The configuration braid short exact sequence $1\to P{B}_n\to B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}\to S_n\to 1$

The quotient is a covering and $p_{\ast }$ is injective. By [L3] the map $p:F_n(D^2)\to C_n(D^2)$ is a covering map; by [L6] the induced map $p_{\ast }:{\pi }_1(F_n(D^2),q)\to {\pi }_1(C_n(D^2),[q])$, $p_{\ast }([\gamma ])=[p\circ \gamma ]$, is a well-defined group homomorphism and is injective. Under [F1] this is a homomorphism $p_{\ast }:P{B}_n\to B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$.

The kernel of $\pi$ lies in the image of $p_{\ast }$. Let $[\alpha ]\in B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ with $\pi ([\alpha ])=e$, and let $\tilde{\alpha }$ be the lift of $\alpha$ starting at $q$; by the definition of $\pi$ in [F2] one has $\tilde{\alpha }(1)={\sigma }_{\alpha }\cdot q=\pi ([\alpha ])\cdot q=e\cdot q=q$. So $\tilde{\alpha }$ is a loop at $q$ in $F_n(D^2)$, and [L6] gives $p_{\ast }[\tilde{\alpha }]=[p\circ \tilde{\alpha }]=[\alpha ]$. Hence $\ker \pi \subseteq \mathrm{im}{p}_{\ast }$.

A model configuration with explicit distances. Assume $n\ge 2$ and put $q_j^{\mathrm{\prime }}:=\frac{2j-n-1}{2n}$ for $1\le j\le n$, and for $1\le i<n$ put $u_i:=\frac{q_i^{\mathrm{\prime }}+q_{i+1}^{\mathrm{\prime }}}{2}=\frac{2i-n}{2n}$, $w:=\frac{1}{2n}$ and $\eta :=\frac{w}{2}$. Then $q_{j+1}^{\mathrm{\prime }}-q_j^{\mathrm{\prime }}=\frac{1}{n}$ and $\mathrm{\mid }{q}_j^{\mathrm{\prime }}\mathrm{\mid }\le \frac{n-1}{2n}<1$ for all $j$, so $q^{\mathrm{\prime }}=(q_1^{\mathrm{\prime }},\dots ,q_n^{\mathrm{\prime }})\in F_n(\mathrm{int}{D}^2)$; moreover $q_i^{\mathrm{\prime }}=u_i-w$ and $q_{i+1}^{\mathrm{\prime }}=u_i+w$, and $\mathrm{\mid }{q}_j^{\mathrm{\prime }}-u_i\mathrm{\mid }=\frac{\mathrm{\mid }2j-2i-1\mathrm{\mid }}{2n}\ge 3w$ for every $j\mathrm{\notin }\{i,i+1\}$.

The first arrow is injective with image the kernel of $p_{\ast }$. The left arrow is the unique homomorphism $1\to P{B}_n$ from the one-element group; its image consists of the identity $e_{P{B}_n}$ alone, so it is injective, and since $p_{\ast }$ is injective by step 1.1 its kernel is $\{e_{P{B}_n}\}$, which is exactly that image.

The image of $p_{\ast }$ lies in the kernel of $\pi$. Let $[\gamma ]\in P{B}_n$ with $\gamma :I\to F_n(D^2)$ a loop at $q$; then $\alpha :=p\circ \gamma$ is a loop at $[q]$, since $p(\gamma (0))=[q]=p(\gamma (1))$. The path $\gamma$ satisfies $p\circ \gamma =\alpha$ and $\gamma (0)=q$, so by [L4] it is the unique lift of $\alpha$ starting at $q$. By [F2] the endpoint of that lift is ${\sigma }_{\alpha }\cdot q$ for the permutation $\pi ([\alpha ])={\sigma }_{\alpha }$, that is ${\sigma }_{\alpha }\cdot q=\gamma (1)=q$; freeness of the action by [L5] gives ${\sigma }_{\alpha }=e$, so $\pi (p_{\ast }[\gamma ])=\pi ([\alpha ])=e$. Hence $\mathrm{im}{p}_{\ast }\subseteq \ker \pi$.

