diffeomorphism groups $$\begin{split}
{{\mathrm{Diff}}}_0(M) &:= 
\left\{{{\varphi}}\in{{\mathrm{Diff}}}(M)\,|\,
{{\varphi}}\mbox{ is isotopic to the identity}\right\}, \\
{{\mathrm{Diff}}}_0(M,{{\omega}}) &:= 
\left\{{{\varphi}}\in{{\mathrm{Diff}}}_0(M)\,|\,
{{\varphi}}^*{{\omega}}={{\omega}}\right\}.
\end{split}$$ Thus ${{\mathrm{Diff}}}_0(M,{{\omega}})$ denotes the group of symplectomorphisms of $M$ that are [*smoothly*]{} isotopic to the identity. Care must be taken to distinguish it from the subgroup of all symplectomorphisms of $M$ that are [*symplectically*]{} isotopic to the identity (i.e. the identity component of ${{\mathrm{Diff}}}(M,{{\omega}})$). Denote by $${{\mathcal S}}_{{\omega}}:= \bigl\{\rho\in{{\mathcal S}}_a\,|\,
\rho\stackrel{(\mbox{\tiny{a}})}{\sim}{{\omega}}\bigr\}$$ the connected component of ${{\omega}}$ in ${{\mathcal S}}_a$. (Here $\stackrel{(\mbox{\tiny{a}})}{\sim}$ denotes the equivalence relation (a) in \[para:equiv\].) By Moser isotopy the group ${{\mathrm{Diff}}}_0(M)$ acts transitively on ${{\mathcal S}}_{{\omega}}$. Hence the map $
{{\mathrm{Diff}}}_0(M)\to{{\mathcal S}}_{{\omega}}:{{\varphi}}\mapsto{{\varphi}}^*{{\omega}}$ induces a homeomorphism $${{\mathrm{Diff}}}_0(M)/{{\mathrm{Diff}}}_0(M,{{\omega}})\cong {{\mathcal S}}_{{\omega}}.$$ Thus the topology of ${{\mathcal S}}_{{\omega}}$ is closely related to the topology of the symplectomorphism group. For example, if there exists a symplectomorphism of $(M,{{\omega}})$ that is smoothly, but not symplectically, isotopic to the identity then ${{\mathrm{Diff}}}_0(M,{{\omega}})$ is not connected and hence ${{\mathcal S}}_{{\omega}}$ has a nontrivial fundamental group (see Example \[ex:seidel\]).

\[para:SW\] Many results about symplectic structures in dimension four rely on Taubes–Seiberg–Witten theory. Here is some relevant background. Let $M$ be a closed oriented smooth four-manifold.

[**1.**]{} Denote the set of equivalence classes of spin$^c$ structures on $M$ by 