Bredon cohomology and robot motion planning

Farber, Michael; Grant, Mark; Lupton, Gregory; Oprea, John

doi:10.2140/agt.2019.19.2023

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Bredon cohomology and robot motion planning

Michael Farber, Mark Grant, Gregory Lupton and John Oprea

Algebraic & Geometric Topology 19 (2019) 2023–2059

DOI: 10.2140/agt.2019.19.2023

Abstract

We study the topological invariant $TC (X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and $TC (X)$ 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 $TC (X)$ depends only on the fundamental group $π = π_{1} (X)$ and we denote it by $TC (π)$ . We prove that $TC (π)$ can be characterised as the smallest integer $k$ such that the canonical $π \times π$ –equivariant map of classifying spaces

E (π \times π) \to E_{D} (π \times π)

can be equivariantly deformed into the $k$ –dimensional skeleton of $E_{D} (π \times π)$ . The symbol $E (π \times π)$ denotes the classifying space for free actions and $E_{D} (π \times π)$ denotes the classifying space for actions with isotropy in the family $D$ of subgroups of $π \times π$ which are conjugate to the diagonal subgroup. Using this result we show how one can estimate $TC (π)$ in terms of the equivariant Bredon cohomology theory. We prove that $TC (π) \leq max {3, {cd}_{D} (π \times π)}$ , where ${cd}_{D} (π \times π)$ denotes the cohomological dimension of $π \times π$ with respect to the family of subgroups $D$ . 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 (2017) are exactly the classes having Bredon cohomology extensions with respect to the family $D$ .

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Keywords

topological complexity, aspherical spaces, Bredon cohomology

Mathematical Subject Classification 2010

Primary: 55M10

Secondary: 55M99

References

Publication

Received: 16 May 2018

Revised: 29 December 2018

Accepted: 15 January 2019

Published: 16 August 2019

Authors

Michael Farber
School of Mathematical Sciences Queen Mary, University of London London United Kingdom

Mark Grant
Institute of Pure and Applied Mathematics University of Aberdeen Aberdeen United Kingdom

Gregory Lupton
Department of Mathematics Cleveland State University Cleveland, OH United States

John Oprea
Department of Mathematics Cleveland State University Cleveland, OH United States

Volume 19, issue 4 (2019)