2010/09/30 by Ibai Basabe, Jesus Gonzalez, Yuli B. Rudyak +1 · 1 citation
Mathematics · #math.AT #msc:55M30 #msc:55R80 #msc:55R05 #msc:57Q40 #msc:68T40
paper · pdf · doi:10.2140/agt.2014.14.2103
published as Algebr. Geom. Topol. 14 (2014) 2103-2124 · The ideas about cellular stratified spaces and its application to the homotopy dimension of configuration spaces on spheres have been removed from this version. The title has changed accordingly. 19 pages. Submitted for publication
arxiv created 2013/08/26 · arxiv updated 2014/11/11
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\hookrightarrow Xn, 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-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.