Suppose we have a sympletic toric manifold $(M,\omega)$ of dimension $4$ and let $\triangle$ be its corresponding Delzant polytope. Suppose that this polytope is "nice" enough so that we are able to defined a map, using action-angle coordinates $\Psi:\triangle \times \mathbb{T}^2\rightarrow \triangle \times \mathbb{T}^2$
\begin{equation}
\Psi(x\_1,x\_2,\theta\_2,\theta\_2)=(-x\_1,-x\_2,-\theta\_1,-\theta\_2).
\end{equation}
Examples of Delzant polytopes where we could define this map would be a square, octagon, etc.… We would just need to have enough symmetries. We are assuming that the polygon is centered at the origin.


Now this map will be a symplectomorphism, and what I have been wondering is if this could be a Hamiltonian symplectomorphism ? I tried some approaches to prove this but I got nowhere. Basically my idea was to use Banyaga's theorem, since we know that $H\_1(M)=\{0\}$, we would just need to prove that $\Psi$ is in the connected component of the identity in $\operatorname{Symp}(M)$, however I was not able to construct such a family of maps.


Then, I remebered the Arnol'd conjecture, and when we can talk about Floer homology, this basically gives us a lower bound on the number of fixed points of an Hamiltonian symplectomorphism in terms of Betti numbers of $M$. So here $\Psi$ has $4$ fixed points, and so if the topology of $M$ is complicated enough we could come into some trouble, assuming that we were in a position to talk about Floer homology.


So right now I'm inclining to the fact that generally this won't be a Hamiltonian symplectomorphism because I'm not sure how $H\_2(M)$ behaves, but I would like to hear some input on this. What do you think ?



Could this be a Hamiltonian symplectomorphism on a symplectic toric manifold
Sometimes it is, sometimes it isn't.


The spheres living over the edges generate the second homology, so you can read off the action on $H\_2$ from that. For $S^2\times S^2$ (square) the action on homology is trivial (because opposite edges are homologous) and the symplectomorphism is indeed Hamiltonian. For a hexagon (3-point blow up of $P^2$) the action on homology is nontrivial.


