Let Πn,k,k and Πn,k,h, h < k, denote the intersection lattices of the k-equal subspace arrangement of type $$\mathcal{D}$$ n and the k,h-equal subspace arrangement of type $$\mathcal{B}$$ n respectively. Denote by $$S_n^B $$ the group of signed permutations. We show that Δ(Πn,k,k)/ $$S_n^B $$ is collapsible. For Δ(Πn,k,h)/ $$S_n^B $$ , h < k, we show the following. If n ≡ 0 (mod k), then it is homotopy equivalent to a sphere of dimension $$\tfrac{{2n}}{k}$$ . If n ≡ h (mod k), then it is homotopy equivalent to a sphere of dimension $$2\tfrac{{n - h}}{k} - 1$$ . Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of $$S_n^B $$ on the homology groups of Δ(Πn,k,k) and Δ(Πn,k,h) are stated.
The collapsibility of Δ(Πn,k,k)/ $$S_n^B $$ is established using a discrete Morse function. The same method is used to show that Δ(Πn,k,h)/ $$S_n^B $$ , h < k, is homotopy equivalent to a certain subcomplex. The homotopy type of this subcomplex is calculated by showing that it is shellable. To do this, we are led to introduce a lexicographic shelling condition for balanced cell complexes of boolean type. This extends to the non-pure case work of P. Hersh (Preprint, 2001) and specializes to the CL-shellability of A. Björner and M. Wachs (Trans. Amer. Math. Soc.4 (1996), 1299–1327) when the cell complex is an order complex of a poset.