An explicit section of the Laudenbach exact sequence of the mapping class group of connect sums of $S^2 \times S^1$

Abstract

Laudenbach proved that the mapping class group of the connect sum of $n$ copies of $S^2 \times S^1$ is an extension of $Out(F_n)$ by a finite group. Brendle-Broaddus-Putman proved that this exact sequence splits. We provide an explicit section $s$ of this split exact sequence.