A function on a space of symplectic forms Let $M$ be a smooth manifold and $\text{symp}(M)$ be the set of all symplectic forms on $M$.
Let $\text{Diff}_{ 0}(M)$ be a connected component of difeomorphisms of $M$.
Then, is there an explicit function (or a invariant) on $\frac{\text{symp}(M)}{\text{Diff}_{\ 0}(M)}$ which is not determined by a second cohomology class represented by a symplectic form $\omega$?
I want to know the difference between $\frac{\text{symp}(M)}{\text{Diff}_{\ 0}(M)}$ and the space of classes represented by symplectic forms.
 A: This is, in my opinion, one of the most interesting question one may ask about symplectic topology. Not only is it related to the classification problem of symplectic structures on manifolds, but the results obtained in this direction show all the complexity and the subtlety of the subject.
Let $M^{2n}$ be a manifold. Write $S(M)$ for the space of symplectic forms on $M$ ; It is an open subset of the set $Z^2(M)$ of closed differential 2-forms on $M$ and, as such, an infinite dimensional manifold. Denote by $q: Z^2(M) \to H_{dR}^2(M) : \alpha \mapsto [\alpha]$ the map which associates to a closed 2-form its cohomology class. The set $\hat{S}(M) = q(S(M))$ is called the symplectic cone ; It is an open subset of $H^2_{dR}(M)$ invariant under the multiplication by $\mathbb{R}^*$. Given $\alpha \in \hat{S}(M)$, write $S_{\alpha} = q^{-1}(\alpha) \cap S(M)$.
We say that two symplectic forms $\omega_0$, $\omega_1$ in the same connected component of some $S_{\alpha}$, respectively of $S(M)$, are isotopic, respectively deformation equivalent.

What is $\hat{S}(M)$?

If $M$ is closed (i.e. compact without boundary), we know from Stoks theorem that $0 \not \in \hat{S}(M)$. More generally, when $M$ is closed, note that $H^{2n}_{dR}(M) \cong \mathbb{R}$, so that $\hat{S}(M)$ is a subset of $N(M) \subset H^2_{dR}(M)$, the set of non-degenerate $\alpha \in H^2{dR}(M)$, i.e. such that $\cup^n \alpha \neq 0$. In fact, $\hat{S}(M)$ is the subset of those $\alpha \in N(M)$ admitting a closed nowhere vanishing representative. In some cases, we know $\hat{S}(M)$ to be stricly smaller than $N(M)$. For instance, this is the case for the twisted bundle $S^2 \times_{\tau} S^2$ (that is, $\mathbb{C}P^2 \sharp \, \overline{\mathbb{C}P^2}$ or again the one point blow-up of $\mathbb{C}P^2$) ; see Theorem 6.11 and the discussion that precedes it in the second edition of McDuff and Salamon's Introduction to Symplectic Topology. 
For the case of $M = \mathbb{R}^{2n}$, we have $\hat{S}(M)= H^2_{dR}(M) = 0$. More generally, when $M$ is connected and open (i.e. not closed, so either non-compact or with a boundary), $H^{2n}_{dR}(M) = 0$ and so $N(M)$ is empty. Remarkably, Gromov showed in his thesis that $\hat{S}(M) = H^2_{dR}(M)$. A proof of this result is given in Theorem 7.34(i) of McDuff-Salamon's book.

Let $G = \mathrm{Diff}_0(M)$ be the group of diffeomorphisms of $M$ isotopic to the identity. It is well-known that for a smooth path $t \mapsto \phi_t \in G$ is such that $q \circ \phi_t^* = q$. As such, the action of $G$ by pullbacks leaves each (connected component of each) $S_{\alpha}$ invariant. Hence, for $\phi \in G$ and any symplectic form $\omega$, $\omega$ and $\phi^*\omega$ are isotopic.

What are the $S_{\alpha}/G$? What is $S(M)/G$ and how is it related to $\hat{S}(M)$?

When $M$ is closed, there is a converse : Moser's stability theorem implies that given a smooth path $t \mapsto \omega_t$ in (a connected component of) some $S_{\alpha}$, there exists a smooth isotopy $t \mapsto \phi_t \in G$ such that $\phi_t^*\omega_t = \omega_0$. Hence, for closed $M$, we can identify the set $\pi_0(S_{\alpha})$ of connected components of $S_{\alpha}$ with the quotient space $S_{\alpha}/G$, which are the 'fibers' of the quotient map $\bar{q} : S(M)/G \to H^2_{dR}(M)$.
When $M$ is open, Gromov showed (again in his thesis) that two cohomologous symplectic forms are isotopic and that two (non-cohomologous) symplectic forms are deformation equivalent ; see again Theeorem 7.34 (ii) and (i) respectively. However, Moser's stability theorem no longer apply, as the non-autonomous vector field obtained by Moser's trick is no longer necessarily complete. So, we cannot identify so easily $S_{\alpha}/G$  with $\pi_0(S_{\alpha})$ and they are in fact distinct for some manifolds. For instance, when $n > 1$, $\mathbb{R}^{2n}$ admits non-symplectomorphic (yet isotopic) symplectic structures ; These are the so-called exotic symplectic structure, whose existence was established by Gromov in his 1985 article and a proof of which is discussed in the 13th chapter of McDuff-Salamon's book.
In dimension 2 (i.e. $n=1$), each space $S_{\alpha}$ is connected and so is $S(M)$. Even in the open case, all the examples I know don't admit exotic structure, so I would conjecture that they don't exist at all. However, looking just at the case $M=\mathbb{R}^2$, we see that $G$ does not act transitively on $S(M)$. The analogue statements in higher dimension are known to be generally false. All sorts of complexity might happen, even in the closed case.
a) Given a diffeomorphism $\phi \in \mathrm{Diff}(M) \backslash G$ and a symplectic form $\omega \in S_{\alpha}$, the symplectic form $\phi^*\omega$ may be non-cohomologous to $\omega$ (that is, not in $S_{\alpha}$), cohomologous but non-isotopic to $\omega$ or cohomologous and isotopic to $\omega$. In the first two situations, the forms may or may not be deformation equivalent. In any case, $\phi^*\omega$ and $\omega$ are symplectomorphic, so they can only be distinguished in an 'extrinsic' manner, whatever it means.
b) Two cohomologous symplectic forms $\omega_0$, $\omega_1 \in S_{\alpha}$ which are not isotopic may be, as we just mentioned, symplectomorphic. They may also be non-symplectomorphic. In any case, they may or may not be deformation equivalent.
Examples along all those lines ares given in section 13.2 of McDuff-Salamon's book. Example 13.20 is particularly relevant for your question, if you expect to have some continuous $G$-invariant function on $S(M)$. Indeed, McDuff has exhibited a 6-manifold $M$ and two families of symplectic forms $\omega_t$ and $\sigma_t$ such that
i) For each $t$, $[\omega_t] = [\sigma_t]$,
ii) there exists $\phi \in \mathrm{Diff}(M) \backslash G$ satisfying $\phi^* \sigma_t = \omega_t$,
iii) For $t > 0$, $\omega_t$ and $\sigma_t$ are isotopic, but not for $t=0$.
This shows that no continuous $G$-invariant function on $S(M)$ can distinguish between $\omega_0$ and $\sigma_0$ and yet there are not $G$-related. Indeed, her proof relies on studying some parametrised (by $t$) families of moduli spaces of holomorphic curves which happen to undergo a 'discontinuous' change at $t=0$.
Wonderfully puzzling questions!
