Lecture notes on ECH 7: The ECH index and the definition of ECH

We come now to the key nontrivial part of the definition in ECH: the ECH index.

What we know from before

Continuing with the setup from the previous installment, let Y be a closed oriented 3-manifold with a nondegenerate contact form \lambda, and let J be a symplectization-admissible almost complex structure on {\mathbb R}\times Y. Let \alpha=\{(\alpha_i,m_i)\} and \beta=\{(\beta_j,n_j)\} be ECH generators in the same homology class

\sum_i[\alpha_i]=\sum_j[\beta_j]=\Gamma\in H_1(Y),

and let C\in{\mathcal M}(\alpha,\beta) be a J-holomorphic curve between them.

Define the Fredholm index of C by

ind(C) = -\chi(C) + 2c_\tau(C) + CZ_\tau^{ind}(C).

Here \tau is a trivialization of the contact plane field \xi over the Reeb orbits \alpha_i and \beta_j; the notation c_\tau(C) is a shorthand for the relative first Chern class c_1(\xi|_C,\tau), which by a previous exercise depends only on the relative homology class [C]\in H_2(Y,\alpha,\beta); and CZ_\tau^{ind}(C) is a shorthand for the Conley-Zehnder term that appears in ind, namely the sum over all positive ends of C at a Reeb orbit \gamma of CZ_\tau(\gamma) (these Reeb orbits are covers of the Reeb orbits \alpha_i), minus the corresponding sum for the negative ends of C. We know that if J is generic and if C is somewhere injective, then {\mathcal M}(\alpha,\beta) is a manifold near C of dimension ind(C). (Different components of {\mathcal M}(\alpha,\beta) can have different dimensions, even for somewhere injective curves.)

Definition of the ECH index

If Z\in H_2(Y,\alpha,\beta), define the ECH index

I(\alpha,\beta,Z) = c_\tau(Z) + Q_\tau(Z) + CZ_\tau^I(\alpha,\beta).

Here Q_\tau is the relative intersection pairing defined in the last installment, and CZ_\tau^I is the Conley-Zehnder term that appears in I, namely

CZ_\tau^I(\alpha,\beta) = \sum_i\sum_{k=1}^{m_i}CZ_\tau(\alpha_i^k) - \sum_j\sum_{k=1}^{n_k}CZ_\tau(\beta_j^k).

Here if \gamma is a Reeb orbit and k is a positive integer then \gamma^k denotes the k-fold iterate of \gamma. If C\in{\mathcal M}(\alpha,\beta) define I(C)=I(\alpha,\beta,[C]).

Note that the Conley-Zehnder terms CZ_\tau^{ind}(C) and CZ_\tau^I(\alpha,\beta) are quite different.  The former just involves the Conley-Zehnder indices of orbits corresponding to ends of C; while the latter sums up the Conley-Zehnder indices of all iterates of \alpha_i up to multiplicity m_i, minus the Conley-Zehnder indices of all iterates of \beta_j up to multiplicity n_j. For example, if C has positive ends at \alpha_i^3 and \alpha_i^5 (and no other positive ends at covers of \alpha_i), then the corresponding contribution to CZ_\tau^{ind}(C) is CZ_\tau(\alpha_i^3)+CZ_\tau(\alpha_i^5), while the contribution to CZ_\tau^I(\alpha.\beta) is \sum_{k=1}^8CZ_\tau(\alpha_i^k). I’ll try to say something about the motivation for the definition of CZ_\tau^I (aside from the fact that it just works) later when I talk about the symmetric product picture.

Basic properties of the ECH index

Here are some basic properties of the ECH index. You can try to prove them as exercises (some are harder than others), and if you get stuck, see my paper “An index inequality for pseudoholomorphic curves in symplectizations”.

1) [Well defined] The ECH index I(Z) does not depend on the choice of trivialization \tau (even though the individual terms in its definition do).

2) [Additivity under gluing] If \gamma is another ECH generator in the same homology class as \alpha and \beta and if W\in H_2(Y,\beta,\gamma), then Z+W\in H_2(Y,\alpha,\gamma) is defined and I(Z+W)=I(Z)+I(W).

3) [Index ambiguity formula] If Z'\in H_2(\alpha,\beta) is another relative homology class, then

I(Z) - I(Z') = \langle Z-Z',c_1(\xi) + 2PD(\Gamma)\rangle.

