Bredon cohomology and robot motion planning

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 spacescan 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 a certain family D of subgroups of π × π. 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. United Kingdom

Higher topological complexity of aspherical spaces

“…Theorem 3.1 and its proof generalise the results of [8] where the case r = 2 was treated. It is known that the classifying space E D (G) admits a realisation as a G-CW-complex of dimension max{3, cd D (G)}, see [22].…”

“…On the face of it, there seems to be no connection between these topics, but in our main result Theorem 3.1 we will describe a path that unites them. In fact, Theorem 3.1 generalizes a similar connection that was displayed in [8] for TC itself. The emergence of equivariant topology as a player in the study of TC allows invariants of the subject such as Bredon cohomology (with respect to a family of subgroups) to be used to estimate TC.…”

“…Let π be a discrete group, r ≥ 2 an integer, and let K ⊂ π r = π × π · · · × π be a subgroup with the property that K ∩ H = 1 for any subgroup H ⊂ π r conjugate to the diagonal ∆ ⊂ π r . Then TC r (π) ≥ cd(K), (6) where cd(K) denotes the cohomological dimension of K. Theorem 2.1 generalises Corollary 3.5.4 from [8] (where the case r = 2 was covered) as well as the result of [18] where the class of subgroups of type K = A × B is considered assuming that r = 2.…”

“…In this paper we generalise the approach of [8] of using Bredon cohomology to detect essential cohomology classes. Namely we show that for any r ≥ 2 the existence of a nonzero cohomology class ξ ∈ H n (π r , M ) which can be extended to a Bredon cohomology class ξ ∈ H n D (π r , M ) (with respect to the family D of subgroups of π r which was described earlier) implies that TC r (π) ≥ n. For r = 2 this was proven in [8].…”

“…Some recent results concerning the invariant TC r (X) with r ≥ 2 can be found in [1], [15], [17], [30]. Also, the introduction to [8] gives an account of most of the significant recent developments regarding TC(X).…”