Higher topological complexity and its symmetrization

TL;DR

This paper explores the properties of higher topological complexity and its symmetrized variants, linking them to classical invariants and computing explicit values for key manifolds, with implications for robotics motion planning.

Contribution

It introduces and analyzes the $n$-th sequential topological complexity $TC_n$, including its symmetrized versions, and computes these invariants for important classes of spaces.

Findings

$TC_n$ relates to Lusternik-Schnirelmann category and cup-length.

Explicit $TC_n$ values for spheres, symplectic manifolds, and quaternionic projective spaces.

A homotopy invariant symmetrized topological complexity is identified.

Abstract

We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of $TC_n(X)$ to the Lusternik-Schnirelmann category of cartesian powers of $X$, to the cup-length of the diagonal embedding $X\hookrightarrow X^n$, and to the ratio between homotopy dimension and connectivity of $X$. We fully compute the numerical value of $TC_n$ for products of spheres, closed 1-connected symplectic manifolds, and quaternionic projective spaces. Our study includes two symmetrized versions of $TC_n(X)$. The first one, unlike Farber-Grant's symmetric topological complexity, turns out to be a homotopy invariant of $X$; the second one is closely tied to the homotopical properties of the configuration space of cardinality-$n$ subsets of $X$. Special attention is given to the case of spheres.