Simplicial and combinatorial versions of higher symmetric topological complexity

TL;DR

This paper introduces new simplicial and combinatorial measures of symmetric topological complexity for complexes and posets, proving their equivalence to existing topological complexity notions.

Contribution

It defines higher symmetric simplicial and combinatorial complexities and establishes their equality with symmetric topological complexity of geometric realizations.

Findings

Higher symmetric simplicial complexity equals symmetric topological complexity of the geometric realization.

Higher symmetric combinatorial complexity equals symmetric topological complexity of the order complex.

Provides a combinatorial framework for symmetric motion planning problems.

Abstract

In this paper, we introduce higher symmetric simplicial complexity $SC_n^{\Sigma}(K)$ of a simplicial complex $K$ and higher symmetric combinatorial complexity $CC_n^{\Sigma}(P)$ of a finite poset $P$. These are simplicial and combinatorial approaches to symmetric motion planning of Basabe - Gonz\'{a}lez - Rudyak - Tamaki. We prove that the symmetric simplicial complexity $SC_n^{\Sigma}(K)$ is equal to symmetric topological complexity $TC_n^{\Sigma}(|K|)$ of the geometric realization of $K$ and the symmetric combinatorial complexity $CC_n^{\Sigma}(P)$ is equal to symmetric topological complexity $TC_n^{\Sigma}(|\mathcal{K}(P)|)$ of the geometric realization of the order complex of $P$.