## 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.