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 ∆̂ws(n) and ∆̂ss(n), respectively. In particular, we find that ∆̂ws(n) is contractible for n ≥ 4, while ∆̂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 ∆̂ws(n) and ∆̂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.