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 $\chi(X)\leq 7$ in terms of the number $N$ of -2 symplectic sphere classes. 
Jun Li, Tian-Jun Li
math.SG math.GT