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



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> 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 
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> 
> * 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*).
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> 


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



> 
> Has anyone made any progress on the strong Arnold conjecture?
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> 
> 


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](https://mathoverflow.net/questions/26203/floer-homology-and-status-of-the-arnold-conjecture)).


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"](http://arxiv.org/abs/dg-ga/9708008), 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"](http://arxiv.org/abs/dg-ga/9708007), 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 $\omega|\_{\pi\_2(M)} = 0$.



What is known about the strong Arnold conjecture?
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.


