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)→ED(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 torsionfree hyperbolic groups as well as all torsionfree 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.
E(π×π)→ED(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 torsionfree hyperbolic groups as well as all torsionfree 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 language  English 

Pages (fromto)  20232059 
Number of pages  37 
Journal  Algebraic & Geometric Topology 
Volume  19 
Issue number  4 
DOIs  
Publication status  Published  16 Aug 2019 
Bibliographical note
Acknowledgements Farber was partially supported by the EPSRC, by the IIAS andby 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.
Keywords
 topological complexity
 aspherical spaces
 Bredon cohomology
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Mark Grant
 School of Natural & Computing Sciences, Mathematical Science  Senior Lecturer
Person: Academic