Where we are
Let be a closed oriented 3-manifold and let be a contact form on . We have seen that we want to define ECH to be the homology of a chain complex which is generated by finite unions of possibly multiply covered Reeb orbits, in which hyperbolic orbits are not multiply covered. We will write such a generator as a finite set where the are distinct embedded Reeb orbits, and the are positive integers, such that whenever is hyperbolic. If is another generator, then the differential coefficient should count -holomorphic curves in (for a generic symplectization-admissible almost complex structure ) with positive ends at covers of with total covering multiplicity , negative ends at covers of with total multiplicity , and no other ends. Let us denote the moduli space of all such -holomorphic curves by . Now we don’t want to count all curves in , but only certain special ones, selected analogously to Taubes’s Gromov invariant of symplectic four-manifolds. Which curves do we select?
The relative adjunction formula
To answer this question, we will need a relative version of the adjunction formula. Recall from an earlier installment that in a closed symplectic four-manifold with an -compatible almost complex structure , if is a somewhere injective -holomorphic curve, then
We want an analogue of this formula in the symplectization setting for a somewhere injective curve . For this purpose we need to re-interpret each term in the formula in the symplectization context; and we will also need to add a new term. The resulting relative adjunction formula is
Here is a trivialization of over the Reeb orbits and ; the left hand side is the relative first Chern class which we defined in the previous installment; is the Euler characteristic of the domain as before; and is the algebraic count of singularities as before (it follows from work of Siefring that this is still finite in the symplectization case, for a reason that I will describe later). The term is the “relative intersection pairing”, which is a symplectization analogue of the term in the closed case. The new term is the “asymptotic writhe”. Let me now explain both of these.
The relative intersection pairing
Let and be ECH generators, and assume that (otherwise there cannot be any holomorphic curves connecting these two generators), and denote this class in by . Let denote the set of -chains in with $\partial\Sigma=\sum_im_i[\alpha_i]-\sum_jn_j[\beta_j]$, modulo the equivalence relation that if and only if . Observe that is an affine space over , and every -holomorphic curve defines a class .
Given a class , we now want to define the relative intersection pairing .
To warm up to this, recall that given a closed oriented 4-manifold , and given a class , to compute one can choose two embedded oriented surfaces representing the class that intersect transversely, and count the intersections of and with signs.
In the symplectization case, we could try to choose two embedded (except at the boundary) oriented surfaces such that , and and intersect transversely (except at the boundary), and algebraically count intersections of the interior of with the interior of . Unfortunately this is not well-defined, because if one deforms or , then intersection points can appear or disappear at the boundary.
To get a well-defined count of intersections, we need to specify something about the boundary behavior. The choice of trivialization allows us to do this. We require that the projections of and to are embeddings near the boundary, and their images in a transverse slice to or are rays which do not intersect and which do not rotate with respect to the trivialization as one goes around the orbit. If we count the interior intersections of two such surfaces and , then we get an integer which depends only on , and we denote this by .
If is a -holomorphic curve, we define .
The asymptotic writhe
Given a somewhere injective -holomorphic curve , consider the slice where . This will be the union, over , of a braid around the Reeb orbit with strands. We can use the trivialization to identify this braid with a link in . The writhe of this link, which we denote by , is defined by projecting to the plane in the usual way and counting crossings with signs. I use the sign convention in which counterclockwise rotations in the direction as one goes counterclockwise around contribute positively to the writhe; this is opposite the usual convention in knot theory, but makes sense in the present context. This writhe is independent of if is sufficiently large (this is equivalent to the previously quoted fact that has only finitely many singularities).
Likewise, the slice for is the union over of a braid around the Reeb orbit with strands. This has a writhe which does not depend on if is sufficiently negative.
Finally, we define the asymptotic writhe of by
We have now defined all of the terms in the relative adjunction formula. Exercise: prove the relative adjunction formula in the case when is an immersion and the only self-intersections are nodes. Hint: let denote the normal bundle of (which can be identified with near the ends of ) and compute the relative first Chern class in two ways: First, generalizing the proof of the adjunction formula in the closed case, use the decomposition to show that
Second, count the intersections of with a nearby surface and compare with the definition of to show that
We are now almost ready to define ECH, which I plan to do in the next installment.