On higher analogs of topological complexity

TL;DR

This paper introduces a sequence of higher topological complexity invariants, extending Farber's original concept, and explores their symmetric versions as higher analogs of symmetric topological complexity, with potential applications in robotics.

Contribution

The paper defines a new series of numerical invariants _n(X) that generalize topological complexity and introduces their symmetric versions, expanding the theoretical framework.

Findings

Defined _n(X) as higher topological complexities

Established _2(X) = (X) and _n(X) _{n+1}(X)

Introduced symmetric versions of these invariants

Abstract

Farber introduced a notion of topological complexity $\TC(X)$ that is related to robotics. Here we introduce a series of numerical invariants $\TC_n(X), n=1,2, ...$ such that $\TC_2(X)=\TC(X)$ and $\TC_n(X)\le \TC_{n+1}(X)$. For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.