A boundary-fixed punctured-disk homeomorphism acting trivially on the fundamental group is isotopic to the identity

Base case and purity. For $n=0$ the conclusion is the boundary-fixed disk Alexander contraction of [F6]; for $n=1$ it follows from [F5]. For $n\ge 2$, [F1] first shows that $h$ fixes every $q_i$, so its class lies in the pure mapping class group and the forgetting map of [F2] applies. We prove the assertion by induction on $n$.

Filling the last puncture and applying induction at the correct configuration. Put $Z=D^2\setminus \{q_1,\dots ,q_{n-1}\}$ and let $j:X\hookrightarrow Z$. The induced $j_{\ast }$ is surjective: any loop in $Z$ is homotopic rel $d$ to a finite polygonal loop avoiding the finite marked set, and a further small detour removes any passage through the single extra point $q_n$. The homotopy and detour stay in $Z$. Since $h_{\ast }{j}_{\ast }=j_{\ast }{h}_{\ast }$ and $h_{\ast }\mathrm{\mid }{\pi }_1(X,d)=\mathrm{id}$, this surjectivity implies $h_{\ast }\mathrm{\mid }{\pi }_1(Z,d)=\mathrm{id}$. To transport the truncation $Q_n^{\mathrm{\prime }}$ to the canonical rank-$(n-1)$ tuple $C=(c_j)$, use the explicit motion $q_j\mapsto (1+t\mathrm{/}n)q_j+t\mathrm{/}(4n)$, $j<n$, on the real axis; it ends at $c_j=(2j-n)\mathrm{/}(4n)$, preserves order, and remains in the interior. Reparametrize smoothly to be constant near the time endpoints and apply [F4], giving a boundary-fixed endpoint homeomorphism $R$ with $R(Q_n^{\mathrm{\prime }})=C$. The map $Rh{R}^{-1}$ induces the identity on the fundamental group of the canonical $(n-1)$-punctured disk. Induction therefore makes its mapping class trivial; conjugating the isotopy back shows that $h$ is isotopic to the identity relative to the actual truncation $Q_n^{\mathrm{\prime }}$. Thus $[h]$ lies in the forgetting kernel of [F2].

An actual point-motion representative of the kernel. By [F2], write $[h]={\mathrm{Push}}_n([\gamma ])$ for a loop $\gamma$ in $Y_n$ based at $q_n$. A compact loop avoiding the finite set of other marked points can be replaced in its based class by a finite polygonal loop, then rounded smoothly and made constant near the time endpoints; each replacement stays in small disks missing those points. Apply [F4] to this last-point motion and the constant motions of all the other points. It gives a jointly continuous ambient isotopy $H_t$ with $H_0=\mathrm{id}$, $H_t(q_j)=q_j$ for $j<n$, and $H_t(q_n)=\gamma (t)$. Put $k=H_1$. By [F3], $[k]=[h{]}^{-1}$ relative to the boundary and all $Q_n$, so $k_{\ast }=\mathrm{id}$ on ${\pi }_1(X,d)$: a marked-set isotopy restricts to a based homotopy on $X$, and the inverse class of $h$ also induces the identity.

The compact tether square detects the moving-point loop. Let $s=s_n$. The map $B(u,t)=H_t(s(u))$ is a continuous map of the compact square into $Z$: its image never meets $q_j$ for $j<n$, since $H_t$ fixes those points and is injective. Its left edge is the constant $d$, its lower edge is $s$, its right edge is $\gamma$, and its upper edge is $k\circ s$. Its boundary relation is therefore $s\cdot \gamma \cdot (k\circ s{)}^{-1}\simeq 1$ in $Z$. Independently, apply [F1] to the endpoint homeomorphism $k$, whose induced action is the identity. This gives a compact relative-endpoint homotopy $k\circ s\simeq s$ with fixed endpoints $d,q_n$. Filling $q_n$ makes this an ordinary relative path homotopy in $Z$. Substitute it into the boundary relation to obtain $s\cdot \gamma \cdot s^{-1}\simeq 1$, hence $[\gamma ]=1$ in ${\pi }_1(Z,q_n)$ by basepoint transport along $s$. This uses the compact endpoint homotopy supplied by [F1], not an extension of an arbitrary homotopy on $X$.

Returning to the interior and closing induction. The loop $\gamma$ lies in the interior and has compact image. Choose $r_0<1$ so that all its points and all $q_j$ have norm less than $r_0$. Compress the outer collar radially by $r\mapsto r$ for $r\le r_0$ and $r\mapsto r_0+(r-r_0)\mathrm{/}2$ for $r\ge r_0$. This continuous map sends $Z$ into $Y_n$, fixes $\gamma$ and $q_n$, and avoids the other marked points because it changes only the outer collar. Composing the nullhomotopy from step 4.1 with it shows $[\gamma ]=1$ already in ${\pi }_1(Y_n,q_n)$. [F3] now gives $[h]={\mathrm{Push}}_n(1)=1$. By [F4] this is precisely an isotopy to the identity relative to $\mathrm{\partial }{D}^2\cup Q_n$. The induction is complete. AC is used through the point-pushing kernel theorem and point-motion extensions; no general arc-tameness or arc-isotopy theorem is needed in this proof.