Do direct calculations of ECH have any value?

The embedded contact homology of a contact 3-manifold (Y,\lambda) is now known to be isomorphic to the Heegaard Floer homology HF^+(-Y).  This follows from work of Taubes identifying ECH with the Seiberg-Witten Floer cohomology \widehat{HM}^*(Y), together with work in progress by Kutluhan-Lee-Taubes and Colin-Ghiggini-Honda using this isomorphism to identify both of these theories with Heegaard Floer.

One may then wonder, since Heegaard Floer theory is extremely computable, is there any purpose to directly calculating ECH, instead of just calculating HF^+ instead?

Well yes, direct computations of ECH still have some value. The reason is that one can use ECH to define the “ECH spectrum” of a contact 3-manifold (Y,\lambda), and the “ECH capacities” of a 4-dimensional “Liouville domain” bounded by (Y,\lambda).  These ECH capacities give obstructions to symplectically embedding one 4-dimensional Liouville domain into another, which are sometimes sharp. And these invariants depend on the contact form \lambda, not just on the contact structure or the three-manifold. To compute these invariants one really needs to understand ECH; they have no known analogues in Heegaard- or Seiberg-Witten-Floer. In particular, my joint paper with Michael Sullivan, “Rounding corners of polygons and the embedded contact homology of T^3” is needed to compute the ECH capacities of a polydisk (although for this calculation one can skip chapters 6 and 7 and most of chapter 8 of that paper).

The four-dimensional symplectic embedding obstruction from ECH capacities is known to be sharp for the following problems:

Embedding a disjoint union of ellipsoids into an ellipsoid (McDuff)

Embedding an ellipsoid into a polydisk (follows from McDuff’s ellipsoid argument together with work of Dorothee Muller)

The ECH obstruction is not sharp for the problem of embedding a polydisk into an ellipsoid.

For more about the above facts see my survey “Recent progress on symplectic embedding problems in four dimensions”.  In other cases I have no idea whether the ECH obstruction is sharp or not.  There is more discussion of these open questions here.

Meanwhile, returning to three dimensions, if one can establish upper bounds on the ECH spectrum of a contact 3-manifold, then these give rise to quantitative refinements of the Weinstein conjecture (proving the existence of Reeb orbits with upper bounds on their length, or symplectic action). In particular, David Farris’s thesis on the ECH of circle bundles provides examples of ECH spectra which disprove some conjectures about upper bounds on the ECH spectrum and may suggest others.  See the end of this post.

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