Sorry, I lied at the end of the last installment of lecture notes when I said that I would give the definition of ECH in the present installment. But I’m going to at least explain some of the ingredients in the definition of ECH. Better not to rush through everything. (By the way, my minicourses are probably not going to cover as much material as announced, for the same reason.)
Contact 3-manifolds and Reeb orbits
Let us begin by reviewing some basic definitions. Let be a closed oriented three-manifold. Recall that a contact form on is a 1-form on such that everywhere. Let ; this is a 2-plane field on , which is oriented by . An oriented 2-plane field which can be obtained this way is called a contact structure. Two contact forms determine the same contact structure if and only if one is obtained from the other by multiplication by a positive function on . The 1-form also determines the Reeb vector field characterized by and . The Reeb flow preserves the contact form, i.e. (one-line exercise).
A Reeb orbit is a closed orbit of , i.e. a map for some such that . We consider two Reeb orbits to be equivalent if they differ by translating the parameter. Every Reeb orbit is either embedded, or an -fold cover of an embedded Reeb orbit for some integer . Given as above, the time flow of determines a diffeomorphism , and if , then restricts to a symplectic linear map . This is called the “linearized return map” of the Reeb flow along , and we denote it by . The eigenvalues of do not depend on the choice of . The Reeb orbit latex is called nondegenerate if does not have 1 as an eigenvalue. One can think of this as meaning that is cut out transversely in an appropriate sense. If is nondegenerate, then it is called elliptic, positive hyperbolic, or negative hyperbolic, according as the eigenvalues of are on the unit circle, positive, or negative respectively. The contact form is called nondegenerate if all Reeb orbits are nondegenerate. It is a standard fact that generic contact forms are nondegenerate (although the only reference I have heard of for a proof is a paper by Cieliebak and Frauenfelder, I need to check this out). Assume below that the contact form is nondegenerate.
This geometric setup is very similar to the mapping torus setup we considered previously. We have just replaced the mapping torus flow with the Reeb vector field , and the vertical tangent bundle of the mapping torus by the contact structure.
For reasons I explained previously, we are led to try to define a chain complex whose generators are unions of Reeb orbits, of which the elliptic ones but not the hyperbolic ones may be multiply covered. The differential should then count certain holomorphic curves in . (Embedded contact homology will be the homology of this chain complex.) Let us now start to discuss the latter.
Holomorphic curves in symplectizations
The symplectization of the contact 3-manifold is the symplectic 4-manifold . Here and below, denotes the coordinate on .
We call an almost complex structure on “symplectization admissible” (sorry for the clunky terminology, but we need some term) if it satisfies the following three conditions: First, . Second, , and rotates positively with respect to the orientation given by . Third, is -invariant. (Symplectization-admissible almost complex structures are quite analogous to the almost complex structures we considered on cross a mapping torus.) Note that a symplectization-admissible is determined by its restriction to , and the space of such (with e.g. the topology) is contractible. Exercise: every symplectization-admissible is -compatible. Fix such a .
Observe that if is an embedded Reeb orbit, then is an embedded -holomorphic cylinder in . More generally, we consider -holomorphic curves of the form where the domain is a punctured compact Riemann surface (i.e. a Riemann surface with finitely points called “punctures” removed). If is a (possibly multiply covered) Reeb orbit, a positive end of is a puncture near which is asymptotic to as . What we mean by this is that a neighborhood of the puncture can be given coordinates with such that and . A negative end is defined analogously with and . We assume that all punctures are positive ends or negative ends as above. We mod out by the usual equivalence relation on holomorphic curves (namely composition with biholomorphic maps between domains).
The dimension formula
Recall that in a closed symplectic four-manifold , if is -compatible and generic, and if is a somewhere injective -holomorphic curve, then the moduli space of -holomorphic curves is a manifold near of dimension
I didn’t explain this before, but when is immersed this is basically just the Riemann-Roch formula for the normal bundle, and the general case can be deduced from this (or a different argument using the tangent bundle instead of the normal bundle).
We will need an analogue of the above formula for holomorphic curves in symplectizations. While there is a general version of this for symplectizations of any dimension, we will stick with the four-dimensional formula that we need. Here it is: If is generic, and if is a somewhere-injective -holomorphic curve in with positive ends at Reeb orbits and negative ends at Reeb orbits , then the moduli space of -holomorphic curves near is a manifold of dimension
I need to explain the terms on the right hand side. First, is a trivialization of over the Reeb orbits and . Second, is the “relative first Chern class” of the complex line bundle over with respect to the trivialization . To define this, one chooses a generic section of over which on each end is nonvanishing and constant with respect to the trivialization . One then defines to be the algebraic count of zeroes of . Note that this definition only uses the fact that is an oriented surface and not that it is -holomorphic.
Exercise: depends only on the Reeb orbits and , the homotopy class of the trivialization , and the relative homology class of (i.e. it is unchanged if you replace by such that is nullhomologous in ). More generally, if is any other oriented surface mapped into with the same boundary conditions, so that is defined, then
where on the right hand side, denotes the usual first Chern class of the complex line bundle in .
Continuing with the explanation of the index formula, if is a Reeb orbit, and if is a trivialization of (which we can take to be symplectic with respect to the symplectic form on ), then is the Conley-Zehnder index of with respect to the trivialization . To define this, pick a parametrization . Let denote the one-parameter group of diffeomorphisms given by the flow of . Then induces a symplectic linear map , which using our trivialization we can regard as a symplectic matrix. Then , and is the linearized return map (in our trivialization), which does not have 1 as an eigenvalue. Then is the Conley-Zehnder index of the family of symplectic matrices , which roughly speaking is twice the rotation number of this family of matrices. While there is a general definition of the Conley-Zehnder index for families of symplectic matrices in any dimension (starting the identity and ending at a matrix which does not have as an eigenvalue), we will just give an explicit formula for the case of two-dimensional symmetric matrices which we need. If is hyperbolic, let be an eigenvector of ; then the family of vectors rotates by angle for some integer , and this integer is the Conley-Zehnder index (which is even in the positive hyperbolic case and odd in the negative hyperbolic case). If is elliptic, then we can change the trivialization so that each is rotation by angle where is a continuous function of and . The number is called the “rotation angle” of with respect to , and the Conley-Zehnder index is .
Exercise: the right hand side of the dimension formula does not depend on (even though the individual terms in it do).
The detailed proof of the dimension formula is beyond the scope of this course. There are two things to prove: first that for generic the moduli space of somewhere-injective -holomorphic curves is a manifold, and second the dimension formula for the moduli space. The former is a standard transversality argument (not to say that this is trivial; it is worked out in a paper by Dragnev). To prove the second part, one considers general operators over punctured Riemann surface of the form that arise as deformations of holomorphic curves and shows that these are Fredholm and proves a general index formula for them. To prove the general index formula, the key is to show that the index is “additive under gluing”. This allows one to reduce to the case of closed Riemann surfaces, for which the general index formula reduces to the Riemann-Roch formula.
As usual, the somewhere injective assumption is necessary; there is no for which transversality holds for all multiply covered curves, and this causes a lot of headaches.
The differential on the ECH chain complex will count certain -holomorphic curves in moduli spaces of dimension . Since is -invariant, there is an -action on the moduli spaces, which we can mod out by to obtain a -dimensional space to count.
But we are not going to count all index 1 curves, but rather certain special ones, which are singled out analogously to the way one singles out certain index 0 curves to count in the definition of Taubes’s Gromov invariant. To explain which curves to single out, the next ingredient we need is a relative version of the adjunction formula.