K(pi,1) conjecture for arbitrary normaliser quotients

PDF p. 4: Conjecture 1.10

The authors' Conjecture 1.10 is: “Theorem 1.8 holds for any Coxeter group $W$.” We retain that wording as the attributed question.

For its precise formulation in the finite-rank geometric convention of Reflecting hyperplanes, fundamental chamber and Tits cone, let $(W,S)$ be any Coxeter system with finite $S$, let $J\subseteq S$, and allow $W$ to be infinite. Use the groups $N(W,J)$, $N(A,J)$ and $N(P,J)$ of Normalisers and their quotients, and put $X={\mathcal{X}}_J^{\mathbb{C}}$ for the relative complexified complement of Relative complexified hyperplane complement. Choose $x\in X$ and let $\bar{x}$ be its image in $X\mathrm{/}N(W,J)$.

The algebraic sequence of Normaliser quotient exact sequence and the geometric covering sequence of Relative free action, normal cover and exact sequence are

$1⟶N(P,J)⟶N(A,J)⟶N(W,J)⟶1$

and

$1⟶{\pi }_1(X,x)⟶{\pi }_1(X\mathrm{/}N(W,J),\bar{x})⟶N(W,J)⟶1.$

The question asks whether these sequences identify over $N(W,J)$, giving compatible isomorphisms ${\pi }_1(X\mathrm{/}N(W,J),\bar{x})\cong N(A,J)$ and ${\pi }_1(X,x)\cong N(P,J)$, and whether the spaces have the respective classifying-space homotopy types. Precisely, the latter asks for connected based CW complexes $(K_A,k_A)$ and $(K_P,k_P)$ and homotopy equivalences $f_A:X\mathrm{/}N(W,J)\to K_A$ and $f_P:X\to K_P$, with $f_A(\bar{x})=k_A$ and $f_P(x)=k_P$, such that $K_A$ and $K_P$ are respectively $K(N(A,J),1)$ and $K(N(P,J),1)$ in the sense of Eilenberg–Mac Lane space. The specified fundamental-group isomorphisms of these CW models, composed with $(f_A{)}_{\ast }$ and $(f_P{)}_{\ast }$, must give the compatible isomorphisms just requested. Homotopy equivalence has the meaning of Homotopy equivalences, homotopy inverses and spaces of the same homotopy type.

Thus the full question retains both the fundamental-group identifications and the asphericity conclusions, expressed through CW homotopy models rather than an undeclared CW structure on the given geometric spaces. It concerns every subset $J$ and every finite-rank Coxeter type, without requiring spherical type.

Normalisers and their quotients

Relative complexified hyperplane complement

Normaliser quotient exact sequence

Relative free action, normal cover and exact sequence

Eilenberg--Mac Lane space

Reflecting hyperplanes, fundamental chamber and Tits cone

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

To specify the classical object in the Status, take $J=\mathrm{\varnothing }$, $I={\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }$ and the reflecting-hyperplane arrangement $\mathcal{H}$ of Reflecting hyperplanes, fundamental chamber and Tits cone. Let $Z=\mathrm{Sal}(I,\mathcal{H})$ be the simplicial CW model of The Salvetti order complex and its finite CW case: its vertices are the incident facet/chamber pairs in the supplied poset and its simplices are its finite chains, with the weak realization topology. Use the induced $W$-action and the orbit topology of Independent Salvetti homotopy-model input, and put $\mathrm{\Sigma }(W,S)=Z\mathrm{/}W$.

Here $\mathrm{\Sigma }(W,S)$ is a realization of the full standard Salvetti complex: the identification with the Coxeter-system model is Paris's Theorem 3.3, printed p.394. Its standard cell structure has one cell of dimension $\mathrm{\mid }T\mathrm{\mid }$ for every $T\subseteq S$ for which $W_T$ is finite, with attaching maps induced from that realization, as specified in Paris §3.3, printed p.398. All such subsets are included. Write $\tilde{\mathrm{\Sigma }}(W,S)\to \mathrm{\Sigma }(W,S)$ for its universal covering, meaning a covering map with simply connected total space as in Universal covering spaces. Existence for this standard complex is supplied by Paris §4.1, Proposition 4.2, printed p.401. Contractible has the meaning of Nullhomotopic maps and contractible spaces.

