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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?

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    $\begingroup$ I am sure your question is open. Very little is known about mapping class groups in 4 dimensions. Some things to look at are work of Ruberman and Konno. These use gauge theory techniques for manifolds with $b_2 >>0$ that are not likely applicable to $S^2 \times S^2$. More hands-on (and for manifolds with relatively few handles) is recent work of Gabai, Watanabe, and Budney-Gabai, which says something about mapping class groups of $D^2 \times S^2, S^4, S^1 \times S^3$ in that order. I've missed some things here, so chase references to see other work. $\endgroup$
    – mme
    Commented Jan 18, 2020 at 19:50
  • $\begingroup$ Isnt the mapping class group of $𝑆^2×𝑆^2$ known already? $\endgroup$
    – wonderich
    Commented Jan 20, 2020 at 2:25
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    $\begingroup$ No, the mapping class group of no closed 4-manifold is known. $\endgroup$
    – archipelago
    Commented Jan 21, 2020 at 8:07

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