The $K(\pi ,1)$-conjecture for Artin groups

Abstract

An Artin group $G_W$ is the fundamental group of the space $Y_W$, which consists of the regular orbits of a Coxeter group $W$ acting on the complexified Tits cone. The best-known example occurs when $W$ is the symmetric group, in which case $G_W$ is the classical braid group, and $Y_W$ is the configuration space of $n$ distinct points in $\mathbb{C}$. A famous long-standing conjecture, attributed to Arnol’d, Brieskorn, Pham, and Thom, asserts that the orbit space $Y_W$ is a $K(\pi ,1)$-space. For finite $W$, the conjecture was proved by Deligne (1972) in a landmark paper. Recently, in collaboration with Giovanni Paolini (2021), we proved the conjecture for the next significant class of Artin groups of affine type.

We discuss various aspects of our proof, providing an overview of the main methods while avoiding the numerous technical details. In particular, we focus on combinatorial methods applied to the so-called dual approach to Artin groups. The latter involves the development of Garside theory and its application to non-crossing partition intervals. Finally, we highlight some naturally related problems and conjectures that can be formulated for any Artin group. Delucchi, Paolini, and Salvetti (2024) have recently resolved these conjectures affirmatively for all rank-three groups.