On the Topology of Weakly and Strongly Separated Set Complexes

Details On the Topology of Weakly and Strongly Separated Set Complexes On the Topology of Weakly and Strongly Separated Set Complexes

2011-10-05

arxiv.org/abs/1110.0880v1

Mathematics

Combinatorics

Scientific paper

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 $n+1-k$.

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