Volume 14, issue 4 (2014)

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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
Abstract
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We develop the properties of the 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 ${TC}_{n}\left(X\right)$ 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 ${TC}_{n}\left(X\right)$. 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.

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