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 Δ ˆ ws ( n ) and Δ ˆ ss ( n ) , respectively. In particular, we find that Δ ˆ ws ( n ) is contractible for n ⩾ 4 , while Δ ˆ ss ( n ) 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 Δ ˆ ws ( n ) and Δ ˆ ss ( n ) : 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 .