The isotopy classes of geometric braids based at $Q$ form a group, and the endpoint permutation is a homomorphism

(a), the identity. Write $\mu (t):=\max (0,2t-1)$ and $\nu (t):=\min (2t,1)$; since the right factor of a stacking runs during the first half of the height interval, $(e\star \beta {)}_j(t)=z_j(2t)=z_j(\nu (t))$ for $t\le \frac{1}{2}$ and $(e\star \beta {)}_j(t)=e_{\pi (\beta )(j)}(2t-1)=q_{\pi (\beta )(j)}=z_j(1)=z_j(\nu (t))$ for $t\ge \frac{1}{2}$, while $(\beta \star e{)}_j(t)=e_j(2t)=q_j=z_j(0)=z_j(\mu (t))$ for $t\le \frac{1}{2}$ and $(\beta \star e{)}_j(t)=z_{\pi (e)(j)}(2t-1)=z_j(2t-1)=z_j(\mu (t))$ for $t\ge \frac{1}{2}$; so $e\star \beta$ and $\beta \star e$ are the reparametrisations $z_j\circ \nu$ and $z_j\circ \mu$ of the tuple $\beta$. Both $\mu$ and $\nu$ are continuous nondecreasing maps of $I$ onto $I$ fixing $0$ and $1$ by [L5], so for $s\in I$ the maps ${\mu }_s(t):=(1-s)t+s\mu (t)$ and ${\nu }_s(t):=(1-s)t+s\nu (t)$ are again of that kind, and $Z_j(s,t):=z_j({\mu }_s(t))$, $W_j(s,t):=z_j({\nu }_s(t))$ are jointly continuous by [L5]; each slice $(z_j({\mu }_s(t)){)}_j$ is a braid based at $Q$, because ${\mu }_s(0)=0$, ${\mu }_s(1)=1$ and the collision-freeness and continuity conditions of [F1] are inherited from $\beta$, and likewise for ${\nu }_s$; the boundary slices are $\beta$, $e\star \beta$ and $\beta \star e$ by ${\mu }_0={\nu }_0=\mathrm{id}$, ${\mu }_1=\mu$, ${\nu }_1=\nu$. Hence $e\star \beta \sim \beta$ and $\beta \star e\sim \beta$, so $[e]$ is a two-sided identity for the operation of [F2].

(a), the inverse is a braid. For ${\bar{\beta }}_j(t):=z_{\pi (\beta {)}^{-1}(j)}(1-t)$ each ${\bar{\beta }}_j$ is a composite of continuous maps with values in $D^{\circ }$, and ${\bar{\beta }}_i(t)\mathrm{\ne }{\bar{\beta }}_j(t)$ for $i\mathrm{\ne }j$ because $\pi (\beta {)}^{-1}$ is injective and the $z_k$ are collision-free by [F1]; its bottom values are ${\bar{\beta }}_j(0)=z_{\pi (\beta {)}^{-1}(j)}(1)=q_{\pi (\beta )(\pi (\beta {)}^{-1}(j))}=q_j$, using the defining property of $\pi (\beta )$ in [F1], and its top values are ${\bar{\beta }}_j(1)=z_{\pi (\beta {)}^{-1}(j)}(0)=q_{\pi (\beta {)}^{-1}(j)}$, which run through the set $\{q_1,\dots ,q_n\}$; hence $\bar{\beta }$ is a braid based at $Q$ with $\pi (\bar{\beta })=\pi (\beta {)}^{-1}$.

(b). The map $\pi$ is well defined on classes $[\beta ]$ by the constancy of the endpoint permutation along isotopies in [F2]; it satisfies $\pi (\gamma \star \beta )=\pi (\gamma )\circ \pi (\beta )$ by [F2], and $\pi (e)=\mathrm{id}$ because $e$ is pure by [F1]; a map of groups that preserves the operation and the identity is a group homomorphism into the symmetric group $S_n$ of The finite symmetric group $S_n$, one-line notation, and cycle notation, so the formula $\pi \text{\hspace{0.17em}⁣}:G_n\to S_n$, $[\beta ]\mapsto \pi (\beta )$, defines a group homomorphism once $G_n$ is known to be a group.

(a), $\bar{\beta }\star \beta \sim e$. By the stacking formula of [F2] and step 1.2, $(\bar{\beta }\star \beta {)}_j(t)=z_j(2t)$ for $t\le \frac{1}{2}$ and $(\bar{\beta }\star \beta {)}_j(t)={\bar{\beta }}_{\pi (\beta )(j)}(2t-1)=z_{\pi (\beta {)}^{-1}(\pi (\beta )(j))}(2-2t)=z_j(2-2t)$ for $t\ge \frac{1}{2}$; that is, $\bar{\beta }\star \beta$ is the out-and-back reparametrisation $z_j\circ \lambda$ of $\beta$ with $\lambda (t):=2t$ for $t\le \frac{1}{2}$ and $\lambda (t):=2-2t$ for $t\ge \frac{1}{2}$. For $s\in I$ put ${\lambda }_s(t):=(1-s)\lambda (t)$; then ${\lambda }_s$ is continuous with ${\lambda }_s(0)={\lambda }_s(1)=0$, so $Z_j(s,t):=z_j({\lambda }_s(t))$ is jointly continuous by [L5], each slice $(z_j({\lambda }_s(t)){)}_j$ is a braid based at $Q$ because it is a reparametrisation of the collision-free tuple $\beta$ with all bottom and top values equal to $q_j$, and the boundary slices are $\bar{\beta }\star \beta$ at $s=0$ and $e$ at $s=1$; hence $\bar{\beta }\star \beta \sim e$.

(a), $\beta \star \bar{\beta }\sim e$. By the same computation with the roles of the two factors exchanged, $(\beta \star \bar{\beta }{)}_j(t)=z_{\pi (\beta {)}^{-1}(j)}(1-2t)$ for $t\le \frac{1}{2}$ and $(\beta \star \bar{\beta }{)}_j(t)=z_{\pi (\beta {)}^{-1}(j)}(2t-1)$ for $t\ge \frac{1}{2}$, which is again an out-and-back parametrisation of $\beta$ with the labels relabelled by $\pi (\beta {)}^{-1}$; putting $\kappa (t):=1-2t$ for $t\le \frac{1}{2}$ and $\kappa (t):=2t-1$ for $t\ge \frac{1}{2}$, and then ${\kappa }_s(t):=(1-s)\kappa (t)+s$, gives a braid isotopy: ${\kappa }_s(0)={\kappa }_s(1)=1$, so every slice begins and ends at $z_{\pi (\beta {)}^{-1}(j)}(1)=q_j$ and is collision-free; at $s=1$ the slice is the constant braid. Thus this is a braid isotopy from $\beta \star \bar{\beta }$ to $e$.

(a), conclusion. By steps 1.1, 2.1 and 3.1 the operation of [F2] on the isotopy classes based at $Q$ is associative, has the two-sided identity class $[e]$, and gives $[\bar{\beta }][\beta ]=[e]=[\beta ][\bar{\beta }]$ for every $[\beta ]$; by [L4] the set $G_n$ of isotopy classes is therefore a group with identity $[e]$ and $[\beta {]}^{-1}=[\bar{\beta }]$.

Assertions (a) and (b) are steps 4.1 and 1.3, the latter now applicable because step 4.1 makes $G_n$ a group; the group structure uses only the explicit stacking, reversal and reparametrisation formulas displayed above. ∎