## Ellipsoid embedding obstructions without Seiberg-Witten

In a previous post, I asked whether the ECH obstruction to symplectically embedding one ellipsoid into another (which McDuff has shown is sharp) can be proved without using Seiberg-Witten theory.  Since then, I figured out how to do this, at least when the target of the embedding is a ball. (Actually I figured this out last April but didn’t get around to writing it, gack!) Namely, I will now sketch a Seiberg-Witten-free proof of:

Theorem. Let $E_1,\ldots,E_n$ be four-dimensional ellipsoids, let $E=\coprod_{i=1}^nE_i$, and let $B$ be a four-dimensional ball.  If there is a symplectic embedding $E\to B$ then $c_k(E)\le c_k(B)$ for all nonnegative integers $k$.  Here $c_k$ denotes the $k^{th}$ ECH capacity, which for $E$ or $B$ we define using the usual combinatorial formula (instead of ECH).

Of course we already know this theorem as a consequence of the monotonicity axiom for ECH capacities (which is proved using ECH cobordism maps which are defined using Seiberg-Witten) as well as the combinatorial formula for the ECH capacities of an ellipsoid (which is also proved using Seiberg-Witten to compute the $U$ map although that is probably not so necessary).  And McDuff in her paper “The Hofer conjecture paper on embedding symplectic ellipsoids” gave another proof of the above theorem, but also using Seiberg-Witten theory, specifically SW=Gr and wall crossing on blown up ${\mathbb C}P^2$.

Why should we go through the exercise of trying to prove the above theorem without using Seiberg-Witten?  Shouldn’t we always use the most effective tools at our disposal?  Yes of course, but by trying to do things without Seiberg-Witten, we can maybe learn more about ECH and holomorphic curves which could possibly be useful in contexts where Seiberg-Witten is not available.  But don’t get too excited: while in my previous post I suggested that this exercise could lead to a holomorphic curve definition of ECH cobordism maps in a special case, I will actually prove the above theorem without using ECH cobordism maps. (Instead I’ll be giving holomorphic curve definitions of certain compositions of ECH cobordism maps with powers of the $U$ map, which turn out to be easier to handle.)

Review of the $U$ map.

To prepare for the proof of the above theorem, we first need to review the definition of the $U$ map in ECH.  Below we always use ${\mathbb Z}/2$ coefficients; while it is possible to use ${\mathbb Z}$ coefficients, this is not necessary to prove the theorem.

Let $(Y,\lambda)$ be a connected nondegenerate contact three-manifold and let $\Gamma \in H_1(Y)$.  Fix a generic almost complex structure $J$ on ${\mathbb R}\times Y$ as needed to define the ECH chain complex.  If $\alpha$ and $\beta$ are chain complex generators, let ${\mathcal M}_k(\alpha,\beta)$ denote the moduli space of $J$-holomorphic curves “from $\alpha$ to $\beta$” (in the ECH sense) with ECH index $k$, regarded as currents (i.e. if there are any multiply covered components we care only about their multiplicities).  For example, the differential coefficient $\langle\partial\alpha,\beta\rangle$ is the ${\mathbb Z}/2$ count of holomorphic currents in ${\mathcal M}_1(\alpha,\beta)/{\mathbb R}$.

We now define

$U: ECH_*(Y,\lambda,\Gamma,J) \to ECH_{*-2}(Y,\lambda,\Gamma,J)$

as follows.  Pick a base point $y\in Y$ which is not on any Reeb orbit. Define a chain map

$U_y: ECC_*(Y,\lambda,\Gamma,J) \to ECH_{*-2}(Y,\lambda,\Gamma,J)$

as follows: If $\alpha$ and $\beta$ are chain complex generators, then the coefficient $\langle U_y\alpha,\beta\rangle$ is the ${\mathbb Z}/2$ count of holomorphic currents in $\mathcal{M}_2(\alpha,\beta)$ that pass through the point $(0,y)\in{\mathbb R}\times Y$. This is a well-defined chain map, see e.g. section 3.8 of my recently posted lecture notes on ECH.  Moreover if $y'$ is a different base point then the maps $U_y$ and $U_{y'}$ are chain homotopic.  To prove this one chooses a path $\gamma$ from $y$ to $y'$ which does not intersect any Reeb orbit.  Define

$K_\gamma: ECC_*(Y,\lambda,\Gamma,J) \to ECH_{*-1}(Y,\lambda,\Gamma,J)$

as follows: the coefficient $\langle K_\gamma\alpha,\beta\rangle$ is the ${\mathbb Z}/2$ count of curves in $\mathcal{M}_1(\alpha,\beta)$ with a marked point mapping to the path $\{0\}\times\gamma$ in ${\mathbb R}\times Y$. If $J$ is generic, then this is a chain homotopy between $U_y$ and $U_{y'}$.  Finally, $U$ is the map on homology induced by any $U_y$.

