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2019 Bredon cohomology and robot motion planning
Michael Farber, Mark Grant, Gregory Lupton, John Oprea
Algebr. Geom. Topol. 19(4): 2023-2059 (2019). 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=π1(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×π–equivariant map of classifying spaces

E ( π × π ) E D ( π × π )

can be equivariantly deformed into the k–dimensional skeleton of ED(π×π). The symbol E(π×π) denotes the classifying space for free actions and ED(π×π) denotes the classifying space for actions with isotropy in the family D of subgroups of π×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) max{3,cdD(π×π)}, where cdD(π×π) denotes the cohomological dimension of π×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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Michael Farber. Mark Grant. Gregory Lupton. John Oprea. "Bredon cohomology and robot motion planning." Algebr. Geom. Topol. 19 (4) 2023 - 2059, 2019. https://doi.org/10.2140/agt.2019.19.2023

Information

Received: 16 May 2018; Revised: 29 December 2018; Accepted: 15 January 2019; Published: 2019
First available in Project Euclid: 22 August 2019

zbMATH: 07121519
MathSciNet: MR3995023
Digital Object Identifier: 10.2140/agt.2019.19.2023

Subjects:
Primary: 55M10
Secondary: 55M99

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 4 • 2019
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