The Fadell-Neuwirth short exact sequence for pure braids

Fibre and base computations. The complement $M_{n-1}$ is homotopy equivalent to a wedge of $n-1$ circles by [F5], hence path-connected, so ${\pi }_0(M_{n-1})$ is a one-point set; also ${\pi }_2(M_{n-1},q_n)=0$ and ${\pi }_1(M_{n-1},q_n)$ is free with the $n-1$ positively oriented meridian classes of $q_1,\dots ,q_{n-1}$ as a free basis, all by [F5]. Since $n\ge 2$ we have $n-1\ge 1$, so [F6] gives ${\pi }_2(F_{n-1}(\mathrm{int}{D}^2),q^{\mathrm{\prime }})=0$.

The open-to-closed comparison. Let ${\iota }^F:F_m(\mathrm{int}{D}^2)\to F_m(D^2)$ be the inclusion for $m=n-1,n$ and let $p^D:F_n(D^2)\to F_{n-1}(D^2)$ be the closed-disc last-coordinate forgetful map. Both $p^D$ and $p$ forget the last coordinate, so $p^D\circ {\iota }^F={\iota }^F\circ p$ as maps; by [F7] the induced maps satisfy $p_{\ast }^D\circ {\iota }_{\ast }^F={\iota }_{\ast }^F\circ p_{\ast }$. By [F3] the maps ${\iota }_{\ast }^F:{\pi }_1(F_n(\mathrm{int}{D}^2),q)\to P{B}_n$ and ${\iota }_{\ast }^F:{\pi }_1(F_{n-1}(\mathrm{int}{D}^2),q^{\mathrm{\prime }})\to P{B}_{n-1}$ are isomorphisms onto the groups in the closed-disc convention, so $P{B}_n$ and $P{B}_{n-1}$ may be computed in the open-disc model.

The forgetful fibration and its fibre. By [F1] the Axiom of Choice [A1] yields the Axiom of Dependent Choice, so the choice hypotheses of [F2] are met; by [F2] the last-coordinate map $p:F_n(\mathrm{int}{D}^2)\to F_{n-1}(\mathrm{int}{D}^2)$, $(x_1,\dots ,x_n)\mapsto (x_1,\dots ,x_{n-1})$, is a Hurewicz, hence Serre, fibration; over $b:=p(q)=q^{\mathrm{\prime }}$ its fibre is $p^{-1}(q^{\mathrm{\prime }})=F_1(M_{n-1})\cong M_{n-1}=\mathrm{int}{D}^2\setminus \{q_1,\dots ,q_{n-1}\}$, which contains $q$ because $q_n\mathrm{\notin }\{q_1,\dots ,q_{n-1}\}$. This use of AC, only to invoke [F2], is the sole choice principle in the proof.

The exact sequence in the open-disc model. Inserting the computations of step 1.1 into the exact sequence of [F4] for the based Serre fibration $p$ of step 1.3 with $e_0=q$, $b_0=q^{\mathrm{\prime }}$ and fibre $M_{n-1}$ gives the exact sequence of groups ${\pi }_2(F_{n-1}(\mathrm{int}{D}^2),q^{\mathrm{\prime }})\stackrel{\mathrm{\partial }}{\to }{\pi }_1(M_{n-1},q_n)\stackrel{i_{\ast }}{\to }{\pi }_1(F_n(\mathrm{int}{D}^2),q)\stackrel{p_{\ast }}{\to }{\pi }_1(F_{n-1}(\mathrm{int}{D}^2),q^{\mathrm{\prime }})\stackrel{\mathrm{\partial }}{\to }{\pi }_0(M_{n-1})$. The left term vanishes by step 1.1, so $\mathrm{im}(i_{\ast })=\ker (p_{\ast })$ is the kernel of $p_{\ast }$ and $i_{\ast }$ is injective; the last term is a one-point set, so the boundary into it is the zero map and exactness at ${\pi }_1(F_{n-1}(\mathrm{int}{D}^2),q^{\mathrm{\prime }})$ makes $p_{\ast }$ surjective. Hence $1\to {\pi }_1(M_{n-1},q_n)\stackrel{i_{\ast }}{\to }{\pi }_1(F_n(\mathrm{int}{D}^2),q)\stackrel{p_{\ast }}{\to }{\pi }_1(F_{n-1}(\mathrm{int}{D}^2),q^{\mathrm{\prime }})\to 1$ is short exact.

Transport to the closed-disc convention. Conjugating the sequence of step 2.1 by the isomorphisms of step 1.2 identifies it with $1\to F_{n-1}\stackrel{\kappa }{\to }P{B}_n\stackrel{\phi }{\to }P{B}_{n-1}\to 1$, where $\kappa ={\iota }_{\ast }^F\circ i_{\ast }$ is the composite of the fibre inclusion with the open-to-closed isomorphism ${\iota }_{\ast }^F$ for $m=n$, and $\phi =p_{\ast }^D$ is the induced map of the closed-disc forgetting map; exactness is preserved by these isomorphisms and $\phi$ is the map induced by forgetting the last strand.

Elementary cases and conclusion. By [F3] the group $P{B}_1$ is trivial, so for $n=2$ the quotient in the displayed sequence is trivial and the sequence reads $1\to F_1\to P{B}_2\to 1\to 1$; the general case $n\ge 2$ is step 3.1, so the theorem is proved.