Symplectic $(-2)$ spheres and the symplectomorphism group of small rational 4-manifolds, I

Let $(X,\omega )$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures ${\mathcal{J}}_{\omega }$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional Alexander duality. This decomposition provides new understandings of both the variation and stability of the symplectomorphism group $Symp(X,\omega )$ when deforming $\omega$. In particular, we compute the rank of ${\pi }_1(Symp(X,\omega ))$ with Euler number $\chi (X)\le 7$ in terms of the number $N$ of -2 symplectic sphere classes.