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 be a closed oriented 3-manifold with a nondegenerate contact form , and let be a symplectization-admissible almost complex structure on . Let and be ECH generators in the same homology class
and let be a -holomorphic curve between them.
Define the Fredholm index of by
Here is a trivialization of the contact plane field over the Reeb orbits and ; the notation is a shorthand for the relative first Chern class , which by a previous exercise depends only on the relative homology class ; and is a shorthand for the Conley-Zehnder term that appears in , namely the sum over all positive ends of at a Reeb orbit of (these Reeb orbits are covers of the Reeb orbits ), minus the corresponding sum for the negative ends of . We know that if is generic and if is somewhere injective, then is a manifold near of dimension . (Different components of can have different dimensions, even for somewhere injective curves.)
Definition of the ECH index
If , define the ECH index
Here is the relative intersection pairing defined in the last installment, and is the Conley-Zehnder term that appears in , namely
Here if is a Reeb orbit and is a positive integer then denotes the -fold iterate of . If define .
Note that the Conley-Zehnder terms and are quite different. The former just involves the Conley-Zehnder indices of orbits corresponding to ends of ; while the latter sums up the Conley-Zehnder indices of all iterates of up to multiplicity , minus the Conley-Zehnder indices of all iterates of up to multiplicity . For example, if has positive ends at and (and no other positive ends at covers of ), then the corresponding contribution to is , while the contribution to is . I’ll try to say something about the motivation for the definition of (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 does not depend on the choice of trivialization (even though the individual terms in its definition do).
2) [Additivity under gluing] If is another ECH generator in the same homology class as and and if , then is defined and .
3) [Index ambiguity formula] If is another relative homology class, then
4) [Parity formula]
where denotes to the number of positive hyperbolic orbits in .
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 where the are distinct embedded Reeb orbits and the are positive integers. An ECH generator is an orbit set in which whenever 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 is somewhere injective. Then .
In particular, , with equality only if is embedded.
The index inequality follows immediately by combining four formulas. The first is the definition of the ECH index,
The second is the formula for the Fredholm index,
The third is the relative adjunction formula from last time,
And the fourth is the writhe bound
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 where is an embedded Reeb orbit.
Proposition. Suppose is generic. Let and be orbit sets and let be any -holomorphic curve, not necessarily somewhere injective. Then:
0. , with equality if and only is a union of covers of trivial cylinders.
1. If , then , where is a union of covers of trivial cylinders, and is embedded and has .
2. If and if and are ECH generators, then , where is a union of covers of trivial cylinders, and is embedded and has .
Let me show you part of the proof of this proposition, because it contains a useful trick. Let be given. Then is the union of covers of somewhere injective -holomorphic curves with connected domains. Let denote the covering multiplicity of in . Let us just consider the special case in which whenever 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 is that any -holomorphic curve can be translated in the -direction to make a new -holomorphic curve. So let us define a new curve to be the union over of the union of different translates of . The curve is now somewhere injective (this is where I am using the simplifying assumption that whenever is a trivial cylinder). So we can apply the index inequality to to get
Now because the Fredholm index is additive under taking unions of holomorphic curves, and because the ECH index depends only on the relative homology class, this gives
Since is generic, we must have , with equality if and only if 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 with . Here one uses the assumption that and 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 coefficients. (ECH can also be defined with coefficients, but I do not want to explain the signs now.)
Let be a closed oriented three-manifold with a nondegenerate contact form . Let . We define as follows.
Let be a generic almost complex structure on . Define a chain complex as follows. This is freely generated by ECH generators with . We define the differential on as follows: If and are generators, then is the mod 2 count of -holomorphic curves with . Here we mod out by the -action on by translation. Also, we declare two elements of to be equivalent if they are equal as currents: that is, if 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 with is a discrete set; and a compactness argument which I will explain later shows that in fact this set is finite, so that is defined. (We also have to check that only finitely many can arise in the differential of ; this follows from symplectic action considerations which I will explain.) The proof that 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 . The proof that this does not depend on 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 . Observe that this is relatively graded, where denotes the divisibility of in mod torsion. That is, if and are two chain complex generators, we can define their “index difference” by choosing an arbitrary and setting
This is well defined by the index ambiguity formula.
Note that if , 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 grading on to an absolute grading in which the empty set has grading zero.
In a series of five papers, Taubes proved that there is a canonical isomorphism of relatively graded modules (over or )
Here denotes the Seiberg-Witten Floer cohomology, defined from the dual of the chain complex that determines Seiberg-Witten Floer homology. Also 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
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 does not depend on the choice of in its definition. It also depends only on and not on (although we will later see a filtered version of ECH which does depend on and is important for applications). In fact all that ECH sees about is the associated spin-c structure , which is determined by the homotopy class of oriented 2-plane fields represented by (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).
It’s time for an example.