Point pushing is the kernel of forgetting the last disk puncture

Choice bookkeeping. By [F1] the Axiom of Choice [A1] yields dependent choice and countable choice, so the short exact sequence of [F2] and the extension lemma of [F7] are both available.

The forgetting homomorphism $\psi$. Let $F_n$ be the pointwise stabiliser of $Q_n$, and let $F^{\mathrm{\prime }}$ be the pointwise stabiliser of $Q_n^{\mathrm{\prime }}$, as in [F6]. A homeomorphism fixing $q_1,\dots ,q_n$ fixes $q_1,\dots ,q_{n-1}$, so the inclusion $\mathrm{inc}:F_n\hookrightarrow F^{\mathrm{\prime }}$ is defined and continuous for the subspace topologies; define $\psi :={\pi }_0(\mathrm{inc})$, that is, $\psi ([f]):=[f]$ for the class of a homeomorphism $f\in F_n$ read in ${\pi }_0(F^{\mathrm{\prime }})=\mathrm{PMod}(D^2,Q_n^{\mathrm{\prime }};\mathrm{\partial }{D}^2)$ by [F6]. This is well defined: if $f$ and $f^{\mathrm{\prime }}$ are joined by a path in $F_n$, the same path lies in $F^{\mathrm{\prime }}$ and joins them there. It is a group homomorphism: for $[f],[g]\in {\pi }_0(F_n)$ one has $[f][g]=[f\circ g]$, and $\mathrm{inc}(f\circ g)=\mathrm{inc}(f)\circ \mathrm{inc}(g)$, so $\psi ([f][g])=[f\circ g]=[f][g]=\psi ([f])\psi ([g])$. Thus $\psi$ is exactly the homomorphism that forgets the last marked point.

An isomorphism from the pure braid group and the identification ${\mathrm{Push}}_n={\mathrm{\Theta }}_n\circ \kappa$. By [F4] the restriction of ${\mathrm{\Psi }}_n^{\mathrm{m}\mathrm{c}}$ to $G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ is a group isomorphism onto $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$, and by [F3] the map ${\mathrm{\Psi }}_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ is a group isomorphism $G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}\to P{B}_n$; so ${\mathrm{\Theta }}_n:={\mathrm{\Psi }}_n^{\mathrm{m}\mathrm{c}}{\mid }_{G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}}\circ ({\mathrm{\Psi }}_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}{)}^{-1}:P{B}_n⟶\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ is a group isomorphism. Now let $[\gamma ]\in {\pi }_1(Y_n,q_n)$. Its ordered lift $L_{\gamma }$ is a pure geometric braid based at $Q_n$: its coordinates are the constant paths at $q_1,\dots ,q_{n-1}$ and the loop $\gamma$, they are pairwise distinct and lie in $\mathrm{int}{D}^2$, and $L_{\gamma }(0)=Q_n=L_{\gamma }(1)$, so $[L_{\gamma }]\in G_n^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$. Writing $i:Y_n\to F_n(\mathrm{int}{D}^2)$, $x\mapsto (q_1,\dots ,q_{n-1},x)$ for the fibre inclusion, we have $i\circ \gamma =L_{\gamma }$, so by [F2] $\kappa ([\gamma ])={\iota }_{\ast }^F(i_{\ast }[\gamma ])={\iota }_{\ast }^F[L_{\gamma }]\mathrm{.}$ The coordinate path of the braid $L_{\gamma }$ is $L_{\gamma }$ itself, so by [F3] $({\mathrm{\Psi }}_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}{)}^{-1}(\kappa ([\gamma ]{)}^{-1})=[L_{\gamma }],$ while [F5] gives ${\mathrm{\Psi }}_n^{\mathrm{m}\mathrm{c}}([L_{\gamma }])={\mathrm{Push}}_n([\gamma ]{)}^{-1}$. Applying the isomorphism ${\mathrm{\Theta }}_n$ to the inverse of $\kappa ([\gamma ])$ we therefore get ${\mathrm{\Theta }}_n(\kappa ([\gamma ]{)}^{-1})={\mathrm{Push}}_n([\gamma ]{)}^{-1},\text{hence}{\mathrm{\Theta }}_n(\kappa ([\gamma ]))={\mathrm{Push}}_n([\gamma ]),$ because a group isomorphism carries inverses to inverses.

Reading ${\mathrm{\Theta }}_m$ off a lifted isotopy. Let $m\ge 1$ and let $x\in P{B}_m$ with $[\beta ]:=({\mathrm{\Psi }}_m^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}{)}^{-1}(x)\in G_m^{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ and coordinate path $z=z_{\beta }$. Suppose $g:I\to {\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2)$ is a lift of the raw slice loop $S(\beta )$ with $g(0)=\mathrm{id}$ and with $g_s(q_j)=z_j(s)$ for all $j$ and $s$. Then $g$ is a lift of $S(\beta )$ with initial value the identity, so [F4] gives ${\mathrm{\Psi }}_m^{\mathrm{m}\mathrm{c}}([\beta ])=[g(1)]$, and hence ${\mathrm{\Theta }}_m(x)=[g(1)]\in \mathrm{PMod}(D^2,Q_m;\mathrm{\partial }{D}^2)\mathrm{.}$ Moreover $g(1)(q_j)=z_j(1)=q_j$ for every $j$, because $[\beta ]$ is pure, so $g(1)\in F_m$ is an element of the pointwise stabiliser and $[g(1)]$ is literally a class of ${\pi }_0(F_m)$.

