Why not develop a Hamiltonian-based Morse theory?

Question

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory am curious why the seemingly more natural analogue of this can not be applied to the study of symplectic manifolds (or indeed maybe it can but is of no use?). By this I mean a sort of finite-dimensional "Hamiltonian Morse theory" where the gradient-like vector field is replaced with a Hamiltonian-gradient-like vector field $X_H:i_{X_H} ω=dH$, and it is the critical points of H that are relevant rather than those of the action functional (as in Floer homology). Any comments would be welcomed.

Answer 1

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (37) (1984), have shown how to extend the Morse inequalities to this more general case. This extension is based on a certain increasing filtration of the space $X$ compatible in a certain way with the flow $\Phi_t$. Such a filtration produces a spectral sequence in homology and the Morse inequalities relate the Betti numbers of the $E_1$ page of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$

In the very special case when $X$ is a compact smooth manifold, and the flow in question is the (negative) gradient flow of a self-indexing Morse function satisfying the Smale transversality conditions several nice things happen.

1. The above Conley-Zehnder filtration can be identified with the filtration by the sublevel sets $\bigl\lbrace f\leq k+\frac{1}{2}\bigr\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.)
2. The $E_1$ page of the canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen applications of the Conley index applied directly to Hamiltonian flows as suggested in your question, though there might exist. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard a problem is until you try to solve it.

Answer 2

Morse Homology has generators being the critical points of $f:M\to\mathbb{R}$, and the differential counts flowlines (integral curves of $\nabla f$) between critical points. But for a Hamiltonian $H$ the integral curves of $X_H$ are contained in the level sets of $H$, so they don't connect critical points. Said differently, $J\nabla H = X_H$ where $J$ is an almost complex structure (built from $\omega$ and a chosen metric), so it's almost Morse theory but "rotated".

Example: $H$ is height function on sphere, critical points are North and South poles, $X_H$ is rotation about vertical axis.

In relation to Floer theory, you take a (periodic) 1-parameter family of Hamiltonians $\lbrace H_t\rbrace$, and study 1-periodic solutions to $\dot x(t)=X_{H_t}(x(t))$. The differential of Floer homology counts $J$-holomorphic strips which solve Floer's equation (and asymptotically approach the 1-periodic orbits). But if you just have a single Hamiltonian, the solutions are the constant ones, i.e. the critical points of $H$. Then Floer's equation becomes the gradient flowlines of $H$ when the pseudholomorphic strips are also independent of time. So you don't gain anything.