4) [Parity formula]

(-1)^{I(Z)} = \epsilon(\alpha)\epsilon(\beta)

where \epsilon(\alpha) denotes -1 to the number of positive hyperbolic orbits in \alpha.

Note that the definition of the ECH index makes sense if we allow mutiply covered hyperbolic orbits, and the above basic properties still hold, except for the parity formula. That is, let us define an “orbit set” to be a finite set of pairs \alpha=\{(\alpha_i,m_i)\} where the \alpha_i are distinct embedded Reeb orbits and the m_i are positive integers. An ECH generator is an orbit set in which m_i=1 whenever \alpha_i is hyperbolic. Everything we have said so far, except for the parity formula, works for any orbit sets, not just ECH generators. This is needed in the foundations of the theory, even though ultimately we only care about the ECH generators.

The index inequality

Now the key result that gets ECH off the ground is the following:

Theorem (index inequality). Let $\alpha$ and $\beta$ be orbit sets and suppose C\in{\mathcal M}(\alpha,\beta) is somewhere injective. Then ind(C)\le I(C)-2\delta(C).

In particular, ind(C)\le I(C), with equality only if C is embedded.

The index inequality follows immediately by combining four formulas. The first is the definition of the ECH index,

I(C) = c_\tau(C) + Q_\tau(C) + CZ_\tau^I(\alpha,\beta).

The second is the formula for the Fredholm index,

ind(C) = -\chi(C) + 2c_\tau(C) + CZ_\tau^{ind}(C).

The third is the relative adjunction formula from last time,

c_\tau(C) = \chi(C) + Q_\tau(C) + w_\tau(C) - 2\delta(C).

And the fourth is the writhe bound

w_\tau(C) \le CZ_\tau^I(\alpha,\beta) - CZ_\tau^{ind}(C).

The proof of the writhe bound is nontrivial, so I will postpone the explanation of it.

Holomorphic curves with low ECH index

The index inequality is most of what is needed to prove the following important result for the definition of ECH. Below, a “trivial cylinder” means a cylinder {\mathbb R}\times \gamma\subset{\mathbb R}\times Y where \gamma is an embedded Reeb orbit.

Proposition. Suppose J is generic. Let \alpha and \beta be orbit sets and let C\in{\mathcal M}(\alpha,\beta) be any J-holomorphic curve, not necessarily somewhere injective. Then:

0. I(C) \ge 0, with equality if and only C is a union of covers of trivial cylinders.

1. If I(C)=1, then C=C_0\sqcup C_1, where C_0 is a union of covers of trivial cylinders, and C_1 is embedded and has ind(C_1)=I(C_1)=1.

2. If I(C)=2 and if \alpha and \beta are ECH generators, then C=C_0\sqcup C_2, where C_0 is a union of covers of trivial cylinders, and C_2 is embedded and has ind(C_2)=I(C_2)=2.

Let me show you part of the proof of this proposition, because it contains a useful trick. Let C\in{\mathcal M}(\alpha,\beta) be given. Then C is the union of covers of somewhere injective J-holomorphic curves C_i with connected domains. Let d_i denote the covering multiplicity of C_i in C. Let us just consider the special case in which d_i=1 whenever C_i is a trivial cylinder. (The proof of the proposition in the general case requires an additional ingredient which we have not explained.) Now a useful fact about a symplectization {\mathbb R}\times Y is that any J-holomorphic curve can be translated in the {\mathbb R}-direction to make a new J-holomorphic curve. So let us define a new curve C' to be the union over i of the union of d_i different translates of C_i. The curve C' is now somewhere injective (this is where I am using the simplifying assumption that d_i=1 whenever C_i is a trivial cylinder). So we can apply the index inequality to C' to get

ind(C') \le I(C') - 2\delta(C').

Now because the Fredholm index ind is additive under taking unions of holomorphic curves, and because the ECH index I depends only on the relative homology class, this gives

\sum_id_iind(C_i) \le I(C) - 2\delta(C').

Since J is generic, we must have ind(C_i)\ge 0, with equality if and only if C_i is a trivial cylinder. Parts (0) and (1) of the Proposition can now be immediately read off from the above inequality! To prove part (2), one still needs to rule out the case where there is one nontrivial C_i with d_i=2. Here one uses the assumption that \alpha and \beta are ECH generators together with parity considerations. (Exercise.)

Definition of ECH over Z/2

We now have enough ingredients in place to give the definition of ECH with {\mathbb Z}/2 coefficients. (ECH can also be defined with {\mathbb Z} coefficients, but I do not want to explain the signs now.)

