given spin$^c$ structure with first Chern class $c$. (This can be proved with a standard Pontryagin manifold type construction as in Milnor [@MILNOR §7].)

\[para:existence\]A natural question is under which conditions a given cohomology class $a\in H^2(M;{{\mathbb R}})$ can be represented by a symplectic form. This is the fundamental existence problem in symplectic topology. The obvious necessary conditions are the existence of an almost complex structure on $M$ and, in the case of a closed manifold, the condition $a^n\ne0$. In the open case a theorem of Gromov [@GROMOV0; @GROMOV2] asserts that the existence of an almost complex structure is also sufficient. In the closed case there are counterexamples in dimension four, based on Seiberg–Witten theory (see Taubes [@TAUBES2]). The simplest example is $M={{{{\mathbb C}}\mathrm{P}}}^2\#{{{{\mathbb C}}\mathrm{P}}}^2\#{{{{\mathbb C}}\mathrm{P}}}^2$ which does not admit a symplectic form because its Seiberg–Witten invariants vanish. In higher dimensions the existence problem is completely open.

\[para:diffeo\]It is also interesting to investigate the topology of the space ${{\mathcal S}}_a$ of symplectic forms in a given cohomology class beyond the question of connectedness. This is closely related to the topology of certain diffeomorphism groups. Let $(M,{{\omega}})$ be a compact symplectic manifold without boundary and denote $a:=[{{\omega}}]\in H^2(M;{{\mathbb R}})$. Consider the 