are related by (a) (i.e. a path of cohomologous symplectic forms), they are called [**homotopic**]{} if they are related by (b) (i.e. a path of symplectic forms), they are called [**diffeomorphic**]{} if they are related by (A) (i.e. a diffeomorphism), and they are called [**deformation equivalent**]{} if they are related by (B) (i.e. a diffeomorphism, followed by a path of symplectic forms).

For closed manifolds these equivalence relations are related as follows $$\begin{array}{ccccccccc}
(a) & \implies & (b) & \implies & (c) & \implies & (d) \\
\Downarrow & & \Downarrow & & \Downarrow & & \Downarrow \\
(A) & \implies & (B) & \implies & (C) & \implies & (D)
\end{array}.$$ In particular, by Moser isotopy, assertion (a) holds if and only if there exists a diffeomorphism ${{\varphi}}:M\to M$ isotopic to the identity such that ${{\varphi}}^*{{\omega}}_1={{\omega}}_0$. Thus (a) implies (A). For (A), (B), (C) it may also be interesting to restrict to diffeomorphisms that act as the identity on homology (see Karshon–Kessler–Pinsonnault [@KKP]). For (A) this would mean that equivalent symplectic forms represent the same cohomology class and for (B) that they have the same first Chern class and the same Gromov–Witten invariants.

\[para:questions\]The uniqueness problem in symplectic topology 