Quotient Complexes and Lexicographic Shellability

Let n,k,k and n,k,h , h < k, denote the intersection lattices of the k-equal subspace arrangement of type Dn and the k, h-equal subspace arrangement of type Bn respectively. Denote by SB n the group of signed permutations. We show that ( n,k,k )/SB n is collapsible. For ( n,k,h )/S B n , h < k, we show the following. If n ≡ 0 (mod k), then it is homotopy equivalent to a sphere of dimension 2n k − 2. If n ≡ h (mod 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 ( n,k,k ) and ( n,k,h ) are stated. The collapsibility of ( n,k,k )/SB n is established using a discrete Morse function. The same method is used to show that ( n,k,h )/SB n , 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.