Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence

Let $\tilde{f}\colon(S^2,\widetilde{A})\circlearrowleft$ be a Thurston map and let $M(\tilde{f})$ its mapping class biset: isotopy classes rel $\widetilde{A}$ of maps obtained by pre- post-composing $\tilde{f}$ the group $(S^2,\widetilde{A})$. $A\subseteq\widetilde{A}$ an $\tilde{f}$-invariant subset, $f\colon(S^2,A)\circlearrowleft$ induced map. We give analogue Birman short exact sequence: just as $\mathbf{Mod}(S^2,\widetilde{A})$ is iterated extension $\mathbf{Mod}(S^2,A)$ fundamental groups punctured spheres, $M(f)$ dynamical biset $f$.