The main question is motivated from the answer of this question https://math.stackexchange.com/questions/2481200/finite-groups-gs-which-acts-freely-on-s2-times-s2
Is it true that every self diffeomorphism of $S^2\times S^2$ either fiber preserving or exchange the two spheres (upto isotopy)?
It is certainly true that the homology classes of $S^2 \times \ pt$ and $\ pt \times S^2$ must be preserved or switched since you can easily check they are the only homology classes with self-intersection 0. And also the complement of an $\epsilon$-nebighbourhood of them is $B^4$. So from here can we conclude anything?
Can anyone share an idea of how to prove or disprove it?