Abstract

On Laudenbach-type exact sequences

Laudenbach proved that the mapping class group of the connected sum of $n$ copies of $S^2 \times S^1$ is an extension of $\text{Out}(F_n)$ by a finite group. Brendle, Broaddus and Putman proved that this exact sequence splits. We provide an explicit section $s$ of this split exact sequence. Given a locally finite graph $\Gamma$, Udall proved that the mapping class group of the doubled handlebody associated to $\Gamma$ is an extension of $\map(\Gamma)$ by a possibly infinite direct product of $\Z_2$, and proved that this exact sequence splits. We provide an explicit formula for this section $s$ of $\Psi$ restricted to $\pmap(\Gamma)$, and in the case that
$E(\Gamma) < \infty$, we provide a formula for a section $s : \map(\Gamma)\xrightarrow{}\map(M_\Gamma)$ of $\Psi$.