Pure braids as pure mapping classes

The lift endpoint induces the endpoint permutation. Fix $[\beta ]\in G_n$ and let $g$ be a lift of the raw slice loop $S(\beta )$ with $g(0)=\mathrm{id}$, so that $\mathrm{\Psi }([\beta ])=[h_{\beta }]$ with $h_{\beta }=g(1)$ and $h_{\beta }(Q_n)=z(1)$ by [L2]; here $z$ is the ordered coordinate lift of $S(\beta )$ from $Q_n$, whose coordinates satisfy $z_j(1)=q_{{\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}([\beta ])(j)}$ by [L4] and [L5]. First, $h_{\beta }$ lies in $F$: indeed $\mathrm{ev}(h_{\beta })=S(\beta )(1)=[Q_n]$, so $[h_{\beta }(Q_n)]=[Q_n]$ and [L3] applies. Therefore the unique permutation $\pi (h_{\beta })$ of [L6] is defined, and evaluating the identity $h_{\beta }(Q_n)=z(1)$ in the $j$-th coordinate gives $h_{\beta }(q_j)=z_j(1)=q_{{\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}([\beta ])(j)}(1\le j\le n),$ so $\pi (h_{\beta })={\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}([\beta ])$: the permutation realised by the evaluation endpoint of the lifted slice is exactly the geometric endpoint permutation of the braid class.

Trivial permutation is exactly purity. By [L7], a class $[f]\in \mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ lies in $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ exactly when the permutation it induces on $Q_n$ is trivial; by [L6] the permutation induced by a representative $f\in F$ is a class invariant, so the condition is $\pi (h_{\beta })=\mathrm{id}$ for the endpoint of any such representative. Combining with step 1.1, for $[\beta ]\in G_n$ we have the equivalence $\mathrm{\Psi }([\beta ])\in \mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)⟺\pi (h_{\beta })=\mathrm{id}⟺{\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}([\beta ])=\mathrm{id}⟺[\beta ]\in G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}},$ the last equivalence being the definition [L8] of the pure subgroup as the kernel of ${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}$.

Conclusion. The equivalence of step 2.1 says that an element $[\beta ]\in G_n$ satisfies $\mathrm{\Psi }([\beta ])\in \mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ if and only if $[\beta ]\in G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$; since $\mathrm{\Psi }$ is a bijection by [L1], it carries $G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ onto $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$. The restriction $\mathrm{\Psi }{\mathrm{\mid }}_{G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}}$ is a group isomorphism onto its image because $\mathrm{\Psi }$ is a group isomorphism by [L1], so the pure geometric braid subgroup equals the pure boundary-fixed mapping class group under this identification. When $n=0$ both groups are trivial and the statement is immediate; when $n=1$ the group $S_1$ is trivial, so every braid class is pure, and by [L7] the setwise and pointwise stabilisers of the one-point marked set coincide, so every mapping class is pure; the equivalence above also holds in these cases because $z_1(1)=q_1$ for every one-strand braid.