Relative family Gromov-Witten invariants and symplectomorphisms

We study the symplectomorphism groups $G_{\lambda}=Symp_0(M,\omega_{\lambda})$ of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms $\omega_{\lambda}$ with variable cohomology class. We show that the existence of nontrivial elements in $\pi_*({\cal A},{\cal A}')$, where $({\cal A},{\cal A}')$ is a suitable pair of spaces of almost complex structures, implies the exiarxiv.org stence of families of nontrivial elements in $\pi_{*-i}G_{\lambda}$, for $i=1$ or 2. Suitable parametric Gromov Witten invariants detect nontrivial elements in $\pi_*({\cal A},{\cal A}')$. By looking at certain resolutions of quotient singularities we investigate the situation $(M,\omega_{\lambda})= (S^2 \times S^2 \times X,\sigma_F \oplus \lambda \sigma_B \oplus \omega_{st})$, with $(X,\omega_{st})$ an arbitrary symplectic manifold. We find families of nontrivial elements in $\pi_k(G_{\lambda}^X)$, for countably many $k$ and different values of $\lambda$. In particular we show that the fragile elements $w_{\ell}$ found by Abreu-McDuff in $\pi_{4 \ell}(G_{\ell+1}^{pt})$ do not disappear when we consider them in $S^2 \times S^2 \times X$.