An explicit section of the Laudenbach-type exact sequence of the big mapping class group of $M_{\mathrm{\Gamma }}$

Given a locally finite graph $\mathrm{\Gamma }$, Udall proved that the mapping class group of the doubled handlebody associated to $\mathrm{\Gamma }$ is an extension of the mapping class group of $\mathrm{\Gamma }$ by a possibly infinite direct product of ${\mathbb{Z}}_2$, and that this extension splits. We give a formula for Udall’s section restricted to the pure mapping class group and, when $\mathrm{\Gamma }$ has finitely many ends, on the whole mapping class group.