This space is nonempty and contractible. In fact the space of nondegenerate $2$-forms on $M$ is homotopy equivalent to the space of almost complex structures on $M$. (The homotopy equivalence is a smooth map $\rho\mapsto J_\rho$, depending on the choice of a background metric, which assigns to every nondegenerate $2$-form $\rho$ an almost complex structure $J_\rho\in{{\mathcal J}}(M,\rho)$; see [@GROMOV2; @MS1].) Thus every nondegenerate $2$-form $\rho$ on $M$ determines a cohomology class $$c_1(\rho):=c_1(TM,J)\in H^2(M;{{\mathbb Z}}),\qquad J\in{{\mathcal J}}(M,\rho),$$ called the [**first Chern class of $\rho$**]{}. Consider the set $${{\mathcal C}}_a := \left\{c_1(\rho)\,|\,\rho\in{{\mathcal S}}_a\right\}
\subset H^2(M;{{\mathbb Z}}).$$ If this set contains more than one element, the answer to Question 1 is negative. Moreover, define $$\begin{split}
{{\mathcal C}}_{{\mathrm{symp}}}&:= \left\{c_1(\rho)\,|\,\rho
\mbox{ is a symplectic form on }M\right\}
= \bigcup_a{{\mathcal C}}_a, \\
{{\mathcal C}}&:= \left\{c_1(\rho)\,|\,\rho
\mbox{ is a nondegenerate $2$-form on }M\right\}.
\end{split}$$

[**Further questions:**]{} What is the set ${{\mathcal C}}_a$? Is ${{\mathcal C}}_{{\mathrm{symp}}}={{\mathcal C}}$? Does the diffeomorphism group act transitively on ${{\mathcal C}}_{{\mathrm{symp}}}$?

\[para:homotopyJ\]Let $M$ be a compact connected oriented smooth four-manifold without boundary. Denote by $\chi$ the Euler characteristic and by $\sigma$ the signature of $M$. By a theorem of Wu an integral cohomology class $c\in H^2(M;{{\mathbb Z}})$ is the first Chern class of 