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

TL;DR

This paper investigates the structure and properties of the mapping class groups of simply-connected smooth 4-manifolds, revealing non-finite generation, extension structures, and the failure of Nielsen realisation in certain cases.

Contribution

It provides new results on the algebraic structure of mapping class groups, including non-finite generation, extension splitting properties, and specific extension classes for key 4-manifolds.

Findings

Mapping class group is non-finitely generated for certain 4-manifolds.

Extension of the mapping class group by the Torelli group can be split or not, depending on the manifold.

Nielsen realisation fails for certain finite subgroups of the mapping class group.

Abstract

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 $\Gamma(X)$ denote the group of automorphisms of the intersection lattice of $X$ that can be realised by diffeomorphisms. Then $M(X)$ is an extension of $\Gamma(X)$ 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 $\mathbb{CP}^2$, 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 $M_1(K3) \to M(K3)$.