Brian Udall proved that there is a short exact sequence of the form:

\(1 \xrightarrow{} Twist(M_{\Gamma})\xrightarrow{} Map(M_{\Gamma})\xrightarrow{\Psi} Map(\Gamma) \xrightarrow{}1,\) where $Twist(M_{\Gamma})$ is the subgroup of $Map(M_{\Gamma})$ generated by sphere twists over sphere systems of $M_{\Gamma}.$ Udall also proved that this short exact sequence splits topologically. The purpose of this talk is to present an explicit formula for a section s of this short exact sequence.