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

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.

On $S_{0,3}$, a simple proper arc joining two distinct specified punctures is unique up to isotopy relative to its ends. Indeed, a small regular neighborhood of the arc and its two ends has one boundary curve separating those two punctures from the third; the Jordan curve theorem gives the unique such separation, and the disc it bounds gives the isotopy between any two choices.

Choose the three joining arcs, one for each puncture pair, with disjoint interiors. They form an ideal triangle graph whose complement consists of two discs. A pure homeomorphism fixes all three endpoints and carries each arc to an isotopic arc. The simultaneous-isotopy lemma makes it fix the three arcs, and the Alexander trick on each complementary disc makes it isotopic to the identity. Therefore

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