6
$\begingroup$

My question concerns Conley theory for topological flows and its connection to classical Morse theory on compact manifolds.

Specifically, I have in mind Conley and Zehnder's seminal paper Morse-type Index Theory for Flows and Periodic Solutions for Hamiltonian Equations (sections 3.3 and 3.6 / pages 60 and 74–76 in printed numbers).

In section 3.6, they want to prove the classical Morse inequalities by applying Conley theory to the gradient flow of a Morse function $f$. Mostly everything is sound about that except for one thing. Namely, in section 3.3, just after equation (3.10), they have made the assumption that certain (Cech) cohomology modules $H^i(X,Y)$ are of finite rank. The $X$ and $Y$ here are sets $N_0 \subset N_1 \subset \cdots N_n$ of a "filtration" of a so-called Morse decomposition of $M$ which is specific to the flow. What bugs me is that this assumption of finite rank is nowhere proved. I know how to prove it in one sufficient special case, but only with the results of classical Morse theory! Meaning, in this manner their treatment does not amount to a proof of the classical Morse inequalities that is independent from the classical treatment (CW complexes etc.) as in Milnor's book.

To elaborate on this, here is how I prove it with the classical results in the sufficient special case: For the gradient flow, the Morse decomposition is the collection of critical points $x_1,\dots,x_n$ of the Morse function $f$, ordered decreasingly by their value under $f$. If no critical value has two different critical points in its pre-image, one can take for $N_1, \dots, N_n$ the super-level sets to the critical values (minus $\varepsilon$), and $N_0=\emptyset$. Then it is easy to see how the finiteness of these cohomology modules follows from the theorems of classical Morse theory. (Perhaps there is a minor mistake in my description here, but this is the idea.)

My question is now: Is there another, independent way to prove the finiteness of rank?

Edit: In section 3.6, the finiteness is proved generally for $H^i(N_j, N_{j-1})$ by using suitable local coordinates. What remains to show is finiteness for $H^i(N_j, N_0)$, with $(N_j, N_0)$ being an index pair not for an idividual critical point but for the set of all points $x_0, \dots, x_j$ together with the flow lines connecting them. //

I mean, we are dealing with a manifold and with a nice flow, so perhaps easier considerations suffice. On the other hand, classical Morse theory is, in a way, as straightforward as it gets.

By the way, the issue is the same in Conley's lecture notes Isolated Invariant Sets and the Morse Index (bottom of page 78 and 79 in printed numbers). Here he also just posits finiteness without proof and says the Morse inequalities follow.

Thanks for reading, I'm very grateful for any help and suggestions you have.

Edit 2 (Solutions): There are at least two solutions. The first was given by Thomas Rot in his post below. It proves the finiteness of rank for arbitrary isolated invariant sets of flows to a smooth vector field on a manifold, if the flow is defined for all time. When talking about filtrations, this is applicable whenever $N_0\subset N_1\subset\cdots N_n$ is a filtration which belongs to a Morse decomposition, as in Theorem 3.1 of 1. While the underlying theorem about the existence of isolating blocks takes some effort to prove (here's a link the paper Thomas references), the reward is a result that is extremely general.

The second solution (by me) to my specific problem is purely algebraic and makes use of the fact that, in my case, $(N_j,N_{j-1})$ is an index pair for the individual, non-degenerate critical point $x_j$. Here finiteness of rank for $H^i(N_j,N_{j-1})$ can be proved readily using local coordinates, by explicit construction of an isolating block which is homotopy equivalent to a pointed sphere (section 3.6 in 1). What is left to show is finiteness for $(N_j,N_0)$. This can be proved by induction over $j$, using the long exact sequence of cohomology of all triples $(A,B,C)=(N_j,N_{j-1},N_0)$: \begin{equation} \overset{\delta^{i-1}}{\to}H^i(N_j,N_{j-1})\to H^i(N_j,N_0) \to H^i(N_{j-1},N_0) \overset{\delta^{i}}{\to} \end{equation} In the base case $j=1$, all of them have finite rank for every $i$. For the induction step, assume that $H^i(N_{j-1},N_0)$ has finite rank for every $i$. The left module of the LES has finite rank by the above remarks, and the right, by the induction hypothesis. This implies that the center one has finite rank as well, completing the induction step. //

$\endgroup$
  • 2
    $\begingroup$ The index pairs can be chosen nicely: In a finite dimensional manifold they can be chosen to be manifolds with a notion of corners. You do this by proving that any Morse decomposition admits smooth Lyapunov functions and looking at sublevel/superlevel sets $\endgroup$ – Thomas Rot Aug 17 '18 at 12:05
  • $\begingroup$ @Thomas Rot: That sounds very promising. In fact, I neglected to mention one thing, which is now under "Edit". Possibly the ideas used in there are similar to the ones you mention. Does your method also prove finiteness for the index pairs $(N_j, N_0)$, which are index pairs for more complicated invariant sets than just rest points? (I try to understand this myself, but so far unsuccessfully, partly because I cannot find good material.) $\endgroup$ – David S. Aug 17 '18 at 15:34
  • $\begingroup$ Yes I think so. "isolating blocks" is maybe a keyword you can google. $\endgroup$ – Thomas Rot Aug 19 '18 at 13:55
1
$\begingroup$

Any isolated invariant set of a $C^1$ flow on a manifold admits an isolating block. This is proven in

MR0279830 (43 #5551) Reviewed Conley, C.; Easton, R. Isolated invariant sets and isolating blocks. Trans. Amer. Math. Soc. 158 1971 35–61. 57.48

I have not read the paper.

An isolating block is a compact manifold $N$ with boundary $\partial N$ with some properties. $N$ should be an isolating neighborhood of the isolated invariant set. On the boundary there are two other submanifolds $\partial N^\pm$ where the flow points inwards or outwards. The $N^\pm$ themselves have boundaries which coincide by assumption. Moreover their union should be the whole of $\partial N$. Compact manifolds with boundary have finite rank and from the long exact sequence of the pair one sees that the cohomology groups $H^*(N,\partial N^-)$ have finite rank. But this is the cohomological Conley index.

$\endgroup$
  • $\begingroup$ This certainly looks like a way to prove my issue. Thank you, I appreciate this a lot. Please also have a look at my upcoming second edit. An hour ago I've found a way to prove it quite easily with the long exact sequences of all triples $(N_j, N_{j-1}, N_0)$, starting at $j=1$, and using that all $(N_j, N_{j-1})$ are known to have finite rank. $\endgroup$ – David S. Aug 26 '18 at 0:19
  • $\begingroup$ Very nice! I thought of that as well, but this seemed to be more general. $\endgroup$ – Thomas Rot Aug 26 '18 at 18:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.