The Euler Characteristic of Spherical Bestvina-Brady Kernels

We compute the Euler characteristic of Bestvina-Brady kernels $\mathrm{BB}(\mathrm{\Gamma })$ for all Artin groups of spherical type. Our method involves discrete Morse theory to describe the cellular structure of a classifying space for the Bestvina–Brady kernel, through the action of $\mathrm{BB}(\mathrm{\Gamma })$ on non-integer level sets of the universal cover of the Salvetti complex. We extend the method of computation of $\chi (\mathrm{BB}(\mathrm{\Gamma }))$ to Artin groups that satisfy the $K(\pi ,1)$-conjecture and whose nerves are rationally acyclic, and we provide an example by realising the projective plane as the nerve of a Coxeter group.