Bredon cohomology and robot motion planning

Abstract

In this paper we study the topological invariant $\mathsf{TC}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and $\mathsf{TC}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number $\mathsf{TC}(X)$ depends only on the fundamental group $\pi ={\pi }_1(X)$ and we denote it $\mathsf{TC}(\pi )$. We prove that $\mathsf{TC}(\pi )$ can be characterised as the smallest integer $k$ such that the canonical $\pi \times \pi$-equivariant map of classifying spaces $$E(\pi \times \pi )\to E_{\mathcal{D}}(\pi \times \pi )$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal{D}}(\pi \times \pi )$. The symbol $E(\pi \times \pi )$ denotes the classifying space for free actions and $E_{\mathcal{D}}(\pi \times \pi )$ denotes the classifying space for actions with isotropy in a certain family $\mathcal{D}$ of subgroups of $\pi \times \pi$. Using this result we show how one can estimate $\mathsf{TC}(\pi )$ in terms of the equivariant Bredon cohomology theory. We prove that $\mathsf{TC}(\pi )\le \max \{3,{\mathrm{cd}}_{\mathcal{D}}(\pi \times \pi )\},$ where ${\mathrm{cd}}_{\mathcal{D}}(\pi \times \pi )$ denotes the cohomological dimension of $\pi \times \pi$ with respect to the family of subgroups $\mathcal{D}$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal{D}$.

This revision contains a few additional comments, among them is Corollary 3.5.4

References in corpus (3)

• On the Berstein-Svarc Theorem in dimension 2
• Motion planning in tori
• Topological Complexity of the Klein bottle

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• Topological complexity of unordered configuration spaces of surfaces
• Higher topological complexity of hyperbolic groups
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• On the sectional category of subgroup inclusions and Adamson cohomology theory
• An upper bound for higher topological complexity and higher strongly equivariant complexity
• Various Topological Complexities of Small Covers and Real Bott Manifolds
• Topological complexity of monotone symplectic manifolds