Embedded contact homology(abbreviated by ECH) is a Floer type theory specially designed for three dimensional contanct manifolds(or generally, manifold with stable Hamiltonian structure) invented by Hutchings and developed by many people.
The generators of ECH is given by orbit sets, which is a multiset of Reeb orbits on the manifold with certain restriction: The hyperbolic orbit must have multiplicitiy one.
I read Hutchings, "Lecture notes on Embedded contact homology" and I've been reading Hutchings and Taubes, "Gluing pseudoholomorphic curves along branched covered cylinders I" to understand the reason, but still don't get the point.
He explains some reasons in "Lecture notes on ECH", but it doesn't seem directly explain the reason.
The restriction comes from the mapping tori example of Taubes' SW = Gr, and some analysis on the setting give the condition on mutliplicity. However, I don't know how to rewrite this result in the language of ECH, since I'm not familiar with Seiberg Witten theory. In addition, the case is a little different, we are dealt with noncompact symplectization on ECH.
To prove $\partial^2 = 0$ in ECH, we investigate the breaking of index $2$ holomorphic currents, and then we find that we don't need any multiply covered hyperbolic orbit at the middle. Howewver, I'm not sure this analysis tells us the condition about the multiplicity is necessary. I thought it only shows it's sufficient condition to construct ECH.
I expect, if there are some reasons, then the reason will be transversality, the dimension of moduli space, caused by some multiply covered pseudoholomorphic curves whose ends contain the multiple hyperbolic orbit. Since there are so many assumption requiring the simpleness and most analytic issues are from multiply covered one.
I would like to have more concrete understanding of the theory, I ask for some help.
