$M$ admits an orientation reversing diffeomorphism that interchanges $L$ and $E$, an orientation preserving diffeomorphism that preserves $L$ and reverses $E$, and an orientation preserving diffeomorphism that reverses $L$ and preserves $E$. By Taubes’ theorem [@TAUBES3; @TAUBES4; @TAUBES5] $L$ or $-L$, and $E$ or $-E$, is represented by a symplectically embedded sphere for every symplectic form on $M$. Hence a class $a\in H^2(M;{{\mathbb R}})$ is represented by a symplectic form if and only if $$\label{eq:LE}
a^2\ne0,\qquad{\langle a, L\rangle}\ne 0,\qquad {\langle a, E\rangle}\ne0.$$ By the theorems of McDuff [@MCDUFF1; @MCDUFF4] and Taubes [@TAUBES5] any two symplectic forms representing the same cohomology class are diffeomorphic, as in Example \[ex:ruled\]. Thus ${{\mathcal S}}_a\cong{{\mathrm{Diff}}}(M,a)/{{\mathrm{Diff}}}(M,{{\omega}})$. By  every diffeomorphism preserving $a$ acts as the identity on homology. Moreover, a theorem of Abreu and McDuff [@A; @AM] asserts that ${{\mathrm{Diff}}}(M,{{\omega}})$ is connected.

[**Conclusion.**]{} [*Questions 2 and 3 have positive answers. Moreover, ${{\mathcal S}}_a$ is connected if and only if every diffeomorphism of $M$ that acts as the identity on homology is smoothly isotopic to the identity.*]{}

\[ex:CP22\]Consider the oriented four-manifold $
M := {{{{\mathbb C}}\mathrm{P}}}^2\#\overline{{{{{\mathbb C}}\mathrm{P}}}}^2\#\overline{{{{{\mathbb C}}\mathrm{P}}}}^2.
$ Let $L\in H_2(M;{{\mathbb Z}})$ be the homology class of the line with self-intersection number $L\cdot L=1$ and $E_1,E_2\in H_2(M;{{\mathbb 