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Higher topological complexity and its symmetrization

Ibai Basabe, Jesús González, Yuli B Rudyak and Dai Tamaki

Algebraic & Geometric Topology 14 (2014) 2103–2124

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 XXn, 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.

Lusternik–Schnirelmann category, Švarc genus, topological complexity, motion planning, configuration spaces
Mathematical Subject Classification 2010
Primary: 55M30
Secondary: 55R80
Received: 31 August 2013
Accepted: 4 January 2014
Published: 28 August 2014
Ibai Basabe
Department of Mathematics
University of Florida
358 Little Hall
Gainesville, FL 32611-8105
Jesús González
Departamento de Matemáticas
A.P. 14-740
México City 07000
Yuli B Rudyak
Department of Mathematics
University of Florida
358 Little Hall
Gainesville, FL 32611-8105
Dai Tamaki
Department of Mathematical Sciences
Shinshu University
Matsumoto 390-8621