## Algebraic & Geometric Topology

### Higher topological complexity and its symmetrization

#### Abstract

We develop the properties of the 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.

#### Article information

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

Dates
Accepted: 4 January 2014
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715959

Digital Object Identifier
doi:10.2140/agt.2014.14.2103

Mathematical Reviews number (MathSciNet)
MR3331610

Zentralblatt MATH identifier
1348.55005

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715959

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