Higher topological complexity and its symmetrization

We develop the properties of the nth sequential topological complexity TCn, 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 TCn(X) to the Lusternik-Schnirelmann category of cartesian powers of X, to the cup length of the diagonal embedding X hooked right arrow X-n, and to the ratio between homotopy dimension and connectivity of X. We fully compute the numerical value of TCn for products of spheres, closed 1-connected symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized versions of TCn(X). 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.