Higher topological complexity of aspherical spaces

Abstract

In this article we study the higher topological complexity TCr(X) in the case when X is an aspherical space, X=K(π,1) and r≥2. We give a characterisation of TCr(K(π,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(π) in terms of cohomological dimension of subgroups of π×π×…×π (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 ∑r=1∞TCr+1(X)xr 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).