Powers of the $U$ map

Next we need to discuss powers of $U$.  Given a positive integer $k$, fix base points $x=(x_1,\ldots,x_k)\in{\mathbb R}\times Y$ which are not in ${\mathbb R}$ cross a Reeb orbit.  Define

$U_x: ECC_*(Y,\lambda,\Gamma,J) \to ECC_{*-2k}(Y,\lambda,\Gamma,J)$

as follows: if $\alpha$ and $\beta$ are chain complex generators, then the coefficient $\langle U_x\alpha,\beta\rangle$ is the ${\mathbb Z}/2$ count of holomorphic currents in $\mathcal{M}_{2k}(\alpha,\beta)$ that pass through $x_1,\ldots,x_k$.  I claim that if $J$ is generic then $U_x$ is a well-defined chain map, and the induced map on homology is $U^k$. To see this, one can show similarly to the definition of $U$ that $U_x$ is a well-defined chain map, and a generic path from $x$ to $x'$ induces a chain homotopy between $U_x$ and $U_{x'}$. Then, if one keeps $x_1,\ldots,x_{k-1}$ fixed and translates $x_k$ up to infinity, one obtains a chain homotopy between $U_x$ and $U_{(x_1,\ldots,x_{k-1})}\circ U_{x_k}$. Since $U_{x_k}$ induces $U$, it follows by induction on $k$ that $U_x$ induces $U^k$.

A special cobordism map

Now suppose we have a symplectic embedding $E\to B$ as in the statement of the theorem. Without loss of generality the ellipsoids $E_i$ are irrational so that the standard contact forms on their boundaries are nondegenerate. Also without loss of generality $E$ maps to the interior of $B$. Then the complement of the image of the interior of $B$ in $E$ is a symplectic cobordism $X$ from the sphere $\partial B$ to the union of three-dimensional ellipsoids $\partial E$.

There is now a standard procedure for attaching symplectization ends to $X$ to obtain a “completion” $\overline{X}$, see e.g. section 5.5 of the lecture notes on ECH. Pick a generic “cobordism-admissible” almost complex structure $J$ on $\overline{X}$, see the above reference for the definition of this. One can then try to define cobordism maps by counting $J$-holomorphic curves in $\overline{X}$. For example one would like to define a cobordism map

$\Phi: ECH_*(\partial B) \to ECH_*(\partial E)$

by counting $I=0$ holomorphic currents in $\overline{X}$. This is fraught with difficulties, see the above reference. However in the present very special case we can actually define something by counting holomorphic currents in the cobordism. This will be, for each nonnegative integer $k$, a map

$\Phi_k: ECH_{2k}(\partial B) \to ECH_0(\partial E) = {\mathbb Z}/2.$

Here by $ECH_{2k}(\partial B)$ we really mean the grading $2k$ ECH of a perturbation of the Morse-Bott contact form on $\partial B$ to a nondegenerate contact form, where the perturbation is small with respect to $k$.

One can think of $\Phi_k$ as the composition of the map $\Phi$ (currently only definable using Seiberg-Witten) with the $k$-fold composition of the $U$ map (on the top and/or the bottom of the cobordism). To define the map $\Phi_k$, pick $k$ generic points $x_1,\ldots,x_k\in\overline{X}$. If $\alpha$ is a generator of the ECH  chain complex for $\partial B$ in grading $2k$, define $\Phi_k(\alpha)$ to be the mod 2 count of holomorphic currents in $\overline{X}$ with ECH index $2k$ from $\alpha$ to the empty set which pass through the points $x_1,\ldots,x_k$.

Why $\Phi_k$ is well-defined

Why is the above count of holomorphic currents well-defined? Why do we not run into multiple cover difficulties here?

