  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 Laudenbach's original
proof, including the identification of the twist subgroup with
$(\mathbb{Z}/2)^n$.
