On the local homology of Artin groups of finite and affine type

TL;DR

This paper investigates the local homology of Artin groups of finite and affine types using weighted discrete Morse theory, revealing square-free torsion properties and computing local homology in exceptional cases.

Contribution

It introduces the concept of precise matchings for Artin groups, extending known torsion properties to affine types and correcting previous results.

Findings

Homology has square-free torsion in all finite and affine cases

Constructed precise matchings for all cases

Computed local homology in exceptional cases

Abstract

We study the local homology of Artin groups using weighted discrete Morse theory. In all finite and affine cases, we are able to construct Morse matchings of a special type (we call them "precise matchings"). The existence of precise matchings implies that the homology has a square-free torsion. This property was known for Artin groups of finite type, but not in general for Artin groups of affine type. We also use the constructed matchings to compute the local homology in all exceptional cases, correcting some results in the literature.