Higher topological complexity and its symmetrization

Abstract

We develop the properties of the $n^{\text{ th}}$ 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 ${TC}_n\left(X\right)$ 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 ${TC}_n\left(X\right)$. The first one, unlike Farber and 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.