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 $(mathbb{Z}/2)^n$ 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 $(mathbb{Z}/2)^n$, which $text{Out}(F_n)$ acts on via the dual of the natural surjection $text{Out}(F_n) rightarrow text{GL}_n(mathbb{Z}/2)$. 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 Laudenbachs original proof, including the identification of the twist subgroup with $(mathbb{Z}/2)^n$.