Proof of Arnold Conjecture for monotone symplectic manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T16:13:22Z http://mathoverflow.net/feeds/question/120210 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120210/proof-of-arnold-conjecture-for-monotone-symplectic-manifolds Proof of Arnold Conjecture for monotone symplectic manifolds nicolas 2013-01-29T13:59:04Z 2013-01-30T00:02:55Z <p>I have a question about the proof of the Arnold conjecture for monotone symplectic manifolds as it is explained in <a href="http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf" rel="nofollow">http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf</a>: 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</p> http://mathoverflow.net/questions/120210/proof-of-arnold-conjecture-for-monotone-symplectic-manifolds/120271#120271 Answer by Michael Hutchings for Proof of Arnold Conjecture for monotone symplectic manifolds Michael Hutchings 2013-01-29T23:54:41Z 2013-01-29T23:54:41Z <p>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.</p> http://mathoverflow.net/questions/120210/proof-of-arnold-conjecture-for-monotone-symplectic-manifolds/120272#120272 Answer by Henry T. Horton for Proof of Arnold Conjecture for monotone symplectic manifolds Henry T. Horton 2013-01-30T00:02:55Z 2013-01-30T00:02:55Z <p>Note that for the Floer boundary map to be well-defined, we require a Floer regular <em>pair</em> $(H_t, J_t)$ (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 <a href="http://www.math.wisc.edu/~oh/all.pdf" rel="nofollow">Oh's notes</a> (these notes focus on the semipositive case, but monotone symplectic manifolds are semipositive).</p>