*Trigger warning: In this post, I am not going to say anything about transversality.*

Actually, I want to advertise a new book about intersection theory in 3-dimensional contact topology, but before I do that, I need to mention two “recent” developments in higher-dimensional contact topology that I am very excited about.

## Contact geometry in dimensions *five* and higher

(1) This is not so recent anymore, but since it’s one of the topics I used to talk about a lot on this blog, it would be criminal of me not to mention that the *symplectic capping problem has now been solved*, in parallel work by Conway-Etnyre and Lazarev. As I discussed in one of the earliest posts on this blog, this is not unexpected, but the fact that it took so long for such a proof to appear makes it a major development. Both proofs are (as far as I can tell) fairly similar and not so hard to comprehend, but they are very much a part of the ongoing revolution that was triggered by Borman-Eliashberg-Murphy’s introduction of overtwistedness in higher dimensions, along with the criteria for overtwistedness subsequently established by Casals-Murphy-Presas. A nontrivial role is also played by Bowden-Crowley-Stipsicz’s topological work on Stein cobordisms via bordism theory. All of these results were published within the last five years, so I think it’s fair to say that the existence of symplectic caps is a fairly deep fact, despite having been expected for a long time. (Note: Lazarev’s paper also answers some questions that I asked on this blog a while back.)

(2) My former student Agustin Moreno has been doing some interesting work lately with Bowden and Gironella, resulting in a new preprint on Bourgeois contact structures that hit the arXiv today. Bourgeois proved in 2002 that for any given contact manifold , there exists a contact structure on that is determined by a choice of supporting open book for ; this was one of the prototypical results suggesting the conjecture (proved in the mean time by BEM) that all manifolds with almost contact structures should also admit contact structures. The new paper by Bowden-Gironella-Moreno shows that in contrast to the overtwisted (and therefore flexible) contact structures produced by BEM, Bourgeois’s contact structures on are more rigid, and this is true *independently* of the choice of contact structure on , e.g. they prove that for any contact 3-manifold , Bourgeois’s contact structure on will be tight. There are also some surprising results about symplectic fillings of such contact structures, including a theorem that for every , all symplectically aspherical strong fillings of the unit cotangent bundle of are diffeomorphic, making this in some ways a natural successor to my paper that proved the uniqueness of fillings of .

Each of those topics probably deserves a post of its own at some point… maybe I will find some time for that now that my daughter has started preschool.

## …and dimension three

But I did actually want to say something today about intersection theory. You are probably aware that the intersection theory of holomorphic curves plays an important role in 4-dimensional symplectic topology, and you may also be aware that an extension of this theory for punctured holomorphic curves in the setting of symplectic field theory exists, and has interesting applications for contact 3-manifolds (e.g. the aforementioned classification of fillings of ). If you’re like most people I know, you are also afraid to read Siefring’s original papers on this subject, which are, well… long. (Also well written, I should add, though perhaps not as user friendly as one might hope.)

I have been on something of a crusade[1] for several years to popularize this intersection theory, and the newest product of that crusade is a book to be published by Cambridge University Press, the latest draft of which has just been updated on the arXiv:

Contact 3-manifolds, holomorphic curves and intersection theory, arXiv:1706.05540

For anyone who already knows standard holomorphic curve theory and wants to learn the facts of Siefring’s intersection theory as efficiently as possible, my recommendation is to turn directly to Appendix C: this is meant as a quick reference guide that states the essential facts as concisely as possible, and I have already gotten into the habit of consulting it myself on a regular basis for various formulas that I sometimes need to use in my papers. If you also want to know why these concisely stated facts are true, you will find them explained in Lectures 3 and 4, though without the analytically intensive proofs of the relative asymptotic formulas that form the basis of the theory. (I suspect that most readers will consider that a feature rather than a bug.)

The book focuses on topological rather than analytical issues, and the main portion of it was written with a student audience in mind, so the amount of background it assumes in symplectic and contact topology is fairly light. Most of the necessary facts from holomorphic curve theory are summarized concisely without proofs in an appendix.

It does also include one thing for readers who specifically enjoy analysis: Appendix B contains a mostly self-contained proof of local positivity of intersections. When I say “self-contained,” I mean that instead of quoting analytically deep results from the late 1980’s by McDuff or Micallef-White, the appendix gives a complete proof of a “weak version” of the Micallef-White theorem to describe the local structure of critical points of holomorphic curves — this is something that one could equally well describe as a “non-asymptotic” variant of Siefring’s relative asymptotic formulas.[2] To understand the proof, you need to be comfortable with distributions and Sobolev spaces, and you need to be able to follow some of the standard arguments of elliptic regularity theory (e.g. the use of difference quotients and the Banach-Alaoglu theorem), but “self-contained” also means that I’ve avoided relying on certain (standard but…) difficult things like the Calderón-Zygmund inequality. This is possible due to some arguments explained to me by Jean-Claude Sikorav, which I’ve written about in previous posts.

You can read the book for free on the arXiv, but it should also be appearing in print sometime in 2020, so if you like it, please buy it! (Yes, that’s right, I get royalties… not much, but something. If you like, think of it as your modest contribution toward my daughter’s bilingual preschool tuition fees.)

[1] My own personal viewpoint on my career path includes the observation that I benefited early on from being one of at most five people to have read and understood Siefring’s thesis. This made it possible for me to pick a certain amount of low-hanging fruit that no one else at the time perceived as low hanging.

[2] The Micallef-White theorem says that in well-chosen coordinates near any critical point of a J-holomorphic curve, the curve looks like a holomorphic polynomial. This makes it possible to prove that if a critical point is present, then any immersed perturbation of the curve will have a well-defined and strictly positive count of double points in a neighborhood of the original critical point. But one doesn’t need the full Micallef-White theorem to prove the latter — it suffices to have a formula presenting the difference between two intersecting curves as a holomorphic polynomial plus a remainder term, and this is what the “weak version” I’m referring to does. The idea to prove positivity of intersections this way is something I originally learned from Hofer, and it is based on the same intuition as Siefring’s Ph.D. thesis which first introduced the punctured intersection theory. For full disclosure, I should mention that the “weak version” of Micallef-White also appears in Chapter 2 of my perpetually unfinished lecture notes on holomorphic curves, but the proof given there has some flaws and will need to be rewritten in a future revision.