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

For $(\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}},\omega)$, let $N_{\omega}$ be the number of $(-2)$-symplectic spherical homology classes.We completely determine the Torelli symplectic mapping class group (Torelli SMCG): the Torelli SMCG is trivial if $N_{\omega}>8$; it is $\pi_0(Diff^+(S^2,5))$ if $N_{\omega}=0$ (by Paul Seidel and Jonathan Evans); it is $\pi_0(Diff^+(S^2,4))$ in the remaining case. Further, we completely determine the rank of $\pi_1(Symp(\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}}, \omega)$ for any given symplectic form. Our results can be uniformly presented regarding Dynkin diagrams of type $\mathbb{A}$ and type $\mathbb{D}$ Lie algebras. We also provide a solution to the smooth isotopy problem of rational $4$-manifolds.

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