On the classifying space of Artin monoids

TL;DR

This paper explores the deep connections between the classifying space of Artin monoids, the Salvetti complex, and the $K(\pi,1)$ conjecture for Artin groups, providing new proofs and insights into their relationships.

Contribution

It demonstrates the existence of connections between the CW model of Artin monoids and the Salvetti complex, leading to an alternative proof of Dobrinskaya's theorem.

Findings

Established links between the CW model and the Salvetti complex.

Derived a new proof of Dobrinskaya's theorem.

Enhanced understanding of the $K(\pi,1)$ conjecture for Artin groups.

Abstract

A theorem proved by Dobrinskaya in 2006 shows that there is a strong connection between the $K(\pi,1)$ conjecture for Artin groups and the classifying space of Artin monoids. More recently Ozornova obtained a different proof of Dobrinskaya's theorem based on the application of discrete Morse theory to the standard CW model of the classifying space of an Artin monoid. In Ozornova's work there are hints at some deeper connections between the above-mentioned CW model and the Salvetti complex, a CW complex which arises in the combinatorial study of Artin groups. In this work we show that such connections actually exist, and as a consequence we derive yet another proof of Dobrinskaya's theorem.