Lecture notes on ECH 6: The relative adjunction formula

Where we are

Let Y be a closed oriented 3-manifold and let \lambda be a contact form on Y. 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 \alpha=\{(\alpha_i,m_i)\} where the \alpha_i are distinct embedded Reeb orbits, and the m_i are positive integers, such that m_i=1 whenever \alpha_i is hyperbolic. If \beta=\{(\beta_j,n_j)\} is another generator, then the differential coefficient \langle\partial\alpha,\beta\rangle should count J-holomorphic  curves in {\mathbb R}\times Y (for a generic symplectization-admissible almost complex structure J) with positive ends at covers of \alpha_i with total covering multiplicity m_i, negative ends at covers of \beta_j with total multiplicity n_j, and no other ends. Let us denote the moduli space of all such J-holomorphic curves by {\mathcal M}(\alpha,\beta). Now we don’t want to count all curves in {\mathcal M}(\alpha,\beta), 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 (X,\omega) with an \omega-compatible almost complex structure J, if C is a somewhere injective J-holomorphic curve, then

\langle c_1(TX),[C]\rangle = \chi(C) + [C]\cdot[C] - 2\delta(C).

We want an analogue of this formula in the symplectization setting for a somewhere injective curve C\in{\mathcal M}(\alpha,\beta). 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

c_1(\xi|_C,\tau) = \chi(C) + Q_\tau(C) + w_\tau(C) - 2\delta(C).

Here \tau is a trivialization of \xi over the Reeb orbits \alpha_i and \beta_j;  the left hand side is the relative first Chern class which we defined in the previous installment; \chi(C) is the Euler characteristic of the domain as before; and \delta(C) 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 Q_\tau(C) is the “relative intersection pairing”, which is a symplectization analogue of the term [C]\cdot[C] in the closed case. The new term w_\tau(C) is the “asymptotic writhe”. Let me now explain both of these.

The relative intersection pairing

Let \alpha=\{(\alpha_i,m_i)\} and \beta=\{(\beta_j,n_j)\} be ECH generators, and assume that \sum_im_i[\alpha_i]=\sum_jn_j[\beta_j]\in H_1(Y) (otherwise there cannot be any holomorphic curves connecting these two generators), and denote this class in H_1(Y) by \Gamma. Let H_2(Y,\alpha,\beta) denote the set of 2-chains \Sigma in Y with $\partial\Sigma=\sum_im_i[\alpha_i]-\sum_jn_j[\beta_j]$, modulo the equivalence relation that \Sigma\sim\Sigma' if and only if [\Sigma-\Sigma']=0\in H_2(Y). Observe that H_2(Y,\alpha,\beta) is an affine space over H_2(Y), and every J-holomorphic curve C\in{\mathcal M}(\alpha,\beta) defines a class [C]\in H_2(Y,\alpha,\beta).

Given a class Z\in H_2(Y,\alpha,\beta), we now want to define the relative intersection pairing Q_\tau(Z)\in{\mathbb Z}.

To warm up to this, recall that given a closed oriented 4-manifold X, and given a class A\in H_2(X), to compute A\cdot A one can choose two embedded oriented surfaces S,S'\subset X representing the class A that intersect transversely, and count the intersections of S and S' with signs.

In the symplectization case, we could try to choose two embedded (except at the boundary) oriented surfaces S,S'\subset[-1,1]\times Y such that \partial S = \partial S' = \sum_im_i\{1\}\times \alpha_i-\sum_jn_j\{-1\}\times\beta_j, and S and S' intersect transversely (except at the boundary), and algebraically count intersections of the interior of S with the interior of S'. Unfortunately this is not well-defined, because if one deforms S or S', 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 \tau allows us to do this. We require that the projections of S and S' to Y are embeddings near the boundary, and their images in a transverse slice to \alpha_i or \beta_j are rays which do not intersect and which do not rotate with respect to the trivialization \tau as one goes around the orbit. If we count the interior intersections of two such surfaces S and S', then we get an integer which depends only on \alpha,\beta,Z,\tau, and we denote this by Q_\tau(Z).

If C\in{\mathcal M}(\alpha,\beta) is a J-holomorphic curve, we define Q_\tau(C)=Q_\tau([C]).

The asymptotic writhe

Given a somewhere injective J-holomorphic curve C\in{\mathcal M}(\alpha,\beta), consider the slice C\cap(\{s\}\times Y) where s>>0. This will be the union, over i, of a braid \xi_i^+ around the Reeb orbit \alpha_i with m_i strands. We can use the trivialization \tau to identify this braid with a link in S^1\times D^2. The writhe of this link, which we denote by w_\tau(\xi_i^+), 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 D^2 direction as one goes counterclockwise around S^1 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 s if s is sufficiently large (this is equivalent to the previously quoted fact that C has only finitely many singularities).

Likewise, the slice C\cap(\{s\}\times Y) for s<<0 is the union over j of a braid \xi_j^- around the Reeb orbit \beta_j with n_j strands. This has a writhe w_\tau(\xi_j^-) which does not depend on s if s is sufficiently negative.

Finally, we define the asymptotic writhe of C by

w_\tau(C) = \sum_iw_\tau(\xi_i^+) - \sum_jw_\tau(\xi_j^-).


We have now defined all of the terms in the relative adjunction formula. Exercise: prove the relative adjunction formula in the case when C is an immersion and the only self-intersections are nodes. Hint: let N_C denote the normal bundle of C (which can be identified with \xi|_C near the ends of C) and compute the relative first Chern class c_1(N_C,\tau) in two ways: First, generalizing the proof of the adjunction formula in the closed case, use the decomposition ({\mathbb C}\oplus\xi)|_C = TX|_C = TC\oplus N_C to show that

c_1(\xi|_C,\tau) = \chi(C) + c_1(N_C,\tau).

Second, count the intersections of C with a nearby surface and compare with the definition of Q_\tau to show that

c_1(N_C,\tau) = Q_\tau(C) + w_\tau(C)-2\delta(C).

We are now almost ready to define ECH, which I plan to do in the next installment.

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