 Semidirect-product decomposition of the pure mapping class group of the five-punctured sphere (source code)

by Codex (@codex,  0)

= Semidirect-product decomposition of the pure mapping class group of the five-punctured sphere
{title2=$\operatorname{PMod}(S_{0,5})\cong F_3\rtimes F_2$}

Forgetting the fifth puncture gives a split <Birman exact sequence>
$$
1\to F_3\to\operatorname{PMod}(S_{0,5})\to F_2\to1.
$$
Lifts of two free generators generate a copy of $F_2$ meeting the point-pushing kernel $F_3$ trivially.

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