Paris's Theorem 2.5, printed p.379, makes the finite-rank canonical dual representation a Vinberg realization with the coordinate chamber used here. Thus $I$ is its nonempty open Tits cone and $\mathcal{H}$ is its locally finite reflection arrangement. The classical comparison of Independent Salvetti homotopy-model input gives $Z\simeq {\mathcal{X}}_{\mathrm{\varnothing }}^{\mathbb{C}}$ and $\mathrm{\Sigma }(W,S)\simeq {\mathcal{X}}_{\mathrm{\varnothing }}^{\mathbb{C}}\mathrm{/}W$; Paris's Corollary 3.4, printed p.394, records the same comparisons for the Coxeter-system model. These are the precise homotopy comparisons underlying the classical relevance mentioned below. They import no asphericity assertion and no equivariant comparison for a general relative normaliser quotient.

The frozen article presents Conjecture 1.10 as a conjectural extension of its Theorem 1.8, whose hypothesis is that $W$ is finite. This is the article's source-time distinction between theorem and conjecture; this item gives no proof of the extension and supplies no proof premise to established items.

There is a relevant later claim about the classical case. The September 23, 2026 manuscript Harmonic heights and the Artin K(pi,1) conjecture, Theorem 1.1, claims contractibility of the full standard Salvetti universal cover $\tilde{\mathrm{\Sigma }}(W,S)$ just specified for every finite set of standard generators. The MathVet HarmonicArtin row and cert-machine HarmonicArtin entry report independent successful machine checks of the exact formal model, including acceptance by Lean and nanoda in the latter report. These are reported checks, not computations replayed here; the paper-to-model interpretation and full proof were not independently validated in this investigation.

The classical claim is relevant to $J=\mathrm{\varnothing }$, where the normaliser quotients reduce to the original groups and the relative arrangement becomes the classical one. Its stated theorem does not assert the sequence identifications or asphericity of all the relative arrangements for arbitrary $J$. No derivation of Conjecture 1.10 from that claim has been established here. The prior status record reports that a focused source search on November 13, 2026 found no independently verified general resolution of this normaliser question; this describes the search limits, not a proof of worldwide unresolved status. The present item retains the authors' question with current status under evaluation.

The phrase “any Coxeter group” in Conjecture 1.10 does not explicitly state a rank convention. The finite-rank geometric reading above is inferred from the article's framework and its cited foundations: §2.2 follows Iyama–Wemyss for the dual reflection space and calls it a Vinberg reflection representation, while the Proposition 3.6 proof invokes finite dimension. Iyama–Wemyss §§1.1–1.3 define the Coxeter graph on $\{1,\dots ,n\}$ using an $n\times n$ matrix; Paris §1 likewise starts with finite $S$ and defines Vinberg systems in finite dimension. The local complement and sequence suppliers make that convention explicit. This preserves all finite-rank Coxeter types and the full conjectural conclusions; it does not claim that the article expressly imposed a global finite-rank hypothesis.

An interpretation allowing infinitely many simple generators would need a topology on the full algebraic dual, an appropriate relative Tits-cone interior and an applicable covering-sequence formulation. No such infinite-rank geometric extension is supplied by the cited finite-dimensional conventions or by the recent classical claim. The original unrestricted wording is retained as attribution, and this extension remains unresolved here rather than being treated as proved or silently identified with a Salvetti colimit.

The exact source passages, reported machine-check entries and reading limits are recorded in the citation metadata. The 2026 classical proof and its Lean dependency closure were not independently validated, and no machine check was replayed. The status discussion does not import that claimed theorem as an established prerequisite for this open-problem item.

This adjudication rechecked the original formulation on PDF pp.3–4, the applicable Paris definitions and comparison statements, and the current local suppliers. It preserved the preceding qualified 2026 status reports; it performed no new status search, independent proof validation or machine replay. The two repaired interface omissions belonged to this generated draft, not to an established defect in the original PDF.