On the classifying space of Artin monoids

ABSTRACT

A theorem proved by Dobrinskaya [9] shows that there is a strong connection between the K(π,1) conjecture for Artin groups and the classifying spaces 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.

KEYWORDS:

• Artin groups
• Artin monoids
• Coxeter groups
• discrete Morse theory

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 20F36
• 55R35
• 55U10
• 52C35

Acknowledgment

This work is based on the material of my master thesis, written under the supervision of Mario Salvetti. Therefore I would like to thank him for introducing me to the topic and for giving me good advice throughout my final year of master degree at the University of Pisa. I would also like to thank the referee, for his careful reading and useful comments.