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In this paper we study the topological invariant ${\sf {TC}}(X)$ reflectingthe complexity of algorithms for autonomous robot motion. Here, $X$ stands forthe configuration space of a system and ${\sf {TC}}(X)$ is, roughly, theminimal number of continuous rules which are needed to construct a motionplanning algorithm in $X$. We focus on the case when the space $X$ isaspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group$\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$can be characterised as the smallest integer $k$ such that the canonical$\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \toE_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the$k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol$E(\pi\times\pi)$ denotes the classifying space for free actions and$E_{\mathcal D}(\pi\times\pi)$ denotes the classifying space for actions withisotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Usingthis result we show how one can estimate ${\sf TC}(\pi)$ in terms of theequivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3,{\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcalD}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ withrespect to the family of subgroups $\mathcal D$. We also introduce a Bredoncohomology refinement of the canonical class and prove its universality.Finally we show that for a large class of principal groups (which includes alltorsion free hyperbolic groups as well as all torsion free nilpotent groups)the essential cohomology classes in the sense of Farber and Mescher are exactlythe classes having Bredon cohomology extensions with respect to the family$\mathcal D$.

Bredon cohomology and robot motion planning