## Erratum to “The ECH index revisited”

You may have noticed that there has been a long hiatus in postings on this blog. Last semester was extraordinarily busy with teaching. Anyway I am now thinking about some exciting (to me at least) research topics, and this led me to notice that there is a small mistake in my paper “The ECH index revisited”.

The mistake is that Proposition 6.14 is false as stated; to get a true statement, one should delete the $N$ term when $C$ and $C'$ have a component in common. (The reason why this is a mistake is that the proof of Proposition 6.14 follows the proof of Theorem 5.1, and in the first line of Case 2 of the proof of Theorem 5.1, which is where the $N$ term appears, one is assuming that $a\neq b$.) Moreover, Proposition 6.14 is used in the proof of Theorem 6.6.

One can easily correct Proposition 6.14 by deleting the $N$ term.

Theorem 6.6 and its proof are then still fine if you add the hypothesis that $C$ goes between orbit sets in which negative hyperbolic orbits never have multiplicity greater than one, for example ECH generators. I’m not sure if Theorem 6.6 still works if you allow orbit sets in which negative hyperbolic orbits have multiplicity greater than one. I could figure this out, but I’m not sure if anyone will ever need it.

The only time I have ever used Theorem 6.6 is in the appendix to “Algebraic torsion in contact manifolds” by Latschev and Wendl. And, there we only considered orbit sets that are ECH generators. Hence, this mistake has no effect on what is written in that appendix.

Phew!

The reason I noticed this error is that I have found a way to use $J_0$ and $J_+$ to get new symplectic embedding obstructions in some cases which go beyond the obstructions given by ECH capacities. Thus I needed to go back and recall the details of how $J_0$ and $J_+$ work. (I like to think that although the math literature is full of mistakes, things that are actually useful get reworked and checked and corrected as needed.) Stay tuned for more about these symplectic embedding obstructions a little later.