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 → 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.

No abstract

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 → 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.

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak with the aim of constructing a cellular model of the configuration space of a sphere. Although the original aim was not achieved in the project, the notion of cellular stratified spaces turns out to be useful, at least, in the study of configuration spaces of graphs. In particular, the notion of totally normal cellular stratified spaces was used successfully in a joint work with the former students of the author [FMT15] to study the homotopy type of configuration spaces of graphs with a small number of vertices.Roughly speaking, totally normal cellular stratified spaces correspond to acyclic categories in the same way regular cell complexes correspond to posets.In this paper, we extend this correspondence by replacing cells by stellar cells and acyclic categories by topological acyclic categories.

No abstract