The [Embedded Contact Homology][1] (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][2] and its [gen][3] [erali][4] [zations][5], the [Arnold chord conjecture][6], are proved based on this invariant. Furthermore, the [ECH capacities][7] are used to establish a [Weyl law][8], which is the foundation of the [equidistribution][9] 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][10] (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][11] 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?


  [1]: https://arxiv.org/abs/1303.5789
  [2]: https://arxiv.org/abs/math/0611007
  [3]: https://arxiv.org/abs/1202.4839
  [4]: https://arxiv.org/abs/1701.02262
  [5]: https://arxiv.org/abs/2001.01448
  [6]: https://arxiv.org/abs/1004.4319
  [7]: https://arxiv.org/abs/1005.2260
  [8]: https://arxiv.org/abs/1210.2167
  [9]: https://arxiv.org/abs/1812.01869
  [10]: https://arxiv.org/abs/math/0010059
  [11]: https://arxiv.org/abs/0811.3985