Second mod 2 homology of Artin groups

Abstract

In this paper, we compute the second mod 2 homology of an arbitrary Artin group, without assuming the K(π, 1) conjecture. The key ingredients are (A) Hopf's formula for the second integral homology of a group and (B) Howlett's result on the second integral homology of Coxeter groups. 20F36,20J06; 20F55

Homological stability for Artin monoids

“…In Second Mod 2 Homology of Artin Groups by Akita and Liu [AL18], homological stability in degree two with Z 2 coefficients was proved, for the sequence of Artin groups {A Wn } relating to Hepworth's sequence {W n }. They proved this by showing the mod 2 homology in degree two of any finite rank Artin group was isomorphic to the mod 2 homology of the corresponding Coxeter group.…”

The minimal genus problem for right angled Artin groups

“…Unfortunately, there is no known formula for the second homology of a general Artin group. Akita and Liu [AL18] give a general formula for the second homology with Z/2Z coefficients, but no integral results are known. Without such a result, a general investigation of the minimal genus problem for Artin groups seems to be out of reach.…”