Higher topological complexity and its symmetrization

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

doi:10.2140/agt.2014.14.2103

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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

DOI: 10.2140/agt.2014.14.2103

Abstract

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We develop the properties of the $n^{th}$ 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 ${TC}_{n} (X)$ 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 ${TC}_{n} (X)$ . 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 55M30

Secondary: 55R80

References

Publication

Received: 31 August 2013

Accepted: 4 January 2014

Published: 28 August 2014

Authors

Ibai Basabe
Department of Mathematics University of Florida 358 Little Hall Gainesville, FL 32611-8105 USA

Jesús González
Departamento de Matemáticas CINVESTAV-IPN A.P. 14-740 México City 07000 México

Yuli B Rudyak
Department of Mathematics University of Florida 358 Little Hall Gainesville, FL 32611-8105 USA

Dai Tamaki
Department of Mathematical Sciences Shinshu University Matsumoto 390-8621 Japan

Volume 14, issue 4 (2014)