is the problem of understanding these equivalence relations on the space of symplectic forms. Here are some questions that may serve as a guideline. Fix a deRham homology class $$a\in H^2(M;{{\mathbb R}})$$ and denote the space of symplectic forms on $M$ representing the class $a$ by $${{\mathcal S}}_a := \left\{\rho\in{{\Omega}}^2(M)\,|\,
\rho^n\ne 0,\,d\rho=0,\,[\rho]=a\right\}.$$ This is an open set in the Fréchet space of all closed $2$-forms on $M$ representing the class $a$. The following three questions correspond to the equivalence relations (a), (A), (B). In all three questions assume that ${{\mathcal S}}_a$ is nonempty (respectively that $M$ admits a symplectic form).

[**Question 1.**]{} [*Is ${{\mathcal S}}_a$ connected?*]{}

[**Question 2.**]{} [*Are any two symplectic forms in ${{\mathcal S}}_a$ diffeomorphic?*]{}

[**Question 3.**]{} [*Are any two symplectic forms on $M$ deformation equivalent?*]{}

Section \[sec:EX\] discusses what is known about these questions in some examples. Section \[sec:D\] includes a brief discussion of the Donaldson geometric flow approach to Question 1 in dimension four (see [@D4]).

\[para:ac\]Denote the set of almost complex structures on $M$ that are compatible with a nondegenerate $2$-form $\rho$ by $${{\mathcal J}}(M,\rho) := \left\{J:TM\to TM\,|\,J^2=-{{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}},\,
\rho(\cdot,J\cdot)\mbox{ is a Riemannian metric}\right\}.$$ 