Sorry I’ve been too busy to post lately. There are a lot of things which I would like to write as soon as I have time. Anyway I wanted to make one brief remark now.
Cylindrical or linearized contact homology does not yet have a rigorous definition in most cases (although the heroic polyfold project of Hofer-Wysocki-Zehnder is supposed to fix this). However, for a contact three-manifold, if there are no contractible Reeb orbits, then there is little difficulty in defining the cylindrical contact homology differential for generic . Since people keep asking me about this, I thought I should explain it here.
The most important point is that any cylinder which is a multiple cover of a nontrivial holomorphic cylinder has Fredholm index at least . To see this, assume that is generic and let be a nontrivial somewhere injective -holomorphic cylinder. Then has index at least . Let and denote the Reeb orbits at the positive and negative end of , respectively. Choose trivializations of the contact plane field over and which extend over . Then
where denotes the Conley-Zehnder index with respect to the above trivialization. Suppose has index . Then we can choose the trivialization so that one of these Conley-Zehnder indices is . Then without loss of generality, and . If denotes the cylinder which is a -fold cover of , then
But is positive hyperbolic, and is elliptic or negative hyperbolic, so it follows from the formulas for the Conley-Zehnder index in this post that and . Thus . We were assuming that , but if then one can get even stronger inequalities.
Since there are no nontrivial multiply covered cylinders of index , there is no difficulty with the compactness to prove that the differential is defined and has square zero.
What about transversality for defining the differential? Note that an immersed cylinder of index 1 is automatically transverse for any . This is because its “normal Chern number” , where denotes the genus and denotes the number of ends at positive hyperbolic orbits, is equal to zero; and automatic transversality holds for immersed curves whenever the normal Chern number is less than or equal to zero. (Incidentally, while curves with nonpositive normal Chern number are good from this perspective, they are bad for ECH, because their multiple covers can have zero or negative ECH index, as I can explain another time.)
What about cylinders which are not immersed? The somewhere injective ones of course will be transverse for generic . The only thing I’m not sure about is multiple covers of cylinders with singularities. I don’t know whether these are transverse for generic or whether one needs to do something fancier to count them. (But I haven’t thought hard about this. Maybe one of the experts can weigh in here.)
So, modulo that last issue, defining the cylindrical contact differential is OK. However, negative index multiple covers become a problem if one tries to define cobordism maps on cylindrical contact homology, or if one allows contractible Reeb orbits and tries to define the contact homology algebra. However I think it is interesting to try to understand these objects without using too much technology, so I might discuss this some more later.