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

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 $\text{Mod}(M_n)$ is an extension of $\text{Out}(F_n)$ by a group $(\mathbb{Z}\mathrm{/}2{)}^n$ generated by sphere twists. We prove that this extension splits, so $\text{Mod}(M_n)$ is the semidirect product of $\text{Out}(F_n)$ by $(\mathbb{Z}\mathrm{/}2{)}^n$, which $\text{Out}(F_n)$ acts on via the dual of the natural surjection $\text{Out}(F_n)\to {\text{GL}}_n(\mathbb{Z}\mathrm{/}2)$. Our splitting takes $\text{Out}(F_n)$ to the subgroup of $\text{Mod}(M_n)$ 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 $(\mathbb{Z}\mathrm{/}2{)}^n$.