## 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

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

Conclusion

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.