Let Y be a closed oriented three-manifold with a nondegenerate contact form \lambda. Let \Gamma\in H_1(Y). We define ECH(Y,\lambda,\Gamma) as follows.

Let J be a generic almost complex structure on {\mathbb R}\times Y. Define a chain complex ECC(Y,\lambda,\Gamma,J) as follows. This is freely generated by ECH generators \alpha=\{(\alpha_i,m_i)\} with \sum_im_i[\alpha_i]=\Gamma. We define the differential \partial on ECC as follows: If \alpha and \beta are generators, then \langle\partial\alpha,\beta\rangle is the mod 2 count of J-holomorphic curves C\in{\mathcal M}(\alpha,\beta) with I(C)=1. Here we mod out by the {\mathbb R}-action on {\mathcal M}(\alpha,\beta) by translation. Also, we declare two elements of {\mathcal M}(\alpha,\beta) to be equivalent if they are equal as currents: that is, if C\in{\mathcal M}(\alpha,\beta) contains a multiply covered trivial cylinder, then we only care about the covering multiplicity, and not about further details of the cover.

The above Proposition implies that the space of curves C\in{\mathcal M}(\alpha,\beta) with I(C)=1 is a discrete set; and a compactness argument which I will explain later shows that in fact this set is finite, so that \partial is defined. (We also have to check that only finitely many \beta can arise in the differential of \alpha; this follows from symplectic action considerations which I will explain.) The proof that \partial^2=0 is much more difficult, for reasons I will explain later.  (The above Proposition gives the compactness part of this; the hard part is gluing theory, and also signs.)  The homology of this chain complex is the embedded contact homology ECH(Y,\lambda,\Gamma). The proof that this does not depend on J is even more difficult, for reasons which I will also explain later. (Currently one needs Taubes’s isomorphism with Seiberg-Witten Floer cohomology for this, although I think it is possible to do this without using Seiberg-Witten theory.)

So I owe you explanations of a number things, but modulo those, we have now defined ECH(Y,\lambda,\Gamma). Observe that this is relatively {\mathbb Z}/d graded, where d denotes the divisibility of c_1(\xi)+2PD(\Gamma) in H^2(Y;{\mathbb Z}) mod torsion. That is, if \alpha and \beta are two chain complex generators, we can define their “index difference” I(\alpha,\beta) by choosing an arbitrary Z\in H_2(Y,\alpha,\beta) and setting

I(\alpha,\beta) = I(\alpha,\beta,Z) \in {\mathbb Z}/d.

This is well defined by the index ambiguity formula.

Note that if \Gamma=0, then the empty set of Reeb orbits is a generator of the chain complex. (It is also a cycle for reasons to be explained later.) There is then a canonical refinement of the relative {\mathbb Z}/d grading on ECH(Y,\lambda,0) to an absolute {\mathbb Z}/d grading in which the empty set has grading zero.

Taubes’s isomorphism

In a series of five papers, Taubes proved that there is a canonical isomorphism of relatively graded modules (over {\mathbb Z}/2 or {\mathbb Z})

ECH_*(Y,\lambda,\Gamma) = \widehat{HM}^{-*}(Y,\frak{s}_\xi+\Gamma).

Here \widehat{HM}^* denotes the Seiberg-Witten Floer cohomology, defined from the dual of the chain complex that determines Seiberg-Witten Floer homology. Also \frak{s}_\xi is a distinguished spin-c structure determined by the contact structure. (I’ll explain this later. So many things to explain! When I turn these blog postings into more formal lecture notes, I hope to make everything much better organized.) This has the property that

c_1(\frak{s}_\xi+\Gamma) = c_1(\xi) + 2PD(\Gamma),

so the index ambiguities on both sides of the isomorphism agree. (The index ambiguity formula from before was an indication that we were on the right track in the definition of the ECH index.)

Taubes’s isomorphism implies that ECH(Y,\lambda,\Gamma) does not depend on the choice of J in its definition. It also depends only on \xi and not on \lambda (although we will later see a filtered version of ECH which does depend on \lambda and is important for applications). In fact all that ECH sees about \xi is the associated spin-c structure \frak{s}_\xi, which is determined by the homotopy class of oriented 2-plane fields represented by \xi (although later we will talk about the ECH contact invariant which can distinguish contact structures in the same homotopy class of oriented 2-plane fields).

What’s next

It’s time for an example.

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