The <pure mapping class group> $\operatorname{PMod}(S)$ consists of orientation-preserving mapping classes fixing every puncture individually. If $S^*$ 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 conventions
$$
\operatorname{Push}(\gamma)=T_{\gamma_+}T_{\gamma_-}^{-1}.
$$
The <Birman exact sequence> is
$$
\boxed{1\longrightarrow\pi_1(S)
\xrightarrow{\operatorname{Push}}\operatorname{PMod}(S^*)
\xrightarrow{\operatorname{Forget}}\operatorname{PMod}(S)
\longrightarrow1},
$$
for the finite-type negative-Euler-characteristic surfaces under consideration.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 156 / 3 / a / Solution

{wiki=Point-pushing_map}

Pushing a distinguished puncture once around a loop $\gamma$ defines a mapping class. For a simple loop, the push is the product of opposite Dehn twists about the two boundary components of a thin annular neighborhood of $\gamma$.


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 Incoming links: Point-pushing map