not homotopic; they have different Hopf invariants. The argument breaks down for symplectic forms where the $S^2$ factors have different areas, because in that case the relevant moduli spaces are no longer compact.

[**Conclusion.**]{} [*For the six-manifold $M={{\mathbb T}}^2\times S^2\times S^2$ question 1 has a negative answer and questions 2 and 3 are open problems.*]{}

\[ex:CP2\]A theorem of Taubes [@TAUBES3; @TAUBES4; @TAUBES5] asserts that any two symplectic forms on $M={{{{\mathbb C}}\mathrm{P}}}^2$ with the same volume are diffeomorphic. The proof uses a theorem of Gromov [@GROMOV1] which requires, as an additional hypothesis, the existence of a symplectically embedded two-sphere (see also [@MS2 Theorem 9.4.1]). The existence of the required symplectic sphere follows from Taubes’ [*“Seiberg–Witten equals Gromov”*]{} theorem [@TAUBES5]. Combining Taubes’ theorem with Moser isotopy one finds that, for every cohomology class $a\in H^2(M;{{\mathbb R}})$ with $a^2\ne0$, there is a bijection $${{\mathcal S}}_a \cong {{\mathrm{Diff}}}(M,a)/{{\mathrm{Diff}}}(M,{{\omega}}),$$ where ${{\omega}}$ is the Fubini–Study form representing the class $a$ and $$\begin{split}
{{\mathrm{Diff}}}(M,a) 
&:= \left\{{{\varphi}}\in{{\mathrm{Diff}}}(M)\,|\,{{\varphi}}^*a=a\right\} \\
&\,= \left\{{{\varphi}}\in{{\mathrm{Diff}}}(M)\,|\,
{{\varphi}}_*={{\mathrm{id}}}:H_*({{{{\mathbb C}}\mathrm{P}}}^2;{{\mathbb Z}})\to H_*({{{{\mathbb C}}\mathrm{P}}}^2;{{\mathbb Z}})\right\}.
\end{split}$$ Another theorem of Gromov [@GROMOV1] asserts that ${{\mathrm{Diff}}}(M,{{\omega}})$ retracts onto the isometry group ${{\mathrm{PU}}}(3)$ of ${{{{\mathbb C}}\mathrm{P}}}^2$ (see also [@MS2 Theorem 9.5.3]).

[**Conclusion.**]{} [*Questions 2 and 3 have positive answers. Moreover, 