Here are the two versions of Arnold's conjecture on Hamiltonian orbits:

Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate Hamiltonian. Then the number of $1$-periodic orbits of the vector field $X_H$ defined by $\omega(X_H, \cdot) = dH$ is bounded below by

  • the sum of the rational Betti numbers of $M$ (weak version)

  • the minimum number of critical points of a Morse function on $M$ (strong version).

The weak version has been settled using Floer homology. My question is:

Has anyone made any progress on the strong Arnold conjecture?

I have asked a couple of experts in person, but they didn't know of anything (see also Tim Perutz's answer to this question).

Also, I wonder if anyone can tell me the right intuition for the strong conjecture. The only idea I have for why the strong conjecture should be true is that when $H$ is time-independent, each critical point of $H$ is a $1$-periodic orbit of $X_H$.

Edit about the version of the conjecture without assuming $H$ nondegenerate: Thanks to Thomas's comment below I looked up Arnold's original statement of the conjecture and realized that it is different than the one I wrote above (which I have never seen in print). Call the conjecture above the nondegenerate Arnold conjecture; here is the original, "possibly degenerate" Arnold conjecture from Mathematical methods of classical mechanics:

Let $(M, \omega)$ be a closed symplectic manifold, and let $H: \mathbb{R}/\mathbb{Z} \times M \to \mathbb{R}$ be a Hamiltonian. Then the number of $1$-periodic orbits of $X_H$ is bounded below by the minimal number of critical points of a smooth function on $M$.

There is also a weak version (not from Arnold's book) where the bound is replaced by $1$ plus the cuplength of $M$.

Some progress has been made toward the strong version of this conjecture. In "On analytical applications of stable homotopy", Yuli Rudyak proves that if $(M,\omega)$ is a closed symplectic manifold such that $\omega|_{\pi_2(M)} = 0$ and $c_1|_{\pi_2(M)} = 0$ and the Lusternik-Schnirelmann category of $M$ is $\dim M$, then the strong version of the "possibly degenerate" Arnold conjecture holds. In "On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture", John Oprea and Rudyak eliminate the LS hypothesis by showing that $\text{LS}(M) = \dim M$ whenever $M$ is a closed symplectic manifold such that $\omega|_{\pi_2(M)} = 0$.

  • $\begingroup$ Hi Nate. The reference you give seems (to me) to prove the cup length conjecture in those case. Ie that the number of fixed points is greater than or equal to the cup length (or something stronger). Does it mention anything about Morse functions? Or does that somehow follow from that? $\endgroup$ May 25 '13 at 13:24
  • $\begingroup$ Hi Thomas! You are right, I was mixing up the versions with and without the "nondegenerate" hypothesis. I will edit my question. $\endgroup$ May 29 '13 at 5:17
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    $\begingroup$ A related but not directly relevant point you may already be aware of: for the Arnold conjecture on cotangent bundles (that the number of intersections between the zero-section and a Hamiltonian pushoff is bounded below by the minimal Morse number of the zero section) one can show using generating functions (Chaperon, Laudenbach-Sikorav, Eliashberg-Gromov) that a lower bound is given by the stable Morse number. This is not necessarily the Morse number, there are examples due to Damian, but it sometimes gives a better bound than the sum of Betti numbers. $\endgroup$ Jun 18 '13 at 19:13
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    $\begingroup$ @Jonny: Yes - absolutely relevant, and the stable Morse number is in fact the number of cells in a minimal cell structure of the suspension spectrum of the manifold. The reason these counter examples by Damian exists are because inherently Floer homology (at least in this example with cotangent bundles) is stable - i.e. it leaves room due to the possibility of having "negative dimensional cells" - that is negative Conley Zehnder index. In fact it is probably possible to construct similar counter examples in the case at hand ($M$ closed), so that the best possible bound is the stable Morseindex $\endgroup$ Jun 18 '13 at 21:43

Assume dim$(M) \geq 6$ and $M$ is simply connected.

By cancelling Morse critical points for a Morse function $f$ on $M$ one can get the number of critical points equal to the minimum number of generators needed in $C_* (M)$ to generate $H_* (M)$. I.e. the sum of the Betti numbers plus 2 for each torsion generator. This is also the number of cells in a minimal cell structure (CW-complex) for $M$.

This means that if one can define Floer homology with $\mathbb{Z}$ coefficients and the PSS isomorphism works with $\mathbb{Z}$ coefficients, then the strong Arnold conjecture is true.

I believe this is the case for e.g. monotone symplectic manifolds.


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