} in h_-(t) 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 S^1\times Y_\phi? Here is one answer: let us put together all of the S^1-invariant Gromov invariants into a single generating function

P(t) = \sum_d\sum_{\Gamma\cdot[\Sigma]=d}Gr(S^1\times Y_\phi,\Omega,[S^1]\times\Gamma)t^d.

Here \Gamma\cdot[\Sigma] denotes the intersection number of the homology class \Gamma with a fiber of Y_\phi\to S^1. Then a fun calculation which I will omit for now shows, using our above calculation of Gr(S^1\times Y_\phi,\Omega,[S^1]\times\Gamma), that

P(t) = \sum_{d\ge 0}L(S^d\phi)t^d,

where L(S^d\phi) denotes the Lefschetz number (i.e. algebraic count of fixed points) of the d^{th} symmetric product of \phi acting on the d^{th} symmetric product of \Sigma. 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!

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