Subspace arrangements of type Bn and Dn

AbstractLet $$\mathcal{D}$$ n,k be the family of linear subspaces of ℝn given by all equations of the form $$\varepsilon _1 x_{l_1 } = \varepsilon _2 x_{l_2 } = \cdots = \varepsilon _k x_{l_k } ,$$ for 1≤il<⋯<ik≤n and (∈1,...,∈k)∈{+1,−1}k. Also let Bn,k,h be $$\mathcal{D}_{n,k} $$ enlarged by the subspaces $$x_{j_1 } = x_{j_2 } = \cdots = x_{j_h } = 0,$$ for 1 ≤ j1 <⋯< jh ≤ n.The special cases Bn,2,1 and $$\mathcal{D}_n $$ ,2 are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type Bn and Dn, respectively.In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of Bn,k,h, 1 ≤ h < k, which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold Mn,k,h=ℝn\∪Bn,k,h. For instance, it is shown that Hd(Mn,k,k−1) is torsion-free and is nonzero if and only if d=t(k−2) for some t, 0 ≤ t ≤[n/k]. Torsion-free cohomology follows also for the complement in ℂn of the complexification Bn,k,hℂ, 1 ≤ h < k.