The K(π, 1) conjecture for Artin groups of spherical type

2026/06/29 by Giovanni Paolini
Mathematics · #math.GR #math.AT #math.CO #math.GT

paper · pdf · doi:10.5802/wbln.44

Abstract

In these notes, we introduce the 50-year-old K(π, 1) conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in ℂⁿ of a "symmetric" configuration of hyperplanes is a K(π, 1) space. Our end goal is to present a proof of the conjecture in the so-called spherical case, where only a finite number of hyperplanes are removed, through methods from combinatorial topology. This proof draws inspiration from the original proof of the spherical case, which is a special case of a celebrated 1972 theorem by Pierre Deligne.

Citations

• Artin-Gruppen und Coxeter-Gruppen
• Die Fundamentalgruppe des Raumes der regul�ren Orbits einer endlichen komplexen Spiegelungsgruppe
• The Complete Enumeration of Finite Groups of the Form Ri2=(RiRj)kij=1
• Discrete Groups Generated by Reflections

Related

• Cycles in spherical Deligne complexes and application to K(π,1)-conjecture for Artin groups
• The K(π, 1) conjecture for affine Artin groups
• K(π,1) conjecture for Artin groups
• The dual approach to the K(π, 1) conjecture
• La conjecture du K(π,1) pour les groupes d'Artin affines (d'après Paolini et Salvetti)