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

We study the symplectic mapping class groups of (\mathbb {C} P^2 \# 5{\overline {\mathbb {C} P^2}},\omega ). Our main innovation is to avoid the detailed analysis of the topology of generic almost complex structures \mathcal {J}_0 as in most of earlier literature. Instead, we use a combination of the technique of ball-swapping (defined by Wu [Math. Ann. 359 (2014), pp. 153–168]) and the study of a semi-toric model to understand a “connecting map”, whose cokernel is the symplectic mapping class group. Using this approach, we completely determine the Torelli symplectic mapping class group (Torelli SMCG) for all symplectic forms \omega. Let N_{\omega } be the number of (-2)-symplectic spherical homology classes. Torelli SMCG is trivial if N_{\omega }>8; it is \pi _0(\text {Diff}^+(S^2,5)) if N_{\omega }=0 (by Seidel [ Lecture notes in Math. , Springer, Berlin, 2008] and Evans [J. Symplectic Geom. 9 (2011), pp. 45–82]); and it is \pi _0(\text {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 in terms of 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 (open problem 16 in McDuff-Salamon’s book 3rd version [ Oxford graduate texts in mathematics , Oxford University Press, Oxford, 2017]).