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The Embedded Contact Homology (ECH), introduced by M. Hutchings, is an invariant of (contact) three-manifolds. Since its introduction, well-known conjectures in symplectic/contact topology in dimension $3$, including the Weinstein conjecture and its gen erali zations, the Arnold chord conjecture, are proved based on this invariant. Furthermore, the ECH capacities are used to establish a Weyl law, which is the foundation of the equidistribution result for generic contact forms. There are also nice applications of ECH to symplectic embedding problems in dimension $4$.

The question is: why is ECH so powerful?

To illustrate the question, the Symplectic Field Theory (SFT) should capture more information from pseudo-holomorphic curves than ECH but the above results were not (yet) proved using SFT. One quick answer to this question would be the celebrated ECH=SWF theorem, which allows one to use powerful results/computations from monopole Floer homology. But eventually, one might hope to stay in the world of pseudo-holomorphic curves without appealing to gauge-theoretic invariants, in order to go to higher dimensions. So the question might be phrased as:

Is ECH a low-dimensional miracle? If not, what lessons should we learn from the selections of pseudo-holomorphic curves which define ECH?

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Without elaborating much there are three key points, with the first two laying the bedrock for the third:

  1. ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).

  2. In dimension 4 (where the J-curves live) we have the adjunction formula.

  3. A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)

Here is another crucial point disguised as an application: On a symplectic 4-manifold with (negative) contact 3-manifold boundary, the standard "ECH curve count" yields a relative invariant in the $ECH_*$ of the boundary, while the standard "SFT curve count" yields a relative invariant in the (ordinary) contact homology $CH_*$ of the boundary. But if the contact structure is overtwisted then $CH_*$ is necessarily trivial, whereas $ECH_*$ can easily be nontrivial.

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  • $\begingroup$ I am not familiar with ECH. So please excuse me if this doesn't make any sense. Can we use ECH to understand the topology a symplectic cap of a contact 3-manifold. Notice that various obstructions such as adjuction inequlaity are valid for the case of symplectic fillings of contact 3-manifolds. And although symlectic filling doesn't always exist, a symplectic cap does exist. $\endgroup$ – Anubhav Mukherjee Feb 17 at 3:00

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