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?

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    $\begingroup$ The theory of Hamiltonian circle actions on symplectic manifolds is closely related to (finite dimensional) Morse theory. In fact, the Hamiltonian is always a perfect Morse-Bott function (perfect Morse function if the fixed point set is finite). This Morse function interplays with the symplectic geometry of the manifold in an interesting way. $\endgroup$ – Nick L Dec 11 '20 at 7:08
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    $\begingroup$ Thanks! :) I've read Cannas da Silva's toric notes and most of her "Lectures on Symplectic Geometry" (and did a summer project on toric things), so I've gotten some taste of this. I'm hoping to dive into the details next semester by reading Audin's book. $\endgroup$ – Nikhil Sahoo Dec 11 '20 at 20:53

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.


Next steps:

(0) (Relative Morse Theory) Geoffrey Mess' paper "Torelli groups of genus two and three surfaces" studies some relative Morse theory of the Abel-Jacobi period locus in the Siegel upper half spaces to deduce that the Torelli group (in genus two) is a free group on countably many generators. I thought his proof was very interesting, and tried to learn more, but hardly made progres...

(1) (Almost Complex Structures) if you're interested in symplectic topology, then Eliashberg-Cielebak's textbook "From Stein to Weinstein and back: Symplectic Geometry of Affine Complex Manifolds" has very interesting treatment of Morse theory, especially as related to almost-complex structures $J$ on symplectic manifolds $(M, \omega)$. I think this textbook eclipses Milnor's texts. Contains very elementary proof that "any $2n$-dimensional complex manifold has the homotopy type of an $n$-dimensional CW-complex". (Indeed the unstable manifold $W^+$ is totally lagrangian with respect to nondegenerate symplectic form $\omega=\omega_f$, and is therefore at most $n$-dimensional). Here $f$ is a real valued Morse function whose restriction to every $J$-invariant two-plane is subharmonic.

(2) Gradient flows to poles (where a potential function $f$ and its gradient $\nabla f$ diverges to $\pm \infty$) appears to have more applications to topology than the conventional gradient flow to zeros. Especially when attempting to strong deformation retract a noncompact source $X$ into a lower dimensional compact spine. Applying gradient flow to zeros requires a Lipschitz continuity-at-infinition condition on the deformation parameter. Here the Lowasiejiwicz inequality typically plays a decisive role in proving the continuity of the reparameterized gradient flow. The biggest problem with "gradient flow to zeros" is that the gradient flow slows down as it approaches its target. In my applications of optimal transport to algebraic topology, I find gradient flow to poles much more convenient, since the gradient enjoys a finite time blow up, and continuity of the reparameterized flow is immediate without any appeal to Lowasiejiwcz. Basically "gradient flow to zeros" is a soft landing, while "gradient flow to poles" accelerates into the target.

More specifically, I'm proposing that "gradient flow to poles" is important next step. And this occurs regularly in optimal transportation, as I describe next.

(3) (Optimal Transportation) Morse theory takes on new form in optimal transportation, where Morse theory plays a role in establishing the regularity/continuity and uniqueness of $c$-optimal transportation plans.

Consider a source probability space $(X, \sigma)$, target $(Y, \tau)$, and cost $c: X\times Y \to \mathbb{R}$. Kantorovich duality characterizes the $c$-optimal transport from $\sigma$ to $\tau$ via $c$-convex potential $\phi=\phi^{cc}$ on $X$ with $c$-transform $\psi=\phi^c$ on $Y$. Kantorovich says the $c$-optimal transport plan $\pi$ is supported on the graph of the $c$-subdifferential $\partial^c \phi$, or equivalently on the graph of $\partial^c \psi$.

The subdifferentials are characterized by the case of equality in $$-\phi(x)+\psi(y)\leq c(x,y).$$ Differentiating the case of equality with respect to $x$ and $y$ yields the equalities $$-\nabla_x \phi(x)=\nabla_x c(x,y)$$ and $$\nabla_y \psi(y)=\nabla_y c(x,y).$$ (R.J.McCann shows these equalities hold almost everywhere under general hypotheses on $c$). For example the (Twist) condition: If $Y\to T_x X$ defined by $y\mapsto \nabla_x c(x,y)$ is injective for every $x\in X$, then $$y=T(x):=\nabla_x c(x, \cdot)^{-1}(\nabla_x \phi(x))$$ defines a $c$-optimal Borel measurable map from $\sigma$ to $\tau:=T\#\sigma$.

Moreover the fibre $T^{-1}(y)$ can be characterized as the set of $x$ satisfying $\nabla_y\psi(y)=\nabla_y c(x,y)$ or $$\nabla_y [c(x,y)-\psi(y)]=0.$$ But observe that differentiating the $c$-Legendre Fenchel inequality a second time we are exlusively studying the global minimums of the potentials $y\mapsto c(x,y)-\psi(y)$, for every $x\in X$.

Using the usual Implicit Function theorem, the fibre $T^{-1}(y)$ is a smooth submanifold of $X$ if $D_x(\nabla_y c(x,y))$ is nondegenerate for every $x\in T^{-1}(y)$. If the target $(Y, \tau)$ is one-dimensional, this requires the function $x\mapsto \nabla_y c(x,y)$ to be critical point free for every $y\in Y$, and $x\in T^{-1}(y)$.

On most source manifolds $(X, \sigma)$ it is difficult to verify the nonexistence of critical points. If $X$ is compact and $c$ is continuous finite valued, then Morse theory (elementary calculus) forbids it. But we happily study costs $c$ with poles if the poles are the only critical values of $c$! For example, the (Twist) hypothesis can be rephrased as saying that the two pointed cross difference $$c_\Delta(x;y,y'):=c(x,y)-c(x,y')$$ is a critical point free function for all $y,y'$,$y\neq y'$ and $x$ on its domain. This cannot be satisfied on compact spaces unless poles are allowed.

(3.1) (Canonical Morse/Cost Functions?) We need distinguish generic and canonical. In my experience, I find generic functions very difficult to write down, or explore, or implement on Wolfram MATHEMATICA. Morse functions are known to be generic (in sense of Sard, Thom, etc.). But personally I prefer canonical Morse functions. Or from mass transport perspective, canonical costs $c$ whose derivatives $\nabla c$ are suitable Morse-type functions.

For example, if you want to study optimal transportation from a closed surface $\Sigma$ to the real line $Y=\mathbb{R}$ (or to circle or to graph), then one seeks an appropriate cost $c: \Sigma \times Y \to \mathbb{R}$ satisfying the above conditions, e.g. that $\frac{\partial c}{ \partial y}(x ,y)$ be critical point free in $x\in \Sigma$ for every $y\in \mathbb{R}$. This is forbidden by Morse theory if $\Sigma$ is compact and $c$ is everywhere finite. (In applications, we allow $c$ to have $+\infty$ poles. Then $\partial c/\partial y$ is possibly critical point free on its domain).

But what is a canonical cost $c: \Sigma \times \mathbb{R} \to \mathbb{R}$ which represents an interesting geometric transport from $\Sigma$ to $\mathbb{R}$? Here the source and target spaces $\Sigma$, $Y=\mathbb{R}$ have no interactions a priori, they are not even embedded in a common background space unless we suppose $Y\subset X$.


For the cup product in the framework of Morse theory, I think Kenji Fukaya studied in in Section 1 of his Morse homotopy and its quantization. Actually to define the cup product we need not one but three Morse functions.

In symplectic geometry, Floer homology can be viewed as an infinite dimensional analogue of Morse theory for the action functional on the path space. See the book Morse Theory and Floer Homology for a detailed introduction.


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