I'm trying to learn about Embedded Contact Homology. To understand the proof of $d^2=0$, I started by watching Hutchings' lectures on Obstruction Bundle Gluing on YouTube ([1](https://www.youtube.com/watch?v=_4rJnq9Fn1Y), [2](https://www.youtube.com/watch?v=FcP8vJSJj3k), [3](https://www.youtube.com/watch?v=0EuXhQpezeE)) and have now started reading the relevant two-part paper by Hutchings-Taubes to learn more about the details of the gluing argument involved ("Gluing pseudoholomorphic curves along branched covered cylinders" [1](https://projecteuclid.org/euclid.jsg/1197491304), [2](https://projecteuclid.org/euclid.jsg/1238592310)). I am stuck at the very beginning of Part I, specifically I'm unable to understand most of the proof of Lemma 1.11 which is about trajectories which are close to breaking (in what follows, I use the notation from the paper). 


Can someone explain (with more details) why one of (i) or (ii) must occur (it's not completely clear to me that this formally follows from the SFT compactness theorem)? I'm guessing that part of the reason for this is that the curves $u\_{\pm}$ are transversally cut out and isolated in $\mathcal M^J(\alpha\_+,\beta\_+)/\mathbb R$ and $\mathcal M^J(\beta\_-,\alpha\_-)/\mathbb R$, and so taking $\delta$ sufficiently small would imply that any index $1$ component in the limit is close to one of $u\_{\pm}$ and thus, is one of $u\_{\pm}$.


Also, why must all the other components in all other levels be branched covers of trivial cylinders (I realize they must be curves whose Fredholm indices add up to $0$, but I'm not able to rule out multiply covered curves which are not trivial cylinders from the limit - and these could have index $<0$). Finally, why is it that in case (ii) there are no other levels?



Question about Obstruction Bundle Gluing paper of Hutchings-Taubes
I was there during this IHES conference to scribe the lectures and also give a discussion session on it. So in case it helps, my notes from both of these are available [here](http://www.math.ias.edu/~joelfish/Hutchings_full.pdf).


Your thought on the reason for (i) and (ii) occurring is correct, in lieu of the fact the sum of the Fredholm indices must equal 2. SFT compactness is only used to get a broken curve from a sequence of curves in $\mathcal{M}$. But we know more about our curves, since they live in $\mathcal{M}\cap\mathcal{C}\_\delta$ for $\delta$ very small.


All other components in other levels are (as maps) going to be covers of trivial cylinders because of the topological type of the image of the curves that live in $\mathcal{C}\_\delta$. The image can be decomposed as $C\_-\cup C\_0\cup C\_+$, such that for $\delta$ very small we have $C\_\pm$ being translates of something that is "very close" to $u\_\pm$ (and that $C\_0$ is "very close" to being a cylinder). For example, in (i) we can't have new genus magically appearing "on top of" $u\_+$ (what's really happening is that the "cylindrical ends" of $u\_+$ are being stretched, and we know that trivial cylinders have energy zero). In the limit, we therefore might get a component which is a multiply covered map over a cylinder, but by Lemma 1.7 it cannot have negative index.


Finally, in (ii) there are no other levels because we already know that one level contains the "nontrivial positive index curve". So any other level would have index 0 and live "at the top or bottom" of the broken curve, which as was explained in the proof, can't happen. (Note that in (i) there could possibly be index 0 levels betwixt the two nontrivial levels.)


