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 .
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