Let P(n,k,k) and P(n,k,h), h < k, denote the intersection
lattices of the k-equal subspace arrangement of type D(n)
and the k,h-equal subspace arrangement of type B(n)
respectively. Denote by SB(n) the group of signed permutations. We
show that the quotient complex D(P(n,k,k))/SB(n) is collapsible. For
D(P(n,k,h))/SB(n), h < k, we show the following. If n is congruent to 0
modulo k, then it is homotopy equivalent to a sphere of dimension
2n/k - 2. If n is congruent to h modulo k, then it is homotopy
equivalent to a sphere of dimension 2(n-h)/k - 1. Otherwise, it is
contractible. Immediate consequences for the multiplicity of the
trivial characters in the representations of SB(n) on the homology
groups of D(P(n,k,k)) and D(P(n,k,h)) are stated.
In order to establish these results, we are led to introduce a
lexicographic shelling condition for a class of not necessarily pure
triangulated spaces. It specializes to the CL-shellability of
Bjorner and Wachs when the triangulated space is an order complex of
a poset.
