Algebraic & Geometric Topology

Bredon cohomology and robot motion planning

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

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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.

Article information

Source
Algebr. Geom. Topol., Volume 19, Number 4 (2019), 2023-2059.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1566439280

Digital Object Identifier
doi:10.2140/agt.2019.19.2023

Mathematical Reviews number (MathSciNet)
MR3995023

Subjects
Primary: 55M10: Dimension theory [See also 54F45]
Secondary: 55M99: None of the above, but in this section

Keywords
topological complexity aspherical spaces Bredon cohomology

Citation

Farber, Michael; Grant, Mark; Lupton, Gregory; Oprea, John. Bredon cohomology and robot motion planning. Algebr. Geom. Topol. 19 (2019), no. 4, 2023--2059. doi:10.2140/agt.2019.19.2023. https://projecteuclid.org/euclid.agt/1566439280


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