My paper with Cliff Taubes, “Proof of the Arnold chord conjecture in three dimensions II”, is finally finished, and should appear on the arXiv shortly. In “Proof of the Arnold chord conjecture in three dimensions I”, we deduced the chord conjecture in three dimensions from another theorem, which asserts that an exact symplectic cobordism between contact three-manifolds induces maps on filtered embedded contact homology satisfying certain axioms. Part II proves the latter theorem and thus completes the proof of the three-dimensional chord conjecture. (These cobordism maps on filtered ECH are also needed to prove the monotonicity property of ECH capacities.)
The construction of the ECH cobordism maps uses Seiberg-Witten theory, together with Taubes’s isomorphism between ECH and Seberg-Witten Floer cohomology. This may raise several questions in the mind of the reader:
- Why do we define the ECH cobordism maps using Seiberg-Witten theory, instead of more directly by counting holomorphic curves in a (completed) cobordism with ECH index zero?
- Why do we not need Seiberg-Witten theory to define the ECH differential?
- What are the prospects for defining ECH cobordism maps using holomorphic curves instead of Seiberg-Witten?
- Would this have any value?
Let me try to answer at least the first two questions (and I might discuss the last two in a later post).
1. Suppose you try to define an ECH cobordism map by counting holomorphic curves in the completed cobordism with ECH index zero. The first thing you need to check is that this count is well-defined, i.e. that for a generic almost complex structure there are only finitely many such curves. For this purpose you would like to prove a compactness statement asserting that the space of ECH index zero curves is a compact zero-dimensional manifold. Also, to prove that the cobordism map is a chain map you would like to similarly prove that the space of ECH index one curves can be compactified to a compact one-manifold with boundary given by broken curves, where a broken curve consists of an ECH index zero level (counted by the chain map) and an ECH index one level (counted by the differential on one side or the other). (Actually, as in the proof that d^2=0, you may also have an index zero level between them consisting of a branched cover of a union of R-invariant cylinders.)
To try to prove this compactness statement, suppose you have a sequence of holomorphic curves of ECH index zero or one (with the same ends, and in an exact symplectic cobordism so that they have the same energy). By SFT compactness, you can pass to a subsequence which converges to a broken holomorphic curve. At this point you would like to say that since the ECH index is additive under gluing and cannot be negative, you must have either an honest curve of ECH index zero or one, or a broken curve with two levels of ECH index zero and one (and maybe a third level between them consisting of branched covers of R-invariant cylinders).
Unfortunately this does not work, because in a cobordism there can be holomorphic curves with negative ECH index, even when the almost complex structure is generic. These can arise as (possibly branched) covers of genus zero curves with ECH index zero. (A detailed classification of the ways this can happen can follows from Theorem 5.1 and Definition 5.3 in my paper “The embedded contact homology index revisited”; I may explain more about this later.) Thus the limit of a sequence of curves with ECH index zero or one could be a broken curve including a negative ECH index level, and also some positive ECH index levels.
To make matters a bit worse, I can show you an example where the cobordism map is nontrivial, but there exists no ECH index zero curve in the cobordism! How is this possible, you may ask? Well, there are still broken curves of total ECH index zero. To define the cobordism chain map, you somehow need to count contributions from the entire moduli space of (possibly broken) curves of total ECH index zero. And as suggested in the previous paragraph, this moduli space can be a huge mess. This is why defining cobordism maps by counting holomorphic curves is hard. The chord conjecture paper circumvents these difficulties by using Seiberg-Witten theory instead to define the ECH cobordism maps, which turns out to be good enough for the applications so far.
2. So why do we not have a similar problem in defining the ECH differential and proving that its square is zero? The reason is that in a symplectization (as opposed to a more general cobordism), if the almost complex structure is generic then there exist no holomorphic curves (not even multiple covers) with negative ECH index. The proof of this last statement makes use of the fact that if C is a holomorphic curve in a symplectization (not containing any R-invariant cylinders), then you can translate C in the R-direction to obtain another holomorphic curve C’, and by intersection positivity the algebraic count of intersections of C with C’ is nonnegative. Now if C is not multiply covered (and not an R-invariant cylinder), then the ECH index of a d-fold cover of C is the same as the ECH index of the union of d different translates of C. If the almost complex structure is generic, then C has ECH index at least one, and you can use this fact together with the aforementioned intersection positivity to show that the ECH index of a d-fold cover of C is at least d. See e.g. Proposition 2 in my ICM article.