Natural finite Salvetti good covers

For a strict facet chain $\gamma =(F_0,\dots ,F_p)$ define $\mathrm{\Delta }(\gamma )=\{y+{\sum }_{i=1}^p{t}_{i}x(F_i):y\in F_0,t_i>0\}$. A strict chain $\delta =(G_0,\dots ,G_q)$ extends $\gamma$ when $G_0=F_0$ and $\{F_1,\dots ,F_p\}\subseteq \{G_1,\dots ,G_q\}$, allowing equality. Define $\omega (\gamma )={\bigcup }_{\delta \text{ extends }\gamma }\mathrm{\Delta }(\delta )$, write $\omega (F)=\omega ((F))$, and set $U(F,C)=\omega (F)\times C_F$. For $p=0$ the empty sum gives $\mathrm{\Delta }((F_0))=F_0$. These are the cited definitions, with valid points in the finite case $I=E$.

The imported cover and chain-intersection assertions apply to step 1.1. Distinct comparable incident pairs have distinct facets, since equal facets force equal localized chambers and then equal incident chambers by [F1]. Thus the facet chain in the intersection formula is strict and its $\omega$ is defined in step 1.1. Each nonempty finite intersection admits a contraction by [F99], so [F2] gives contractibility under the stated convention. This supplies the open good cover, injective indexing, and exact chain criterion.

Linearity and $x(gF)=gx(F)$ give $g\mathrm{\Delta }(F_0,\dots ,F_p)=\mathrm{\Delta }(g{F}_0,\dots ,g{F}_p)$. The facet action preserves incidence and carries extensions bijectively to extensions, so $g\omega (\gamma )=\omega (g\gamma )$. Permuting the hyperplanes sends the chamber of ${\mathcal{A}}_F$ containing $C$ to the chamber of ${\mathcal{A}}_{gF}$ containing $gC$, so $g{C}_F=(gC{)}_{gF}$. Hence $gU(F,C)=U(gF,gC)$. Applying $g$ to the two defining containments of the incident-pair order, and then applying $g^{-1}$, shows that the pair action preserves and reflects that order. There are finitely many facets and chambers, as each is a sign fiber for the finite arrangement; therefore the cover is finite.

The nerve defined in the Statement is an abstract simplicial complex: it contains the empty simplex, every singleton because every cover member is nonempty, and every subset of a simplex because a subfamily of an intersecting family still intersects. Injective indexing gives a vertex bijection $b:U(F,C)\mapsto (F,C)$. The chain criterion sends its nonempty simplices precisely to the nonempty chains; it also sends the empty simplex to the empty simplex. Thus $b$ is a simplicial isomorphism, meaning a vertex bijection preserving and reflecting simplices. Naturality of the cover gives $b(gU)=gb(U)$, so this isomorphism is $G$-equivariant.

For a barycentric point $\alpha$ of $\mathrm{\mid }N(\mathcal{U})\mathrm{\mid }$, define $B(\alpha )$ by $B(\alpha )(b(U))=\alpha (U)$ for each vertex $U$. The vertex bijection and simplex correspondence give a well-defined bijection onto $K$, with inverse given by the inverse coordinate permutation. On every closed simplex both maps are Euclidean coordinate permutations and hence continuous. By the weak topology, the inverse image of an open set under $B$ meets each simplex openly, so $B$ is continuous; the same argument gives continuity of its inverse. Therefore $B$ is a homeomorphism. Equivariance of $b$ implies $B(g\alpha )=gB(\alpha )$ coordinate by coordinate, proving the claimed equivariant identification of realizations.