Equivariant equivalence with fundamental groupoids

By [F1] and [F8], choose $a\in E\setminus \bigcup {\mathcal{H}}^J$; then $(a,0)\in X$, so $X$ is nonempty. If $E$ has dimension zero, there are no hyperplanes and $X$ is a singleton, hence is locally path-connected. Otherwise, fix $z=(x,y)\in X$. For each $H$, at least one of ${\mathrm{\ell }}_H(x)$ and ${\mathrm{\ell }}_H(y)$ is nonzero. The bound in [F9] gives $K_H>0$ with $\mathrm{\mid }{\mathrm{\ell }}_H(v)\mathrm{\mid }\le K_H\mathrm{\parallel }v{\mathrm{\parallel }}_2$. If, for example, ${\mathrm{\ell }}_H(x)\mathrm{\ne }0$, every perturbation of $z$ of norm less than $\mathrm{\mid }{\mathrm{\ell }}_H(x)\mathrm{\mid }\mathrm{/}(2K_H)$ still has nonzero first-coordinate value under ${\mathrm{\ell }}_H$; use the analogous radius for $y$ otherwise. Taking the minimum of these finitely many positive radii gives an open Euclidean ball about $z$ contained in $X$; if the arrangement is empty, any radius works. Thus $X$ is open. Inside any neighbourhood of $z$ open in $X$, a sufficiently small Euclidean ball is still contained in that neighbourhood. Such balls are open and path-connected by [F9], so they give the required neighbourhood base and prove local path connectedness.

There are finitely many chambers by [F1]. Choose a real point $a_C\in C$ for one chamber in each $G$-orbit and extend by $a_{gC}=g{a}_C$. By [F2], the action preserves chambers and is free on them, so this is well-defined. A real point of $C$ lies in its chart: its real part lies on no arrangement hyperplane, making the chart conditions in [F3] vacuous. Apply the path assignment of [F3] to these points.

Let $p,q\in X$. For $H\in {\mathcal{H}}^J$ put $L_H=H\times H$. The linear map $f_H:E\times E\to {\mathbb{R}}^2$ defined by $f_H(u,v)=({\mathrm{\ell }}_H(u),{\mathrm{\ell }}_H(v))$ is surjective, since ${\mathrm{\ell }}_H$ is nonzero, and has kernel $L_H$. For $s=p$ or $s=q$, $f_H(s)\mathrm{\ne }0$, so $B_{H,s}=L_H+\mathbb{R}s$ is a proper linear subspace: its image under the surjective $f_H$ is just the one-dimensional line $\mathbb{R}{f}_H(s)$. By [F8], choose $r$ outside all these finitely many $B_{H,s}$. The segment $[p,r]$ cannot meet any $L_H$: if $(1-t)p+tr\in L_H$, then $t\mathrm{\ne }0$ because $p\mathrm{\notin }{L}_H$, and solving for $r$ puts it in $L_H+\mathbb{R}p$, a contradiction. The identical argument with $q$ shows that $[r,q]$ avoids every $L_H$. Both segments are continuous paths by [F9], and traversing them successively gives a path in $X$ from $p$ to $q$. An empty arrangement gives straight segments directly; in dimension zero the constant path suffices. Thus $X$ is path-connected.

Compare the two crossing conventions. The map $\kappa :X\to X$, $\kappa (x+iy)=x-iy$, is a homeomorphism and its own inverse: it preserves the excluded sets $H\times H$ and is an isometry of $E\times E$. It fixes the real chamber points and commutes with the real linear $G$-action. Composing paths and endpoint-fixed homotopies with $\kappa$ therefore induces a groupoid automorphism ${\kappa }_{\ast }$ of ${\mathrm{\Pi }}_1(X)$. For adjacent $C,C^{\mathrm{\prime }}$, take a short transverse real segment through the relative interior of their common wall, avoiding every other hyperplane, and give it a small imaginary displacement on the $C^{\mathrm{\prime }}$ side near its crossing, tapering to zero at its endpoints. Join these endpoints to $a_C,a_{C^{\mathrm{\prime }}}$ in the respective real chambers. Choose an intermediate real point $d$ before the crossing, on the $C$ side. Then $d\in X_{C,C^{\mathrm{\prime }}}$; the portion up to $d$ lies in $X_C$, and the portion after $d$ lies in $X_{C^{\mathrm{\prime }}}$, since at the crossed wall its imaginary part is on the $C^{\mathrm{\prime }}$ side and no other wall meets this small segment. Hence [F3] identifies its class with the assigned crossing. Its conjugate under $\kappa$ instead has imaginary part on the preceding chamber's $C$ side, exactly the convention in [F4]. If $\mathrm{\Phi }$ denotes that supplier's isomorphism on real chamber basepoints, the assigned crossing class is therefore ${\kappa }_{\ast }\mathrm{\Phi }(C\to C^{\mathrm{\prime }})$.

By [F4], $\mathrm{\Phi }$ is an isomorphism from $\mathcal{D}$ to the full fundamental subgroupoid on the selected real points $a_C$. Step 2.2 shows that the graph assignment is ${\kappa }_{\ast }\mathrm{\Phi }$ on every generator. This composite is again an isomorphism onto that subgroupoid, since ${\kappa }_{\ast }$ fixes its objects and is invertible. Thus the assignment respects the equal-endpoint geodesic relations and extends over formal inverses. Composing with inclusion gives $p$. It is equivariant by [F3] on generating crossings and objects, and hence on formal inverses and composites. It is fully faithful, and it is essentially surjective because $X$ is path-connected by step 2.1. Using [F7], choose for each point of $X$ a path to a selected chamber point, taking the constant path at selected points. Transporting arrows along these paths and through the inverse of ${\kappa }_{\ast }\mathrm{\Phi }$ defines a quasi-inverse; the chosen path classes give the natural isomorphism to the identity, while the other composite is the identity on $\mathcal{D}$. Hence $p$ is an equivalence in the stated sense.

Let $A$ be the full subgroupoid on the points $a_C$. The free action on chambers makes $C\mapsto a_C$ injective: distinct real chambers are disjoint. Thus [F2] makes the action on the objects of $A$ free, and [F5] makes the action on all points of $X$ free. By [F6], $A\mathrm{/}G$ is the full subgroupoid of ${\mathrm{\Pi }}_1(X)\mathrm{/}G$ on the orbits of the $a_C$: any arrow orbit between two such orbits can be translated so that its endpoints are selected points. It is essentially surjective because every point of $X$ is joined by a path to a selected point by step 2.1. Consequently $\mathcal{D}\mathrm{/}G\cong A\mathrm{/}G$ maps by an equivalence to ${\mathrm{\Pi }}_1(X)\mathrm{/}G$; a quasi-inverse is obtained by choosing connecting arrows in that quotient as in step 3.1.

Steps 1.1 and 2.1 establish nonemptiness, path connectedness and local path connectedness of $X$. Together with the action properties in [F5], these verify every hypothesis of the topological orbit interface, so projection induces an isomorphism ${\mathrm{\Pi }}_1(X)\mathrm{/}G\cong {\mathrm{\Pi }}_1(X\mathrm{/}G)$ and $q$ is a normal covering. Define $\bar{p}$ as this isomorphism after the equivalence in step 4.1. On each object and arrow, both $\bar{p}Q$ and $q_{\ast }p$ take the assigned point or path class to its projection, so the square commutes strictly. Composition with an isomorphism preserves the quasi-inverse data, proving that $\bar{p}$ is an equivalence.