# Could this be a Hamiltonian symplectomorphism on a symplectic toric manifold

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 ?

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.