The $K(π, 1)$ 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

Group Theory 2026-07-27 v1 Algebraic Topology Combinatorics Geometric Topology

Authors: Giovanni Paolini

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Abstract

In these notes, we introduce the 50-year-old $K(\pi, 1)$ conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in $\mathbb{C}^n$ of a "symmetric" configuration of hyperplanes is a $K(\pi, 1)$ 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.

Keywords

k(pi,1) conjecture artin groups coxeter groups spherical type hyperplane arrangement combinatorial topology

Cite

@article{arxiv.2607.24659,
  title  = {The $K(π, 1)$ conjecture for Artin groups of spherical type},
  author = {Giovanni Paolini},
  journal= {arXiv preprint arXiv:2607.24659},
  year   = {2026}
}