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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 anexplicit section $s$ of this split exact sequence.

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