A swap move at the model configuration. Assume $n\ge 2$ and fix $i$ with $1\le i<n$, keeping the notation of step 1.3. Define paths $a_i,b_i:I\to \mathbb{C}$ by the two-part formulas $a_i(t)=u_i-w(1-2t)+2t\eta \mathrm{i}$ and $b_i(t)=u_i+w(1-2t)-2t\eta \mathrm{i}$ for $0\le t\le \frac{1}{2}$, and $a_i(t)=u_i+w(2t-1)+2(1-t)\eta \mathrm{i}$ and $b_i(t)=u_i-w(2t-1)-2(1-t)\eta \mathrm{i}$ for $\frac{1}{2}\le t\le 1$, and let $x^{(i)}(t):=(x_1(t),\dots ,x_n(t))$ be the tuple with $x_i(t):=a_i(t)$, $x_{i+1}(t):=b_i(t)$ and $x_j(t):=q_j^{\mathrm{\prime }}$ for $j\mathrm{\notin }\{i,i+1\}$. The two parts of each formula agree at $t=\frac{1}{2}$, so $a_i$ and $b_i$ are continuous by [L10], and $a_i(0)=q_i^{\mathrm{\prime }}$, $a_i(1)=q_{i+1}^{\mathrm{\prime }}$, $b_i(0)=q_{i+1}^{\mathrm{\prime }}$, $b_i(1)=q_i^{\mathrm{\prime }}$.

Exactness at $B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$. Steps 2.2 and 1.2 together give $\mathrm{im}{p}_{\ast }=\ker \pi$.

The tuples $x^{(i)}(t)$ are collision-free. With the notation of step 2.3, one has $\mathrm{Im}{a}_i(t)=2t\eta \ge 0$ for $0\le t\le \frac{1}{2}$ and $\mathrm{Im}{a}_i(t)=2(1-t)\eta \ge 0$ for $\frac{1}{2}\le t\le 1$, while $\mathrm{Im}{b}_i(t)=-2t\eta \le 0$ and $\mathrm{Im}{b}_i(t)=-2(1-t)\eta \le 0$ on the same intervals; equality holds only at $t=0$ and $t=1$, where $a_i(0)=q_i^{\mathrm{\prime }}\mathrm{\ne }{q}_{i+1}^{\mathrm{\prime }}=b_i(0)$ and $a_i(1)=q_{i+1}^{\mathrm{\prime }}\mathrm{\ne }{q}_i^{\mathrm{\prime }}=b_i(1)$ by step 1.3. Hence $a_i(t)\mathrm{\ne }{b}_i(t)$ for every $t$. For $j\mathrm{\notin }\{i,i+1\}$ one has $\mathrm{\mid }\mathrm{Re}{a}_i(t)-u_i\mathrm{\mid }\le w$ and $\mathrm{Re}{q}_j^{\mathrm{\prime }}-u_i$ real with $\mathrm{\mid }\mathrm{Re}{q}_j^{\mathrm{\prime }}-u_i\mathrm{\mid }=\mathrm{\mid }{q}_j^{\mathrm{\prime }}-u_i\mathrm{\mid }\ge 3w$ by step 1.3, so $a_i(t)\mathrm{\ne }{q}_j^{\mathrm{\prime }}$, and the same argument gives $b_i(t)\mathrm{\ne }{q}_j^{\mathrm{\prime }}$; finally $\mathrm{\mid }{a}_i(t)\mathrm{\mid }\le \mathrm{\mid }{u}_i\mathrm{\mid }+w+\eta \le \frac{n-2}{2n}+\frac{1}{2n}+\frac{1}{4n}<1$ and likewise for $b_i$, while $\mathrm{\mid }{q}_j^{\mathrm{\prime }}\mathrm{\mid }<1$, so every coordinate lies in $\mathrm{int}{D}^2$. Thus $x^{(i)}(t)\in F_n(\mathrm{int}{D}^2)$ for every $t$.

