On the mapping class groups of simply-connected smooth 4-manifolds

The mapping class group $M(X)$ of a smooth manifold $X$ is the group of smooth isotopy classes of orientation preserving diffeomorphisms of $X$. We prove a number of results about the mapping class groups of compact, simply-connected, smooth $4$-manifolds. We prove that $M(X)$ is non-finitely generated for X = 2n \mathbb{CP}^2 # 10n \overline{\mathbb{CP}^2}, where $n\ge 3$ is odd. Let $\mathrm{\Gamma }(X)$ denote the group of automorphisms of the intersection lattice of $X$ that can be realised by diffeomorphisms. Then $M(X)$ is an extension of $\mathrm{\Gamma }(X)$ by $T(X)$, the Torelli group of isotopy classes of diffeomorphisms that act trivially in cohomology. We prove that this extension is split for connected sums of ${\mathbb{C}\mathbb{P}}^2$, but is not split for 2\mathbb{CP}^2 # n \overline{\mathbb{CP}^2}, where $n\ge 11$. We prove that the Nielsen realisation problem fails for certain finite subgroups of M( p \mathbb{CP}^2 # q \overline{\mathbb{CP}^2} ) whenever $p+q\ge 4$. Lastly we study the extension $M_1(X)\to M(X)$, where $M_1(X)$ is the group of isotopy classes of diffeomorphisms of $X$ which fix a neighbourhood of a point. When $X=K3$ or K3 # (S^2 \times S^2) we prove that $M_1(X)\to M(X)$ is a non-trivial extension of $M(X)$ by ${\mathbb{Z}}_2$. Moreover, we completely determine the extension class of $M_1(K3)\to M(K3)$.