Transport to the truncated configuration. Write $C=(c_1,\dots ,c_{n-1})$ for the canonical rank-$(n-1)$ configuration. The affine motion ${\eta }_j(t)=(1+t\mathrm{/}n)q_j+t(h_{n-1},0),h_{n-1}=1\mathrm{/}(4n),$ carries $q_j$ to $c_j$: $q_j=(2j-n-1)\mathrm{/}(4(n+1))$ gives ${\eta }_j(1)=(2j-n)\mathrm{/}(4n)$. The points remain ordered and inside the disc, since each coordinate is a convex combination of its initial and terminal positions. Reparametrize by a smooth nondecreasing function equal to $0$ near $0$ and $1$ near $1$, and extend constantly outside $I$. By [F7] and step 1.1 this smooth separated motion extends to a boundary-fixed disk isotopy with endpoint $R$ satisfying $R(q_j)=c_j$ for $j<n$. Conjugation $f\mapsto Rf{R}^{-1}$ identifies the pointwise stabiliser of $Q_n^{\mathrm{\prime }}$ with that of $C$, continuously in both directions by [F10]. It induces an isomorphism $C_R$ of their component groups. Also $R$ acts coordinatewise on configuration spaces and induces an isomorphism $R_{\ast }$ of their fundamental groups at these basepoints, commuting with the open-to-closed inclusions by [F8]. Define ${\mathrm{\Theta }}^{\mathrm{\prime }}:=C_R^{-1}\circ {\mathrm{\Theta }}_{n-1}\circ R_{\ast }:{\pi }_1(F_{n-1}(D^2),Q_n^{\mathrm{\prime }})⟶\mathrm{PMod}(D^2,Q_n^{\mathrm{\prime }};\mathrm{\partial }{D}^2),$ where ${\mathrm{\Theta }}_{n-1}$ is the canonical isomorphism of step 1.3. This is an isomorphism. If $z^{\mathrm{\prime }}$ is any ordered loop at $Q_n^{\mathrm{\prime }}$ lifted by an ambient isotopy $g$ from the identity, then $Rg{R}^{-1}$ lifts $R{z}^{\mathrm{\prime }}$ from the canonical configuration $C$. The inverse-slicing formula and step 1.4 give ${\mathrm{\Theta }}^{\mathrm{\prime }}(({\iota }_{\ast }^F[z^{\mathrm{\prime }}]{)}^{-1})=[g_1]\mathrm{.}$ This transported formula, rather than a canonical rank-$(n-1)$ identification at $Q_n^{\mathrm{\prime }}$, will be used below.

Naturality at the actual truncation. Let $x\in P{B}_n$ and choose its pure geometric representative $\beta =({\mathrm{\Psi }}_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}{)}^{-1}(x)$. By [F7] and [F9] its ordered path $z$ may be taken smooth and constant near the endpoints without changing its class. By step 1.1 and [F7], lift it to a boundary-fixed smooth isotopy $g$ from the identity with $g_s(q_j)=z_j(s)$; this is a continuous path of homeomorphisms by [F10]. Step 1.4 gives ${\mathrm{\Theta }}_n(x)=[g_1]$. The same isotopy lifts the truncated loop $z^{\mathrm{\prime }}=(z_1,\dots ,z_{n-1})$, based at $Q_n^{\mathrm{\prime }}$. By [F3] and [F8] the forgetting map of [F2] satisfies $\phi (x)=({\iota }_{\ast }^F[z^{\mathrm{\prime }}]{)}^{-1}\in {\pi }_1(F_{n-1}(D^2),Q_n^{\mathrm{\prime }})\mathrm{.}$ Step 1.5 therefore gives ${\mathrm{\Theta }}^{\mathrm{\prime }}(\phi (x))=[g_1]$ in the component group of the pointwise stabiliser of $Q_n^{\mathrm{\prime }}$. Step 1.2 identifies this class with $\psi ({\mathrm{\Theta }}_n(x))$. Thus $\psi \circ {\mathrm{\Theta }}_n={\mathrm{\Theta }}^{\mathrm{\prime }}\circ \phi$, with every map based at the specified configuration.

Exactness of the Birman sequence. By step 1.3, ${\mathrm{Push}}_n={\mathrm{\Theta }}_n\circ \kappa$ with ${\mathrm{\Theta }}_n$ an isomorphism and $\kappa$ injective by [F2], so ${\mathrm{Push}}_n$ is injective: if ${\mathrm{Push}}_n([\gamma ])=1$, then $\kappa ([\gamma ])={\mathrm{\Theta }}_n^{-1}(1)=1$ and hence $[\gamma ]=1$. Its image is ${\mathrm{\Theta }}_n(\mathrm{im}\kappa )={\mathrm{\Theta }}_n(\ker \phi )$ by the exactness in [F2]. By step 2.1 and the injectivity of ${\mathrm{\Theta }}^{\mathrm{\prime }}$, ${\mathrm{\Theta }}_n^{-1}(\ker \psi )=\ker (\psi \circ {\mathrm{\Theta }}_n)=\ker ({\mathrm{\Theta }}^{\mathrm{\prime }}\circ \phi )=\ker \phi ,$ so ${\mathrm{\Theta }}_n(\ker \phi )=\ker \psi$ and $\mathrm{im}{\mathrm{Push}}_n=\ker \psi$; this proves claims 1 and 2. Finally $\psi \circ {\mathrm{\Theta }}_n={\mathrm{\Theta }}^{\mathrm{\prime }}\circ \phi$ is the composite of the surjection $\phi$ of [F2] with the isomorphism ${\mathrm{\Theta }}^{\mathrm{\prime }}$, hence surjective, and therefore $\psi$ itself is surjective. Inserting these three facts into the sequence displayed in the statement gives a short exact sequence, with $F_{n-1}={\pi }_1(Y_n,q_n)$ the free group of [F2] on the $n-1$ puncture meridians.