Higher topological complexity and its symmetrization

Basabe, Ibai¹ ; González, Jesús² ; Rudyak, Yuli B¹ ; Tamaki, Dai³

¹Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA
²Departamento de Matemáticas, CINVESTAV-IPN, A.P. 14-740, México City 07000, México
³Department of Mathematical Sciences, Shinshu University, Matsumoto 390-8621, Japan

Algebraic and Geometric Topology, Tome 14 (2014) no. 4, pp. 2103-2124

Résumé

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↪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 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.

Détail

DOI : 10.2140/agt.2014.14.2103

Classification : 55M30, 55R80
Keywords: Lusternik–Schnirelmann category, Švarc genus, topological complexity, motion planning, configuration spaces

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