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

First forget one puncture from the four-punctured sphere. Parts a and b giveNow forget the fifth puncture of $\Sigma =S_{0,5}$. The Birman exact sequence gives

Choose a minimal free generating pair $x,y$ of $F_2$ and lifts $\tilde{x},\tilde{y}$ in $\mathrm{PMod}(\Sigma )$. Let $F=⟨\tilde{x},\tilde{y}⟩$. The restriction $\pi {|}_F:F\to F_2$ is surjective. Since $F$ is generated by two elements and $F_2$ has minimal generator number two, the stated Hopf-type fact makes $\pi {|}_F$ an isomorphism. Thus $F\cong F_2$ and $F\cap K=1$.

For any $g\in \mathrm{PMod}(\Sigma )$, choose $f\in F$ with $\pi (f)=\pi (g)$. Then $f^{-1}g\in K$, so $g\in FK$. The kernel is normal by exactness. We have proved the semidirect-product decomposition of the pure mapping class group of the five-punctured sphereThe minimal generating-set sizes are therefore