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 ?
