# Why is embedded contact homology so powerful?

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?

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.