A loop at $[q]$ with monodromy $s_i$. Assume $n\ge 2$ and fix $i$. By step 3.2 the formula ${\beta }_i:=p\circ x^{(i)}$ defines a continuous loop in $C_n(D^2)$ at $[q^{\mathrm{\prime }}]$, because $x^{(i)}(0)=q^{\mathrm{\prime }}$ and $x^{(i)}(1)=s_i\cdot q^{\mathrm{\prime }}$ is the tuple $q^{\mathrm{\prime }}$ with its $i$-th and $(i+1)$-st entries exchanged by [L9], whence $p(x^{(i)}(1))=[s_i\cdot q^{\mathrm{\prime }}]=[q^{\mathrm{\prime }}]$ by [L5]. By [L3] $F_n(D^2)$ is path-connected, so there is a path $\gamma :I\to F_n(D^2)$ with $\gamma (0)=q$ and $\gamma (1)=q^{\mathrm{\prime }}$; define ${\alpha }_i:I\to C_n(D^2)$ by ${\alpha }_i(t):=(p\circ \gamma )(3t)$ for $0\le t\le \frac{1}{3}$, ${\alpha }_i(t):={\beta }_i(3t-1)$ for $\frac{1}{3}\le t\le \frac{2}{3}$ and ${\alpha }_i(t):=(p\circ \gamma )(3-3t)$ for $\frac{2}{3}\le t\le 1$. Then ${\alpha }_i$ is a loop at $[q]$, and the path ${\tilde{\alpha }}_i$ given by ${\tilde{\alpha }}_i(t):=\gamma (3t)$, ${\tilde{\alpha }}_i(t):=x^{(i)}(3t-1)$, ${\tilde{\alpha }}_i(t):=s_i\cdot \gamma (3-3t)$ on the same three intervals is a lift of ${\alpha }_i$ starting at $q$: it is continuous, takes values in $F_n(D^2)$ by [L5], and $p({\tilde{\alpha }}_i(t))={\alpha }_i(t)$ on each piece, since $p(s_i\cdot y)=p(y)$. By [L4] it is the lift of ${\alpha }_i$ starting at $q$, so its endpoint is ${\tilde{\alpha }}_i(1)=s_i\cdot \gamma (0)=s_i\cdot q$, and therefore $\pi ([{\alpha }_i])=s_i$ by [F2].

The endpoint monodromy is surjective. Let $n\ge 2$ and let $T:=\{s_1,\dots ,s_{n-1}\}\subseteq S_n$. Step 4.1 exhibits for each $s_j\in T$ a class in $B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ with $\pi$-image $s_j$, so $T\subseteq \mathrm{im}\pi$, and $\mathrm{im}\pi$ is a subgroup of $S_n$ by [L7]; since $⟨T⟩$ is the smallest subgroup containing $T$ by [L8], it follows that $S_n=⟨T⟩\subseteq \mathrm{im}\pi$, so $\mathrm{im}\pi =S_n$ and $\pi$ is surjective by [L7]. For $n=0,1$ the group $S_n$ is trivial by [L8], so $\pi$ is surjective there as well, its image being a subgroup of a one-element group. Hence $\pi$ is surjective for every $n\ge 0$.

Conclusion. Step 1.1 shows that $p_{\ast }$ is an injective homomorphism $P{B}_n\to B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$, step 2.1 that the left arrow from the one-element group is injective with image $\ker p_{\ast }$, step 3.1 that $\mathrm{im}{p}_{\ast }=\ker \pi$, and step 5.1 that $\pi$ is surjective. By the definition of a short exact sequence in [L7], the displayed sequence is short exact for every $n\ge 0$, including the one-point cases $n=0$, where $F_0(D^2)$ and $C_0(D^2)$ are one-point spaces so that $P{B}_0$ and $B_0^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ are one-element groups by [F1] and [L11].