I have a question about the proof of the Arnold conjecture for monotone symplectic manifolds as it is explained in http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf: Namely the author on page 32 says that the Arnold conjecture would immediately follow from a theorem 3.7 on the same page. But as far as I see, the theorem requires a further assumption on the Hamiltonian, namely that it has to be contained in some dense set $H_{\mbox{reg}}$ introduced on page 13. My question therefore is: How can we deduce the Arnold conjecture for a general Hamiltonian and not just for a generic one from this construction of Floer homology? Or is the intention of the above paper to just prove the Arnold Conjecture for such a set of Hamiltonians and not necessarily for every Hamiltonian? Also I have found another point in the proof, which I am not sure about: Namely the author on page 23 has to introduce the Conley-Zehnder index and thus to assume that the $1$-periodic orbit is contractible to assign an index to him. I would then like to know: Does this really mean, we have to make a further restriction on the Hamiltonian and have to assume that every $1$-periodic orbit is contractible? Every help with one of these questions will be appreciated. It somehow looks to me as if ther
2 Answers
Note that for the Floer boundary map to be well-defined, we require a Floer regular pair $(H_t, J_t)$ (we want a family of $\omega$-compatible $J_t$ so we have a nice metric with which to take gradients for Floer's equation). So if we're given an $H_t$ to define the Floer homology groups with, we can hope that maybe there is a family $J_t$ of almost complex structures such that $(H_t, J_t)$ is Floer regular. This turns out to be possible -- see for example Exercise 19.22 in Oh's notes (these notes focus on the semipositive case, but monotone symplectic manifolds are semipositive).
To prove the Arnold conjecture for monotone symplectic manifolds, it is enough to assume that the Hamiltonian satisfies the genericity condition that all (contractible) 1-periodic solutions are regular. You can then define Floer homology for a generic J. The periodic orbits produced by the proof are all contractible.
