Bredon cohomology and robot motion planning

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

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Abstract

In this paper 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 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 torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher [17] are exactly the classes having Bredon cohomology extensions with respect to the family D.
Original languageEnglish
JournalAlgebraic & Geometric Topology
Publication statusAccepted/In press - 15 Jan 2019

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Motion Planning
Classifying Space
Cohomology
Robot
Denote
Torsion-free Group
Subgroup
Equivariant Map
Cohomological Dimension
Free Action
Equivariant Cohomology
Hyperbolic Groups
Autonomous Robots
Topological Invariants
Isotropy
Nilpotent Group
Fundamental Group
Skeleton
Configuration Space
Universality

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Farber, M., Grant, M., Lupton, G., & Oprea, J. (Accepted/In press). Bredon cohomology and robot motion planning. Algebraic & Geometric Topology.

Bredon cohomology and robot motion planning. / Farber, Michael (Corresponding Author); Grant, Mark; Lupton, Gregory; Oprea, John.

In: Algebraic & Geometric Topology, 15.01.2019.

Research output: Contribution to journalArticle

Farber, Michael ; Grant, Mark ; Lupton, Gregory ; Oprea, John. / Bredon cohomology and robot motion planning. In: Algebraic & Geometric Topology. 2019.
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