Let $C$ be the cobordism level of a possibly broken $I=2k$ holomorphic current in $\overline{X}$ from $\alpha$ to the emptyset which passes through the points $x_1,\ldots,x_k$. We can write $C=\{(C_i,d_i)\}$ where the $C_i$ are somewhere injective curves in $\overline{X}$ and the $d_i$ are positive integer multiplicities. Write $C = C_{good} \cup C_{bad}$ where $C_{good}$ denotes the union of those pairs $(C_i,d_i)$ such that $C_i$ contains one of the points $x_j$, and $C_{bad}$ denotes the union of those pairs $(C_i,d_i)$ such that $C_i$ does not contain one of the points $x_j$.

The reason why $C_{good}$ is “good” is the following. If $C_i$ contains one of the points $x_j$, then the Fredholm index $ind(C_i)\ge 2$. In particular, one can perturb $C_i$ to a nearby holomorphic curve which intersects it in only finitely many points. One can now carry over the argument in Proposition 3.7 of the lecture notes on ECH to show that $I(C_{good})\ge 2k$, and if equality holds then $d_i=1$ for each $(C_i,d_i)$ in $C_{good}$.

The reason why $C_{bad}$ is “bad” is the following. If $C_i$ contains none of the points $x_j$, then it is possible that the Fredholm index $ind(C_i)=0$. In this case $C_i$ cannot be perturbed to a nearby holomorphic curve. One then cannot carry over the argument in Proposition 3.7 of the lecture notes on ECH. In particular one could have $d_i>1$ and $I(C_{bad})<0$, which is the usual trouble with multiple covers.

The crux of the argument is now that $C_{bad}=\emptyset$. To see this, note that the good part $C_{good}$ goes from some orbit set $\alpha_{good}$ in $\partial B$ to some orbit set $\beta_{good}$ in $\partial E$.  Because the ${\mathbb R}\times \partial B$ levels, if any, of our broken holomorphic current decrease symplectic action, we have

$\mathcal{A}(\alpha) \ge \mathcal{A}(\alpha_{good}).$

Now it is a very special property of the ECH chain complex of an ellipsoid – “monotonicity” if you like – that the ECH index increases with symplectic action. (Also the ECH chain complex has a canonical ${\mathbb Z}$-grading so that this is a meaningful statement.) Thus we have

$I(\alpha) \ge I(\alpha_{good}).$

By the same reasoning applied to ${\mathbb R}\times \partial E$ levels (if any) of our broken holomorphic current, we have

$I(\beta_{good}) \ge 0.$

Since $I(\alpha) = 2k$, it follows that

$I(\alpha_{good}) - I(\beta_{good}) \le 2k.$

Now there is no ECH index ambiguity in our cobordism, so it follows that

$I(C_{good})\le 2k.$

But we saw before that $I(C_{good})\ge 2k$. Therefore equality holds in all the above inequalities. Consequently, $\alpha_{good}=\alpha$ and $\beta_{good}=\emptyset$, so $C_{bad}=\emptyset$. Also, since $I(C_{good})=2k$, we know from above that each $d_i=1$ so we do not have to worry about any multiple covers whatsoever.

The map $\Phi_k$ is nontrivial

The above argument shows that $\Phi_k$ is well-defined, and similar reasoning shows that it does not depend on the choice of points $x_1,\ldots,x_k$.

Moving the points $x_1,\ldots,x_k$ up the ${\mathbb R}\times \partial B$ end of the completed cobordism $\overline{X}$ shows that

$\Phi_k = \Phi_0 \circ U^k$.

Now $\Phi_0$ sends the empty set to the empty set because there is exactly one $I=0$ holomorphic current from the empty set to the empty set, namely the emptyset.

And $U^k$ sends the grading $2k$ generator to the empty set; there is a Seiberg-Witten-free proof of this in section 4.1 of the lecture notes on ECH.

Conclusion of the proof of the theorem

Now move the points $x_1,\ldots,x_k$ down the ${\mathbb R}\times\partial E$ end of the cobordism. There is always at least one curve counted by $U_k$, and as the points go down to minus infinity this family of curves must break into something, where the ${\mathbb R}\times\partial E$ levels have total ECH index at least $2k$. And the cobordism level must decrease symplectic action by the usual Stokes’s theorem argument, see e.g. section 5.5 of the ECH lecture notes.

Conclusion: any $I=2k$ generator of the ECH chain complex for $\partial E$ must have symplectic action less than or equal to that of the $I=2k$ generator of $\partial B$. And that is what was to be shown.