 Solution

The pure mapping class group $\mathrm{PMod}(S)$ consists of orientation-preserving mapping classes fixing every puncture individually. If $S^{\ast }$ adds a distinguished puncture and $\gamma$ is a loop in $S$, the point-pushing map moves that puncture once around $\gamma$ while leaving the old punctures fixed. For a simple loop, a thin annular neighborhood of $\gamma$ has boundary curves ${\gamma }_{+}$ and ${\gamma }_{-}$, and with consistent twist conventionsThe Birman exact sequence isfor the finite-type negative-Euler-characteristic surfaces under consideration.