On the Topology of Weakly and Strongly Separated Set Complexes

TL;DR

This paper investigates the topological properties of complexes formed by weakly and strongly separated subsets of a finite set, revealing contractibility and sphere-like structures, with symmetry considerations.

Contribution

It characterizes the homotopy types of these complexes, showing contractibility for weakly separated complexes and sphere homotopy equivalence for strongly separated complexes, including symmetry actions.

Findings

or n  4, ully contractible or weakly separated complexes.

or all n, ully homotopy equivalent to an (n-3)-sphere for strongly separated complexes.

ound equivariant homotopy equivalences respecting specific symmetries.

Abstract

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 $\hat{\Delta}_{ws}(n)$ and $\hat{\Delta}_{ss}(n)$, respectively. In particular, we find that $\hat{\Delta}_{ws}(n)$ is contractible for $n\geq4$, while $\hat{\Delta}_{ss}(n)$ 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 $\hat{\Delta}_{ws}(n)$ and $\hat{\Delta}_{ss}(n)$: one induced by the set complementation action on subsets of $[n]$ and another induced by the action on subsets of $[n]$ which replaces each $k\in[n]$ by $n+1-k$.