Symplectomorphism Group of Rational 4-Manifolds

Abstract

We develop techniques for studying the symplectomorphism group of rational 4-manifolds. We study the space of tamed almost complex structures $\mJ_{\w}$ 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 $Symp(X,\w)$ when deforming $\w$. In particular, we compute the rank of $\pi_0(Symp(X,\w))$, with Euler number less than 8 in terms of the number $N$ of -2 symplectic sphere classes. In addition, using the above decomposition and coarse moduli of rational surfaces with a given symplectic form, we are able to determine $\pi_0(Symp(X,\w))$, the symplectic mapping class group (SMC). Our results can be uniformly presented regarding Dynkin diagrams of type $\aA$ and type $\DD$ Lie algebras. Applications of $\pi_0(Symp(X,\w))$ and $\pi_0(Symp(X,\w))$ includes the classification of symplectic spheres and Lagrangian spheres up to Hamiltionian isotopy and a possible approach to determine the full rational homotopy type $Symp(X,\w)$.