Bredon cohomology and robot motion planning

Abstract

We study the topological invariant [math] reflecting the complexity of algorithms for autonomous robot motion. Here, [math] stands for the configuration space of a system and [math] is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in [math] . We focus on the case when the space [math] is aspherical; then the number [math] depends only on the fundamental group [math] and we denote it by [math] . We prove that [math] can be characterised as the smallest integer [math] such that the canonical [math] –equivariant map of classifying spaces\n¶\n<math display="block">\n<mrow>\n<mi>E</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>π</mi>\n<mo class="MathClass-bin">×</mo>\n<mi>π</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">→</mo>\n<msub>\n<mrow>\n<mi>E</mi>\n</mrow>\n<mrow>\n<mi mathvariant="bold-script">D</mi>\n</mrow>\n</msub>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>π</mi>\n<mo class="MathClass-bin">×</mo>\n<mi>π</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n</mrow>\n</math>\n¶ can be equivariantly deformed into the [math] –dimensional skeleton of [math] . The symbol [math] denotes the classifying space for free actions and [math] denotes the classifying space for actions with isotropy in the family [math] of subgroups of [math] which are conjugate to the diagonal subgroup. Using this result we show how one can estimate [math] in terms of the equivariant Bredon cohomology theory. We prove that [math] , where [math] denotes the cohomological dimension of [math] with respect to the family of subgroups [math] . 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 [math] .