## Strong cobordisms and the ECH contact invariant

I haven’t yet finished my end of semester duties, but as procrastination, here is a math post. If I am not mistaken, there is a simple argument to prove the following, answering a question which Chris Wendl asked me a while ago (inspired by some related work of Cieliebak and Latschev in symplectic field theory):

Let $(Y_+,\lambda_+)$ and $(Y_-,\lambda_-)$ be closed contact three-manifolds, not necessarily connected.  I assume for convenience that all contact forms are nondegenerate.  Define a “strong symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$” to be a compact symplectic four-manifold $(X,\omega)$, not necessarily exact, with oriented boundary $\partial X = Y_+ - Y_-$ such that $\omega|_{Y_\pm}=\lambda_\pm$.  (My use of the words “to” and “from” here is nonstandard, sorry.) Recall that the ECH contact invariant of a closed contact three-manifold $(Y,\lambda)$ is a canonical element of $ECH(Y,\lambda,0)$ represented by the empty set of Reeb orbits. (Here we use ECH with integer coefficients.)

Theorem. Let $(X,\omega)$ be a strong symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$.  If the ECH contact invariant of $(Y_+,\lambda_+)$ is zero, then the ECH contact invariant of $(Y_-,\lambda_-)$ is also zero.

We will prove this using a suitable cobordism map on ECH, defined using Seiberg-Witten theory, by a slight modification of the construction in my joint paper with Taubes “Proof of the Arnold chord conjecture in three dimensions II”, referred to as [CC2] below. To set this up, fix a relative homology class $A\in H_2(X,\partial X)$.  Let $\Sigma$ be a compact oriented smooth surface in $X$, with boundary on $\partial X$, representing the class $A$. Write $\partial_\pm\Sigma = Y_\pm\cap\partial\Sigma$.

Lemma. There is a real number $\rho(A)$ such that for any surface $\Sigma$ as above representing the class $A$, we have

$\int_\Sigma\omega = \int_{\partial_+\Sigma}\lambda_+ - \int_{\partial_-\Sigma}\lambda_- + \rho(A).$

Moreover, $\rho:H_2(X,\partial X)\to{\mathbb R}$ is a homomorphism.

The proof of the lemma is an exercise using Stokes’s theorem.  Note that one automatically has $\rho(A)=0$ for exact cobordisms, as well as for “weakly exact” cobordisms (defined in my paper “Quantitative ECH”) when $\Gamma_\pm=0$ (see the definition of $\Gamma_\pm$ below). These are the cases where we previously defined ECH cobordism maps, see Theorem 1.9 in [CC2]. Now I claim that the proof of that theorem carries over mutatis mutandis to prove the following:

Let $A\in H_2(X,\partial X)$ and write $\partial A = \Gamma_+ - \Gamma_-$ with $\Gamma_\pm\in H_1(Y_\pm)$.  Also fix a “homology orientation” of $X$ as defined in Kronheimer-Mrowka’s book, to determine the signs in Seiberg-Witten cobordism maps. (I suspect that there is a canonical homology orientation in the present situation, but this is not important for now.) Then for each $L\in{\mathbb R}$, there is a map on filtered ECH,

$\Phi^L_{X,A}: ECH^L(Y_+,\lambda_+,\Gamma_+) \to ECH^{L+\rho(A)}(Y_-,\lambda_-,\Gamma_-)$

with the following properties: (Below I am omitting some properties which are not relevant for this discussion, and I am using some terminology from [CC2] without reviewing all of it.)

