Higher topological complexity of aspherical spaces

In this article we study the higher topological complexity TCr(X) in the case when X is an aspherical space, X = K(pi, 1) and r >= 2. We give a characterisation of TCr(K(pi, 1)) in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper [7], joint with M. Grant and G. Lupton, treats the special case r = 2. We also obtain in this paper useful lower bounds for TCr(pi) in terms of cohomological dimension of subgroups of pi x pi x ... x pi (r times) with certain properties. As an illustration of the main technique we find the higher topological complexity of Higman's group. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in [16] by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the TC-generating function Sigma(infinity)(r=1) TCr+1(X)x(r) encoding the values of the higher topological complexity TCr(X) for all values of r. We show that in many examples (including the case when X = K(H, 1) with H being a RAA group) the TC-generating function is a rational function of the form P(x)/(1 - x)(2) where P(x) is an integer polynomial with P(1) = cat(X). (C) 2019 Published by Elsevier B.V.