The $K(\pi , 1)$ conjecture for Artin groups of spherical type

Giovanni Paolini

doi:10.5802/wbln.44

The

K (π, 1)

conjecture for Artin groups of spherical type

Giovanni Paolini

Winter Braids Lecture Notes, Winter Braids XII, Tome 9 (2023), Exposé no. 3, 11 p.

Résumé

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 $ℂ^{n}$ 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.

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Publié le : 2026-06-29

DOI : 10.5802/wbln.44

Giovanni Paolini. The $K(\pi , 1)$ conjecture for Artin groups of spherical type. Winter Braids Lecture Notes, Winter Braids XII, Tome 9 (2023), Exposé no. 3, 11 p.. doi: 10.5802/wbln.44

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