${{\mathcal S}}_a$ is connected if and only if every diffeomorphism of ${{{{\mathbb C}}\mathrm{P}}}^2$ that acts as the identity on homology is isotopic to the identity*]{}.

\[ex:S2S2\]The discussion of Example \[ex:CP2\] carries over to $M:= S^2\times S^2$ as follows. First, every class $a\in H^2(M;{{\mathbb R}})$ with $a^2\ne 0$ is represented by a symplectic form. Second, theorems of Gromov [@GROMOV1] and McDuff [@MCDUFF1] assert that every symplectic form for which the homology classes $A:=[S^2\times\{\mathrm{pt}\}]$ and $B:=[\{\mathrm{pt}\}\times S^2]$ (with either orientation) are represented by symplectically embedded spheres is diffeomorphic to a standard form (see [@MS2 Theorem 9.4.7]). Third, Taubes’ theorem [@TAUBES5] establishes the existence of the required symplectic spheres. Fourth, a theorem of Gromov [@GROMOV1] asserts that the group of symplectomorphisms that preserve $A,B$ retracts onto the isometry group ${{\mathrm{SO}}}(3)\times{{\mathrm{SO}}}(3)$ when $A,B$ have the same area (see [@MS2 Theorem 9.5.1]). Fifth, a theorem of Abreu and McDuff [@A; @AM] asserts that the symplectomorphism group is connected when $A,B$ have different areas. Thus ${{\mathcal S}}_a\cong{{\mathrm{Diff}}}(M,a)/{{\mathrm{Diff}}}(M,{{\omega}})$ and ${{\mathrm{Diff}}}(M,{{\omega}})$ has two connected components when ${\langle a, A\rangle}={\langle a, B\rangle}$, and is connected otherwise.

[**Conclusion.**]{} [*Questions 2 and 3 have positive answers. Moreover, ${{\mathcal S}}_a$ is connected if and only if every diffeomorphism of ${S^2\times S^2}$ that 