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 [8], 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 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 [17] 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)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).

Generating functions and topological complexity

“…The goal of this paper is to show that the above conjecture is false in the form it was stated in [1], namely we show that for a specific finite CWcomplex X the power series F X (x) is a rational function of the form (2) however the value P X (1) is distinct from cat(X).…”

“…Theorem 1 suggests how to produce an example contradicting the rationality conjecture as stated in [1]. Namely, suppose that X is a finite CW-complex satisfying TC r (X) = zcl r (X; k) for all large r although cl(H * (X; k)) < cat(X).…”

“…the TC-generating function of X. It was observed in [1] that in many examples this series represents a rational function of a special type. Based on M. Farber was partially supported by a grant from the Leverhulme Foundation.…”

“…The authors thank Fields Institute for Research in Mathematical Sciences for hospitality. these computations, it was conjectured in [1] that for any finite CW-complex X the power series F X (x) is a rational function of the form…”

Higher topological complexity of hyperbolic groups

“…Recall that a group Γ is said to be of type F (or geometrically finite) if it admits a finite model for KpΓ, 1q. Following [FO19], we say that a finite CW-complex X (resp. a group Γ of type F ) satisfies the rationality conjecture if the TC-generating function f X ptq (resp.…”