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Pure mapping class group

Point-pushing map

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For a finite-type surface $S$ of negative Euler characteristic and the surface $S^*$ obtained by adding a puncture, forgetting that puncture gives
$$
1\longrightarrow\pi_1(S)\xrightarrow{\operatorname{Push}}
\operatorname{PMod}(S^*)\longrightarrow\operatorname{PMod}(S)\longrightarrow1.
$$


Birman exact sequence

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

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

Pure mapping class group of the four-punctured sphere

Semidirect-product decomposition of the pure mapping class group of the five-punctured sphere

Birman exact sequence

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Birman exact sequence

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