We study the topological invariant reflecting the complexity of algorithms for autonomous robot motion. Here, stands for the configuration space of a system and is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in . We focus on the case when the space is aspherical; then the number depends only on the fundamental group and we denote it by . We prove that can be characterised as the smallest integer such that the canonical –equivariant map of classifying spaces
can be equivariantly deformed into the –dimensional skeleton of . The symbol denotes the classifying space for free actions and denotes the classifying space for actions with isotropy in the family of subgroups of which are conjugate to the diagonal subgroup. Using this result we show how one can estimate in terms of the equivariant Bredon cohomology theory. We prove that , where denotes the cohomological dimension of with respect to the family of subgroups . 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 .
"Bredon cohomology and robot motion planning." Algebr. Geom. Topol. 19 (4) 2023 - 2059, 2019. https://doi.org/10.2140/agt.2019.19.2023