• (Inclusion axiom) If one replaces $L$ with $L'$ then the maps $\Phi^L_{X,A}$ and $\Phi^{L'}_{X,A}$ are compatible with the maps on filtered ECH induced by the inclusions of chain complexes.
• (Direct limit axiom) If $Y_+$ and $Y_-$ are connected (so that their Seiberg-Witten Floer cohomology is defined), then the direct limit over $L$ of $\Phi^L_{X,A}$ corresponds under Taubes’s isomorphism to the map on Seiberg-Witten Floer cohomology

$\widehat{HM}^*(X,\frak{s}_\omega+A): \widehat{HM}^*(Y_+,\frak{s}_{\xi_+}+\Gamma_+) \to \widehat{HM}^*(Y_-,\frak{s}_{\xi_-}+\Gamma_-).$

Here $\frak{s}_{\xi_\pm}$ denotes the spin-c structure on $Y_\pm$ determined by the contact structure $\xi_\pm$, and $\frak{s}_\omega$ denotes the spin-c structure on $X$ determined by the symplectic form $\omega$.

• (Holomorphic curves axiom) Let $J_\pm$ be almost complex structures on ${\mathbb R}\times Y_\pm$ as needed to define the ECH chain complex for $(Y_\pm,\lambda_\pm)$, and extend these to a “cobordism-admissible” almost complex structure $J$ on $X$. Then $\Phi^L_{X,A}$ is induced by a chain map $\phi$ such that if $\alpha_\pm$ are chain complex generators for $Y_\pm$, then the following hold:
1. If $\langle\phi\alpha_+,\alpha_-\rangle\neq 0$, then there is a (possibly broken) $J$-holomorphic curve in the symplectization completion of $X$ from $\alpha_+$ to $\alpha_-$.
2. If the only (possibly broken) $J$-holomorphic curve in the symplectization completion of $X$ from $\alpha_+$ to $\alpha_-$ is a union of “product cylinders”, then $\langle\phi\alpha_+,\alpha_-\rangle=\pm 1$.

If you believe all of that, then the proof of the theorem on the contact invariants is very short. Namely take $A=0$, and take any $J$ on $X$. Then $\rho(A)=0$, so every nonempty $J$-holomorphic curve in the class $A$ must have a least one positive end, since otherwise the total symplectic action of the negative ends would be negative, a contradiction. (Note that I regard “holomorphic curves” here as currents, so constant holomorphic curves are not allowed.)  Thus the only $J$-holomorphic curve in the class $A$ is the empty set. This means that the chain map $\phi$ in the Holomorphic Curves axiom sends the empty set to plus or minus the empty set. It follows that the direct limit over $L$ of $\Phi^L_{X,A}$ sends the ECH contact invariant for $(Y_+,\lambda_+)$ to plus or minus the ECH contact invariant for $(Y_-,\lambda_-)$. QED.

There are more fun things we can do with the maps $\Phi^L_{X,A}$, but those will have to wait for another post. Meanwhile, let us briefly compare with Seiberg-Witten theory and Heegaard Floer theory.

It was shown by Taubes in “ECH=SWF V” that the ECH contact invariant agrees with a counterpart contact invariant in Seiberg-Witten Floer theory, whose definition is implicit in “Monopoles and contact structures” by Kronheimer-Mrowka. It follows from this and the above proof that if $Y_+$ and $Y_-$ are connected, then $\widehat{HM}^*(X,\frak{s}_\omega)$ sends the Seiberg-Witten contact invariant for $(Y_+,\lambda_+)$ to the Seiberg-Witten contact invariant for $(Y_-,\lambda_-)$. I learned from Steven Sivek that this was proved by Mrowka-Rollin in an unpublished preprint, under the additional assumption that the map $H^1(X;{\mathbb Z})\to H^1(Y_+;{\mathbb Z})$ is surjective; see Theorem 2.4 in his paper “Monopole Floer homology and Legendrian knots”.

For the Heegaard Floer contact invariant (is this known to agree with the contact invariants in the other two theories?), I learned from Paolo Ghiggini that apparently not much is known for strong cobordisms that are not Stein, except when the negative boundary is $S^3$, as shown in Remark 2.14 of his paper “Ozsvath-Szabo invariants and fillability of contact structures”.