Today I gave the first of five lectures at the SFT VI workshop in Munich. Thanks to the excellent minicourses over the weekend, I could assume some background about holomorphic curves and Seiberg-Witten theory (and I’ll be able to assume some background about contact geometry when we come to that later). I reviewed the definition of Taubes’s Gromov invariant of symplectic four-manifolds, and computed this for the example of cross a mapping torus (although I rushed through the latter a bit due to lack of time). More details about this are in installments 2 and 3 of the lecture notes previously posted here.
Tim Perutz asked an interesting question: What can we say about a contact three manifold if all of its Reeb orbits (are nondegenerate and) have odd Conley-Zehnder index (i.e. are elliptic or negative hyperbolic)? Taubes and I showed that if all Reeb orbits are (nondegenerate and) elliptic then the three-manifold is a lens space. I suspect that similar arguments might prove the same conclusion when there can also be negative hyperbolic orbits. However this requires checking because there are a number of subtle points in that argument.
Chris Wendl talked about (his interpretation of) Richard Siefring’s intersection theory for punctured holomorphic curves in four-dimensional symplectizations (or more generally symplectic cobordisms). Since this is closely related to the intersection theory in the foundations of ECH, but has different purposes, I would like to try to clarify how these two approaches compare.
First let’s review the intersection theory for ECH. Suppose is somewhere injective (see the lecture notes for this notation). Recall from installment 6 of the lecture notes that there is a relative adjunction formula
where is a count of the singularities of with positive integer weights. Furthermore, the ECH index inequality (see installment 7 of the lecture notes) uses the writhe bound
Equality holds in this bound only if the multiplicities of the ends of (i.e. the covering multiplicities of the Reeb orbits associated to each end) satisfy the partition conditions described in installment 9 (in which case it is known that equality holds “generically”, e.g. for curves of index in a symplectization if is generic; this follows from part II of my gluing paper with Taubes).
Now here is my understanding of what Chris Wendl and Richard Siefring are doing. Note that if you combine the relative adjunction formula and writhe bound above, you get an upper bound for . This bound is usually not sharp, because my writhe bound is only sharp if the multiplicities of the ends of satisfy the partition conditions. On the other hand one can study the writhe more carefully to get a sharp upper bound of the form
where is the maximum conceivable value of , given the Euler characteristic of , the relative homology class of , and the multiplicities of the ends of (and the structure of the Reeb orbits). In particular, while can vary as one deforms , the number is a topological invariant depending only on , the relative homology class of , and the end multiplicities. When equality does not hold, one would interpret as the count of actual singularities , plus “virtual” singularities “hidden at infinity”. (Chris didn’t say actually this, but I am guessing that he would have said this if he had more time, since he said something analogous for the intersection of two curves.) And one would expect that equality holds generically. (Perhaps I am the only one who expects this, but I would like this to be true for something else I have been thinking about, involving ruling out certain kinds of degenerations of holomorphic curves; more about this later.)
Summary: Siefring/Wendl get a sharp writhe bound, while the writhe bound that enters into the ECH index inequality is only sharp for certain end multiplicities.
Also: all writhe bounds depend on difficult asymptotic analysis of holomorphic curves proved by Siefring. (I originally proved the ECH writhe bound only under an analytical simplifying assumption.)
I hope that was a fair comparison of the two approaches; if not, please feel free to complain here.