I don't know if this question is really appropriate for MO, but here goes: I quite like Morse theory and would like to know what further directions I can go in, but as a complete non-expert, I'm having trouble seeing forward to identify these directions and where I should be reading. Below, I will mention my background and particular interests, then mention things that I've heard of or wondered about. I would appreciate references appropriate for my level, or even better, sketches of any historical or recent Morse-y trajectories.
I have read Milnor's Morse Theory and Lectures on the H-cobordism Theorem (the latter was the subject of my undergraduate thesis). I have also read a little bit about Morse homology. I think the issue is that my knowledge of Morse theory ends there, not only in detailed knowledge, but also in terms of themes and trajectories. That makes it difficult to know where to look next. My main interests (at the current time) are in differential topology and symplectic stuff. To give this question a reasonable range, here are a couple restrictions:
- This question concerns topics in "Morse theory" (in some broad sense), not applications of Morse theory to other things. I am definitely interested in those as well, but that list would be unending. In particular, I'm moving my toric curiosities to a different question.
- I'm mainly interested in manifold-y stuff, as opposed to say, discrete or stratified Morse theory.
- Restricting to finite dimensions is perfectly fine for this context. I am aware that there are Hilbert/Banach manifolds and such to be discussed, but I don't know anything about them. Perhaps I can't outlaw Floer theory entirely, but I'll just say that while I plan to learn about it eventually, I think it's beyond my present scope.
Here are some specific things that I have wondered about:
Cohomology products: I imagine that for a Morse-Smale pair, the cup product (or its Poincaré dual) could be computed by intersection numbers of the un/stable manifolds, though I haven't read an account of this in detail. Near the end of Schwarz's Morse Homology (which I have not read), he defines the cup product in an analogous style to the usual singular cohomology construction. Perhaps most interesting are the products in Chapter 1 of Fukaya's "Morse Homotopy, $A^\infty\!$-Category, and Floer homologies." I have not read this yet, but hope to do so in the near-ish future. Are there any other major view of the cup product in Morse cohomology that I have missed here?
CW Structure: In Morse Theory, Milnor describes manifolds by adding cells and then sliding them around to get an actual CW structure (i.e. cells only attach to lower-dimensional cells). This is useful, but quickly leaves the manifold behind and just becomes a question about homotoping attaching maps. The un/stable manifolds add an important layer of detail about handle decompositions, but even with a Morse-Smale pair, the "attaching" maps notoriously fail continuity. Fixing this seems to be a finicky question and I'm not sure where the answer lies. If I understand correctly, this is related to compactifying moduli spaces of flow lines, so perhaps the answer can be found in Schwarz's book or Hutchings' notes? (Although a comment on this MO question purports that Hutchings' assertion is mis-stated.) Is a bona fide CW structure related to what Cohen-Jones-Segal were looking for in "Morse theory and classifying spaces"? (Yet again, I have not read, but I am intrigued and hope to.)
Finite volume flows: Another paper that I have been intrigued by, but have not read is Harvey and Lawson's "Finite volume flows and Morse theory." It seems like a beautiful way to circumvent the aforementioned issues of discontinuity and create a whole new schema of Morse theory in the process. However, reading it would probably involve learning about currents first… It seems very elegant in and of itself, but it might be interesting to know where this theory goes and what is being done with it, as motivation to learn the necessary background.
Cerf theory: I've heard a little bit about Cerf theory, but I can't really find any references on it (in English, since I don't speak French). As a way to understand the relationship between different handle decompositions, it seems like a very natural thing to pursue. Perhaps it's unpopular because of the difficulty/length of Cerf's paper? Or because it was later subsumed by the framed function work of Hatcher, Igusa and Klein (and maybe others, I just don't know anything about this area), as mentioned in this MO question? I really don't even know enough about this to ask a proper question, but I would love any suggestions for how to learn more.
Other: Any other major directions that you would suggest to a Morse theory enthusiast?
Best Answer
A recent breakthrough result which uses Morse theory in a substantial manner is Watanabe's disproof of Smale conjecture in dimension 4. In it, he provides a method to compute Kontsevich's configuration space integrals by counting certain broken flowlines for gradients of Morse functions. These Morse-theoretic invariants are used to prove that certain 4-dimensional disk bundles with trivialized are not trivial bundles. There is still much to do in developing the properties of these types of invariants, and in using them to detect non-trivial homotopy groups of the diffeomorphism groups of other manifolds.