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 long-standing 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 EL-shellable; 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.
Highlights
The long-standing K (π, 1) conjecture for Artin groups states that the orbit configuration space YW associated with a Coxeter group W is always a K (GW, 1) space
We prove the conjecture for the important family of Artin groups, namely for all affine Artin groups
The following are some consequences of the K (π, 1) conjecture: affine Artin groups are torsion-free; they have a classifying space with a finite number of cells; the well studied homology and cohomology of YW coincides with the homology and cohomology of the corresponding affine Artin group GW; the natural map between the classifying space of an affine Artin monoid and the classifying space of the corresponding Artin group is a homotopy equivalence
Summary
The long-standing K (π, 1) conjecture for Artin groups states that the orbit configuration space YW associated with a Coxeter group W is always a K (GW , 1) space. The conjecture was proved for spherical Artin groups (i.e. if W is finite) by Deligne [29], after being proved by Fox and Neuwirth in the case An [40] and by Brieskorn in the cases Cn, Dn, G2, F4, and I2( p) [16]. After the fundamental contribution of Deligne, the conjecture was proved for the affine Artin groups of type An, Cn [51], Bn [21], and G 2 [23]. Unlike the proofs for previously known cases, our approach is essentially “case-free,” some partial results use the classification of reflection groups. Besides Euclidean cases, the conjecture was proved for Artin groups of dimension ≤ 2 and for those of FC type [23,43]. In [49], finite-dimensional classifying spaces for affine Artin groups were constructed, but with an infinite number of cells. The following are some consequences of the K (π, 1) conjecture: affine Artin groups are torsion-free (this already follows from the construction of McCammond and Sulway [49]); they have a classifying space with a finite number of cells (see [60]); the well studied homology and cohomology of YW coincides with the homology and cohomology of the corresponding affine Artin group GW (see [19,20,21,55,57,61]); the natural map between the classifying space of an affine Artin monoid and the classifying space of the corresponding Artin group is a homotopy equivalence (see [33,34,52,53])
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