K(π, 1) for Artin Groups of Finite Type

This paper is a continuation of a programme to construct new K(π, 1)’s for Artin groups of finite type which began in [4] with Artin groups on 2 and 3 generators and was extended to braid groups in [3]. These K(π, 1)’s differ from those in [6] in that their universal covers are simplicial complexes. In [4] a complex is constructed whose top-dimensional cells correspond to minimal factorizations of a Coxeter element as a product of reflections in a finite Coxeter group. Asphericity is established in low dimensions using a metric of non-positive curvature. Since the nonpositive curvature condition is difficult to check in higher dimensions a combinatorial approach is used in [3] in the case of the braid groups. It is clear from [3] that the techniques used can be applied to any finite Coxeter group W. When W is equipped with the partial order given by reflection length and γ is a Coxeter element in W, the construction of the K(π, 1)’s is exactly analogous provided that the interval [I, γ] forms a lattice. In dimension 3, see [4], establishing this condition amounts to observing that two planes through the origin meet in a unique line. In the braid group case, see [3], where the reflections are transpositions and the Coxeter element is an n-cycle this lattice property is established by identifying [I, γ] with the lattice of noncrossing partitions of {1, 2, . . . , n}. In this paper, we consider the Artin groups of type Cn and Dn. Thus, for each finite reflection group W of type Cn or Dn, partially ordered by reflection length, we identify a lattice inside W and use it to construct a finite aspherical complex K(W). In the Cn case this lattice coincides with the lattice of noncrossing partitions of {1, 2, . . . , n,−1, . . . ,−n} studied in [8]. The final ingredient is to prove that π1(K(W)) is isomorphic to A(W), the associated finite type Artin group. As in [4] and [3] this involves a lengthy check that the obvious maps be