Simplicial and combinatorial versions of higher symmetric topological complexity

Abstract

In this paper, we introduce higher symmetric simplicial complexity SCnΣ(K) of a simplicial complex K and higher symmetric combinatorial complexity CCnΣ(P) of a finite poset P. These are simplicial and combinatorial approaches to symmetric motion planning of Basabe - González - Rudyak - Tamaki. We prove that the symmetric simplicial complexity SCnΣ(K) is equal to symmetric topological complexity TCnΣ(|K|) of the geometric realization of K and the symmetric combinatorial complexity CCnΣ(P) is equal to symmetric topological complexity TCnΣ(|K(P)|) of the geometric realization of the order complex of P.