  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.
