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If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold itself from relatively basic information about the gradient flow diffeomorphisms of $f$. To sketch briefly: for each pair of critical points $p$ and $q$ of $f$ (i. e., fixed points of the diffeomorphisms), we can consider the subset $S\_{p,q}$ of $M$ that is attracted to $p$ and repelled from $q$ under the diffeomorphisms. ("Repelled from $q$" just means attracted to $q$ under the inverse of the diffeomorphisms.) These $S\_{p,q}$ essentially constitute a decomposition of $M$ as a cell complex. If you just want the homology groups, then you can get away with just considering pairs of critical points whose indices differ by one, and if you just want the Euler characteristic, then you only need local information around each critical point (to define its index). The index of a critical point $p$ is the number of negative eigenvalues of the Hessian (which does not actually depend on the coordinates chosen or even the metric), and it is also the dimension of the submanifold of points in any small neighborhood of $p$ that are attracted to $p$ under the gradient flow diffeomorphisms.

I want to know how much of that can be done if we don't have $f:M\to \mathbb{R}$, but just some transformation $F:M\to M$ homotopic to the identity, and if $M$ isn't necessarily even a manifold (but probably compact and metrizable). Given information about the fixed points of $F$ (or other dynamical information?), how much of the topology of $M$ can be recovered? (Can we still try to define the "index" of a fixed point of $F$ by looking at the set of points that are attracted to it as $F$ is iterated?)

Some thoughts:

  • For some $M$, there might well be maps $F$ that have no fixed points at all. If the Euler characteristic can be recovered from the fixed points of $F$, then such $M$ would have to have an Euler characteristic of zero. (Is that the case??) So the fixed points of $F$ are not very useful in such cases, but are there more general dynamical features of $F$ that relate to the topology of $M$?

  • Some $M$ might admit perfectly continuous, even smooth, $F$ with chaotic dynamics.

  • If $F$ has a unique fixed point $x\_0$ and for every $x\in M$, $F^n(x)\to x\_0$, then $M$ is contractible (recalling our assumption that $F$ is homotopic to the identity).

  • Can we get better results by considering an even more restrictive class of transformations? Of course, I don't want to go as far as to say that $F$ belongs to some group of gradient flow diffeomorphisms on a manifold, but maybe we can try to relax that by supposing there exists $f:M\to \mathbb{R}$ such that $f\circ F \geq f$. (That condition makes sense even if $M$ is not a manifold.)

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I asked a somewhat similar question here: mathoverflow.net/questions/13291/… –  Qiaochu Yuan Feb 16 '10 at 17:43
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3 Answers 3

Have you looked at Robin Forman's work on Discrete Morse Theory?

His Rice page has lots of beautiful work as he extends Morse Theory and all the classical results for simplicial complexes (CW too). He also defines a notion of a discrete vector field and a flow so that you can re-run a lot of the Morse-Smale type dynamics and homology arguments that you can use in the smooth category. I know that you didn't want to look at Morse-type functions, but maybe this will be useful to you

Also, your "homotopic to the identity condition" reminds me a lot of the exact symplectic diffeomorphisms that come up in Symplectic Geometry and Floer theory more specifically. If you have some time-varying diffeomorphism you could ask for fixed points of the time-1 map (your F I suppose). If your F is some gradient flow then you could look at the PDE you get from this and trying doing Morse Homology ala Floer, but maybe you are trying to avoid this well-trodden area.

If you are trying to extract topology from invariant sets of such a map, Conley Index Theory also seems like a good place to have a look. If this is all old hat to you, then I apologize for the lack of fresh ideas.

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It's not old hat at all. Conley index theory seems like it might be relevant, so I'll try to learn about it. –  Darsh Ranjan Nov 9 '09 at 5:00
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Morse theory can be generalised in many directions. The generalisation of which you speak would cover Morse-Novikov homology; this is where there is a function which is locally Morse, but it might not integrate to a full Morse function (there may be loops of 'flow').

Another generalisation it could cover is Morse-Bott homology and this will cover the situation where you don't have isolated fixed points, but the fixed points form nice enough shapes.

Both of these generalisations may be studied by taking cellular decompositions (the 'flow' out of a fixed point), although in the Novikov case things are more complicated as you have to take the limit of relative decompositions, but I wont try to make that precise here. These cellular decompositions give a filtration (by index) of the cell complexes of the underlying space. All of the Morse conditions are carefully chosen to make the associated spectral sequences nice (converging to the homology in the nicest case!). So an approach to generalisation would normally involve relaxing these conditions and seeing what happens to the spectral sequence.

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Regarding 1: The Euler characteristic of a compact manifold M is the self-intersection of the diagonal $\Delta\subset M\times M$. If F is homotopic to the identity, then the diagonal (the graph of the identity) can be deformed to the graph of F. If F has no fixed points, this graph is disjoint from the diagonal; so the self-intersection, and thus the Euler characteristic, is 0.

Note that for any M, the product $M\times\mathbb{R}$ admits such an F, so the condition that M be compact is necessary.

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