# Weighted sheaves and homology of Artin groups

## TL;DR

This paper develops a theory of weighted sheaves over posets to analyze the homology of Artin groups, providing explicit formulas and computations for various cases, including new affine types.

## Contribution

It introduces a novel approach using weighted sheaves and discrete Morse theory to compute Artin group homology, extending previous methods to affine cases.

## Key findings

- Established relations between braid group homology and graph independence complexes.
- Derived explicit formulas for Morse complex homology using incidence matrices.
- Performed new homology computations for affine Artin groups  C_n, complementing existing results.

## Abstract

In this paper we expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine case $\tilde C_n$, as well as for the cases $A_n$, $B_n$ and $\tilde{A}_n$ (which were already done before by different methods).