Algebraic & Geometric Topology

Higher topological complexity and its symmetrization

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

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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 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.

Article information

Algebr. Geom. Topol., Volume 14, Number 4 (2014), 2103-2124.

Received: 31 August 2013
Accepted: 4 January 2014
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space
Secondary: 55R80: Discriminantal varieties, configuration spaces

Lusternik–Schnirelmann category Švarc genus topological complexity motion planning configuration spaces


Basabe, Ibai; González, Jesús; Rudyak, Yuli B; Tamaki, Dai. Higher topological complexity and its symmetrization. Algebr. Geom. Topol. 14 (2014), no. 4, 2103--2124. doi:10.2140/agt.2014.14.2103.

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