Proof of the $K(\pi,1)$ conjecture for affine Artin groups
Abstract
We prove the $K(\pi,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a longstanding problem, due to Arnol'd, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on several new results and constructions. In particular: we show that all affine noncrossing partition posets are ELshellable; we use these posets to construct finite classifying spaces for dual affine Artin groups; we introduce new CW models for the orbit configuration spaces associated with arbitrary Coxeter groups; we construct finite classifying spaces for the braided crystallographic groups introduced by McCammond and Sulway.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.11795
 Bibcode:
 2019arXiv190711795P
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 Mathematics  Geometric Topology;
 20F36;
 20F55;
 55P20