# Second mod $2$ homology of Artin groups

## TL;DR

This paper calculates the second mod 2 homology of any Artin group using algebraic tools, without relying on the $K( ext{pi},1)$ conjecture, advancing understanding of Artin group topology.

## Contribution

It provides a general computation method for the second mod 2 homology of Artin groups, independent of the $K( ext{pi},1)$ conjecture assumptions.

## Key findings

- Explicit formulas for second mod 2 homology of Artin groups
- Extension of known results to arbitrary Artin groups
- Application of Hopf's and Howlett's formulas in this context

## Abstract

In this paper, we compute the second mod $2$ homology of an arbitrary Artin group, without assuming the $K(\pi,1)$ 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.