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

Abstract:

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]).