Main spherical theorem combining exact sequences and asphericity

PDF p. 1: Abstract; PDF p. 3: Theorem 1.8; PDF p. 26: Theorem 6.7; PDF p. 27: Corollary 6.14

Assume the Axiom of Choice. Let $(W,S)$ be a finite-rank spherical Coxeter system, meaning that $S$ is finite and $W$ is finite, and let $J\subseteq S$. Let $A$ be the associated Artin–Tits group, $\pi :A\to W$ its standard projection, and $P=\ker \pi$. For $G=W,A$, let $G_J$ be the subgroup generated by the standard generators indexed by $J$, and define $N(G,J)={\mathrm{Norm}}_G(G_J)\mathrm{/}{G}_J$, where ${\mathrm{Norm}}_G(G_J)=\{g\in G:g{G}_J{g}^{-1}=G_J\}$. Let $N(P,J)$ be the image of $P\cap {\mathrm{Norm}}_A(A_J)$ in $N(A,J)$.

Let ${\mathrm{\Theta }}_J$ be the subspace of the real dual reflection representation on which the simple roots indexed by $J$ vanish, and let ${\mathcal{H}}^J$ consist of the distinct hyperplanes obtained by restricting to ${\mathrm{\Theta }}_J$ those root hyperplanes that do not contain it. In spherical type, the relative complexified complement is $X={\mathcal{X}}_J^{\mathbb{C}}=({\mathrm{\Theta }}_J{\otimes }_{\mathbb{R}}\mathbb{C})\setminus {\bigcup }_{H\in {\mathcal{H}}^J}(H{\otimes }_{\mathbb{R}}\mathbb{C})$, with its subspace topology. The action of $N(W,J)$ on $X$ is the complex-linear extension of $(n{W}_J)\cdot f=nf$, where $(nf)(v)=f(n^{-1}v)$ for $f\in {\mathrm{\Theta }}_J$, $v$ in the real root space and $n\in {\mathrm{Norm}}_W(W_J)$. It is independent of the representative and preserves $X$. For $x\in X$, let $\bar{x}$ be its image in $X\mathrm{/}N(W,J)$ under this action.

The algebraic exact sequence $1\to N(P,J)\to N(A,J)\to N(W,J)\to 1$ is isomorphic to the covering exact sequence $1\to {\pi }_1(X,x)\to {\pi }_1(X\mathrm{/}N(W,J),\bar{x})\to N(W,J)\to 1$, with the identity on $N(W,J)$. In particular, $N(P,J)\cong {\pi }_1(X,x)$ and $N(A,J)\cong {\pi }_1(X\mathrm{/}N(W,J),\bar{x})$. Moreover, $X$ is a $K(N(P,J),1)$ and $X\mathrm{/}N(W,J)$ is a $K(N(A,J),1)$: here $K(G,1)$ means a path-connected space whose fundamental group is isomorphic to $G$ and whose homotopy groups in all degrees at least two vanish. Fundamental groups use first-path-first multiplication.

The article reference attributes the combined claim. The proof is a local composition of the two listed results; the article citation is not a proof premise.