Higher topological complexity and its symmetrization

Abstract

We develop the properties of the [math] sequential topological complexity [math] , 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 [math] to the Lusternik–Schnirelmann category of cartesian powers of [math] , to the cup length of the diagonal embedding [math] , and to the ratio between homotopy dimension and connectivity of [math] . We fully compute the numerical value of [math] for products of spheres, closed [math] –connected symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized versions of [math] . The first one, unlike Farber and Grant’s symmetric topological complexity, turns out to be a homotopy invariant of [math] ; the second one is closely tied to the homotopical properties of the configuration space of cardinality- [math] subsets of [math] . Special attention is given to the case of spheres.