Isomorphism of algebraic and topological normaliser sequences

Let $V$ be the underlying finite-dimensional real vector space of the ambient complexification in [F2], and let $L$ range over the deleted complexified hyperplanes. Each $L$ has real codimension two: a real basis of the corresponding real hyperplane, together with its multiples by $i$, is a real basis of $L$. For $p\in X$, the subspace $L+\mathbb{R}p$ has dimension ${\dim }_{\mathbb{R}}V-1$, since $p\mathrm{\notin }L$, and is proper. For two endpoints $p,q\in X$, [F7] therefore gives $r\in V$ outside all the finitely many subspaces $L+\mathbb{R}p$ and $L+\mathbb{R}q$. If there are no deleted hyperplanes, no avoidance condition on $r$ is needed.

The segment from $p$ to $r$ avoids every deleted $L$: if $(1-t)p+tr\in L$ for $0\le t\le 1$, then $t\mathrm{\ne }0$ because $p\mathrm{\notin }L$, and solving for $r$ gives $r\in L+\mathbb{R}p$, contrary to step 1.1. The same argument applies to the segment from $q$ to $r$. Both segment maps are continuous, since their real coordinates are affine functions of $t$; traversing the first and the reverse of the second gives a path in $X$ from $p$ to $q$. The construction also covers an empty arrangement, while in dimension zero $X$ is a point. Hence $X$ is path-connected. Take $p=x_0$ from [F4] and $q=x$, and choose the resulting path $c:x_0\to x$.

Write $\bar{c}=q\circ c$. By [F5], conjugation by path classes gives isomorphisms $u:{\pi }_1(X,x_0)\to {\pi }_1(X,x)$ and $v:{\pi }_1(X\mathrm{/}N(W,J),q(x_0))\to {\pi }_1(X\mathrm{/}N(W,J),\bar{x})$: with first-path-first products they send $[\alpha ]$ to $[c^{-1}\ast \alpha \ast c]$ and $[\beta ]$ to $[{\bar{c}}^{-1}\ast \beta \ast \bar{c}]$, respectively. Their inverses use the reversed path. Projection preserves concatenation and reversal, so $v{q}_{\ast }=q_{\ast }u$.

Let a loop $\beta$ at $q(x_0)$ lift from $x_0$ to $g{x}_0$, where $g\in N(W,J)$. The lift of ${\bar{c}}^{-1}\ast \beta \ast \bar{c}$ starting at $x$ is $c^{-1}$ followed by that lift of $\beta$ and then by $gc$. Indeed, $gc$ projects to $\bar{c}$ and starts at $g{x}_0$; uniqueness in [F5] identifies it with the final lifted segment. Its endpoint is $gx$. By the endpoint description in [F3], the last maps of the two covering sequences therefore agree after applying $v$.

Steps 3.1 and 4.1 give an isomorphism of the two covering exact sequences, with identity on $N(W,J)$. Compose it with the comparison at $x_0$ from [F4]. The result is an isomorphism from the algebraic sequence of [F1] to the covering sequence based at $x$, with identity on the rightmost nontrivial group. Its kernel and middle components give the two stated group isomorphisms.