## 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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