Higher topological complexity and its symmetrization

Abstract

We develop the properties of the $n$-th sequential topological complexity $T{C}_n$, 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 $T{C}_n(X)$ to the Lusternik-Schnirelmann category of cartesian powers of $X$, to the cup-length of the diagonal embedding $X\hookrightarrow X^n$, and to the ratio between homotopy dimension and connectivity of $X$. We fully compute the numerical value of $T{C}_n$ for products of spheres, closed 1-connected symplectic manifolds, and quaternionic projective spaces. Our study includes two symmetrized versions of $T{C}_n(X)$. The first one, unlike Farber-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.

The ideas about cellular stratified spaces and its application to the homotopy dimension of configuration spaces on spheres have been removed from this version. The title has changed accordingly. 19 pages. Submitted for publication

References in corpus (4)

• Topological Complexity is a Fibrewise L-S Category
• Symmetric products, duality and homological dimension of configuration spaces
• Symmetric topological complexity of projective and lens spaces
• Estimating the higher symmetric topological complexity of spheres

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• A Mapping Theorem for Topological Complexity
• Multitasking collision-free motion planning algorithms in Euclidean spaces
• On sequential versions of distributional topological complexity
• Equivariant parametrized topological complexity
• Sequential collision-free optimal motion planning algorithms in punctured Euclidean spaces
• Higher (equivariant) topological complexity of Milnor manifolds
• LS-category and sequential topological complexity of symmetric products
• On the higher topological complexity of manifolds with abelian fundamental group
• An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs