Homology of even Artin kernels

On the Σ$\Sigma$‐invariants of Artin groups satisfying the K(π,1)$K(\pi,1)$‐conjecture

“…Given a discrete character 𝜒 ∶ 𝐴 Γ → ℤ and a field 𝔽, the associated 𝔽-chain complex of the 𝜒-cyclic cover of Sal(Γ) can be computed in the same way as in [8], that is, as:…”

“…Given a discrete character $$\chi:A_\Gamma \rightarrow \mathbb {Z}$$ and a field $$\mathbb {F}$$, the associated $$\mathbb {F}$$‐chain complex of the $$\chi$$‐cyclic cover of $$\text{Sal}(\Gamma)$$ can be computed in the same way as in [8], that is, as: $$$$\begin{equation*} \mathbb {F}[t^{\pm 1}]\otimes _{\mathbb {F}[A_\Gamma]}C_*(\widetilde{\text{Sal}(\Gamma)}), \end{equation*}$$$$…”

“…If 𝐴 Γ satisfies the 𝐾(𝜋, 1)-conjecture, given a commutative ring 𝑅 and 𝜒 ∶ 𝐴 Γ → ℤ a surjective discrete character with kernel 𝐴 𝜒 Γ , the homology groups 𝐻 * (𝐴 𝜒 Γ , 𝑅) are the 𝑅-homology groups of the 𝜒-cyclic cover of the Salvetti complex of 𝐴 Γ (the fact that 𝜒 needs to be surjective is not a problem because by [8, Remark 2.3] we may assume that 𝜒 is surjective). In [8], Blasco-Cogolludo-Martinez computed these homology groups with coefficients in a field for some Artin kernels for even Artin groups of FC-type.…”

“…Recall that an Artin kernel $$A_\Gamma ^\chi$$ is the kernel of some nontrivial discrete character $$\chi:A_\Gamma \rightarrow \mathbb {Z}$$. If $$A_\Gamma$$ satisfies the $$K(\pi,1)$$‐conjecture, given a commutative ring $$R$$ and $$\chi:A_\Gamma \rightarrow \mathbb {Z}$$ a surjective discrete character with kernel $$A_\Gamma ^\chi$$, the homology groups $$H_\ast (A_\Gamma ^\chi,R)$$ are the $$R$$‐homology groups of the $$\chi$$‐cyclic cover of the Salvetti complex of $$A_\Gamma$$ (the fact that $$\chi$$ needs to be surjective is not a problem because by [8, Remark 2.3] we may assume that $$\chi$$ is surjective). In [8], Blasco–Cogolludo–Martinez computed these homology groups with coefficients in a field for some Artin kernels for even Artin groups of FC‐type.…”