The mapping class group of connect sums of $S^2 \times S^1$

Abstract:

Let $M_n$ be the connect sum of $n$ copies of $S^2 \times S^1$. A classical theorem of Laudenbach says that the mapping class group $\operatorname {Mod}(M_n)$ is an extension of $\operatorname {Out}(F_n)$ by a group $(\mathbb {Z}/2)^n$ generated by sphere twists. We prove that this extension splits, so $\operatorname {Mod}(M_n)$ is the semidirect product of $\operatorname {Out}(F_n)$ by $(\mathbb {Z}/2)^n$, which $\operatorname {Out}(F_n)$ acts on via the dual of the natural surjection $\operatorname {Out}(F_n) \rightarrow GL_n(\mathbb {Z}/2)$. Our splitting takes $\operatorname {Out}(F_n)$ to the subgroup of $\operatorname {Mod}(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbach’s original proof, including the identification of the twist subgroup with $(\mathbb {Z}/2)^n$.