Morse theory in infinite dimensions

Question

It seems that people often talk of "doing Morse theory" on loop spaces in two quite different contexts.

Case 1: When one does Morse theory on a loop space $\Omega(M; p,q)$ using the energy functional as the Morse function, then critical points are geodesics and the index is the index is computed via the second variation formula. As explained in Milnor's classic book, one can recover the homology of the loop space by this technique.

Case 2: On the other hand, when one does Morse theory on the space of (unbased) contractible loops of a symplectic manifold endowed with a Hamiltonian H by using the symplectic action functional, then Floer famously showed that (under suitable hypotheses) one recovers the singular homology of the underlying symplectic manifold.

It seems to me that Case 1 can be interpreted as an infinite dimensional generalization of finite dimensional Morse theory in a very straightforward way. However, Case 2 seems to be of a different nature since the Morse function is not computing the homology of the space on which this function is defined.

Is there an intuitive way to explain this difference? Given a functional on an infinite dimensional space, can we reasonably "guess" what kind of homology theory we will get by "doing Morse theory" with this functional?

Answer 1

The first case has finite indices and parabolic gradient flow; the second infinite (co)indices and elliptic gradient flow.

In more detail, the Morse theory of the energy functional $E$ on $X:=\Omega(M;p,q)$ has the following behavior:

1) For generic metrics it's a Morse function (and for other special metrics of interest, it's Morse-Bott with finite-dimensional critical manifolds).

2) Morse indices of critical manifolds are finite.

3) The downward gradient flow amounts to a parabolic PDE; the flow exists for all positive times. The unstable manifold at a point in a critical manifold is a cell whose dimension is the index.

4) There's reasonable control over the limiting behavior of such flowlines (Palais-Smale condition).

So we can build an increasing sequence of finite-dimensional subspaces $X_k$ of $X$, where $X_k$ is the union of the critical manifolds of index $\leq k$ and their descending manifolds. Reasonably enough, the union $\bigcup X_k$ has the homotopy type of $X$.

The symplectic action functional (actually closed 1-form) on the free loopspace $LM$ of a compact symplectic manifold $M$ (or on its component $(LM)_0$ of nullhomotopic loops) shares property 1), but in other respects is very different:

2') All Morse indices and co-indices are infinite.

3') The downward gradient "flow" amounts to an elliptic PDE. It isn't a flow; you can't flow from an arbitrary initial loop.

4') There are well-behaved spaces of gradient flowlines from one critical point $x_-$ to another $x_+$; these are (virtual) manifolds whose dimension at a flowline $u$, the Fredholm index of the linearization at $u$ of the flowline equation, is finite.

These properties are shared by other Floer theories. You can try to build some kind of a homotopy type from this flow (a "Floer homotopy type") as envisioned by Cohen-Jones-Segal, though the technical difficulties are considerably greater than in setting up Floer homology. But there's no reason for it to look anything like $LM$.