On the topology of weakly and strongly separated set complexes

We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set [n]={1,2,…,n}, which, after deleting all cone points, we denote by and , respectively. In particular, we find that is contractible for n⩾4, while is homotopy equivalent to a sphere of dimension n−3. We also show that our homotopy equivalences are equivariant with respect to the group generated by two particular symmetries of and : One induced by the set complementation action on subsets of [n] and another induced by the action on subsets of [n] which replaces each k∈[n] by n+1−k.