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.