Flexible Surfaces in $\mathbb{C}{P}^2$ and $S^2\times S^2$

A surface $\mathrm{\Sigma }$ in a 4-manifold $M$ is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism $(M,\mathrm{\Sigma })\to (M,\mathrm{\Sigma })$. We construct flexible surfaces in $\mathbb{C}{P}^2$ and $S^2\times S^2$ within any prescribed non-characteristic homology class. Within characteristic homology classes there is a spin structure obstructing flexibility and we construct so-called spin-flexible representatives.