Bredon cohomology and robot motion planning

Michael Farber (Corresponding Author), Mark Grant, Gregory Lupton, John Oprea

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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


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.
Original languageEnglish
Pages (from-to)2023-2059
Number of pages37
JournalAlgebraic & Geometric Topology
Issue number4
Publication statusPublished - 16 Aug 2019

Bibliographical note

Acknowledgements Farber was partially supported by the EPSRC, by the IIAS and
by the Marie Curie Actions, FP7, in the frame of the EURIAS Fellowship Programme.
Lupton and Oprea were partially supported by grants from the Simons Foundation
(# 209575 and # 244393).
This research was supported through the programme Research in pairs by the Mathematisches Forschungsinstitut Oberwolfach in 2017.


  • topological complexity
  • aspherical spaces
  • Bredon cohomology


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