Higher topological complexity of aspherical spaces

Abstract

In this article we study the higher topological complexity ${\sf TC}_r(X)$ in the case when $X$ is an aspherical space, $X=K(\pi, 1)$ and $r\ge 2$. We give a characterisation of ${\sf TC}_r(K(\pi, 1))$ in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper \cite{FGLO}, joint with M. Grant and G. Lupton, treats the special case $r=2$. We also obtain in this paper useful lower bounds for ${\sf TC}_r(\pi)$ in terms of cohomological dimension of subgroups of $\pi\times\pi\times \dots\times \pi$ ($r$ times) with certain properties. As an illustration of the main technique we find the higher topological complexity of the Higman's groups. 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 \cite{GGY} by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the ${\sf TC}$-generating function $\sum_{r=1}^\infty {\sf TC}_{r+1}(X)x^r$ encoding the values of the higher topological complexity ${\sf TC}_r(X)$ 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 ${\sf TC}$-generating function is a rational function of the form $\frac{P(x)}{(1-x)^2}$ where $P(x)$ is an integer polynomial with $P(1)={\sf cat}(X)$.