Higher topological complexity and its symmetrization

By I. Basabe, Jesús González, Yuli B. Rudyak, D. Tamaki. 2010

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We develop the properties of the n th 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 ,! 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. 55M30; 55R80

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/227bbf42f814e391a1a3252ae30feaa2083c159c