General K(pi,1) Conjecture for Coxeter and Artin Groups

Establish that for every Coxeter group W, the orbit space Y_W = Y/W is a K(pi, 1) space, where Y = U_C \setminus \bigcup_{H \in A} H_C is the complement of the complexified reflection hyperplanes H_C in U_C = U + iV, with V the standard real representation of W, U the Tits cone, and A the collection of all hyperplanes fixed by orthogonal reflections in W. Equivalently, show that the hyperplane complement Y is aspherical for all Coxeter groups.

Background

The K(pi,1) conjecture connects singularity theory with the topology of complements of hyperplane arrangements and can be formulated for arbitrary Coxeter and Artin groups. It has been proved in the spherical case by Deligne (1972) and in the affine case by Paolini and Salvetti (2021), but the general case remains unresolved.

In the paper, the authors summarize their proof for affine Artin groups and survey combinatorial, algebraic, and topological tools involved. The general conjecture seeks to establish asphericity for all such orbit spaces across all Coxeter groups, including indefinite types beyond spherical and affine.