Independent Salvetti homotopy-model input

Let $I$ be a nonempty open convex cone in a finite-dimensional real vector space $E$ and $\mathcal{A}$ a locally finite arrangement of linear hyperplanes meeting $I$. Let $M(I,\mathcal{A})$ be the product complement of Complexified complements in an open convex cone, and let $\mathrm{Sal}(I,\mathcal{A})$ be the simplicial CW model of The Salvetti order complex and its finite CW case.

There exists a homotopy equivalence $f:\mathrm{Sal}(I,\mathcal{A})\to M(I,\mathcal{A})$, with homotopy equivalence understood as in Homotopy equivalences, homotopy inverses and spaces of the same homotopy type. If $\mathcal{A}$ is finite, the model is finite; in particular this holds for a finite arrangement in the whole vector space.

For the additional equivariant assertion, let $(W,S)$ be a Vinberg system in $E$ with closed fundamental chamber $\bar{C}$, in the sense specified in Independent Artin fundamental-group identification. Require $I=\mathrm{int}({\bigcup }_{w\in W}w\bar{C})$ and $\mathcal{A}=\{\mathrm{Fix}(r):r\in W\text{ is a linear reflection}\}$, the Coxeter arrangement of this realization. For the actions of Linear symmetries act on the Salvetti model, $f$ can be chosen $W$-equivariant: $f(wx)=wf(x)$ for every $w\in W$ and $x\in \mathrm{Sal}(I,\mathcal{A})$, as in Equivariant maps and isomorphisms of group actions.

Under these additional Vinberg hypotheses, for each of $X=\mathrm{Sal}(I,\mathcal{A})$ and $X=M(I,\mathcal{A})$, let $X\mathrm{/}W$ be the set of orbits for the relation $x\sim y$ if $y=wx$ for some $w\in W$. Give it the quotient topology: $U\subseteq X\mathrm{/}W$ is open exactly when $q_X^{-1}(U)$ is open in $X$, where $q_X(x)=[x]$. With these topologies, the induced map $\bar{f}:\mathrm{Sal}(I,\mathcal{A})\mathrm{/}W\to M(I,\mathcal{A})\mathrm{/}W$, $\bar{f}([x])=[f(x)]$, is a homotopy equivalence.

Complexified complements in an open convex cone

The Salvetti order complex and its finite CW case

Linear symmetries act on the Salvetti model

Homotopy equivalences, homotopy inverses and spaces of the same homotopy type

Independent Artin fundamental-group identification

Equivariant maps and isomorphisms of group actions

The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

Paris, §3.1, Theorem 3.1, printed p.386, supplies the homotopy equivalence to the product complement of Complexified complements in an open convex cone for the simplicial model of The Salvetti order complex and its finite CW case, with homotopy equivalence as in Homotopy equivalences, homotopy inverses and spaces of the same homotopy type.

Its equivariant and orbit-space clauses apply to the Vinberg data of Independent Artin fundamental-group identification, with $I$ equal to the interior of the union of translates of the closed fundamental chamber and $\mathcal{A}$ its reflection hyperplanes. The actions are those of Linear symmetries act on the Salvetti model, equivariance means Equivariant maps and isomorphisms of group actions, and the orbit spaces carry The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection. These clauses are imported in this precise setting; they are not deduced merely from equivariance of an arbitrary homotopy equivalence.

Finiteness for finite $\mathcal{A}$ is the separately proved elementary assertion of The Salvetti order complex and its finite CW case.

The five local helpers spell out the construction and elementary properties; they do not reconstruct Paris's homotopy-equivalence proof. The independent source was read to the limits recorded in its reference metadata. This input supplies neither asphericity nor equivariance under a general normalizer action. The latter requires its own construction in the article consumers. The helpers and this target remain drafts pending ordinary native author adoption and later scientific review.