Right-angled Artin groups, polyhedral products and the TC-generating function

Abstract

For a graph Γ, let K(HΓ, 1) denote the Eilenberg-Mac Lane space associated to the right-angled Artin (RAA) group HΓ defined by Γ. We use the relationship between the combinatorics of Γ and the topological complexity of K(HΓ, 1) to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer n, we construct a graph On whose TC-generating function has polynomial numerator of degree n. Additionally, motivated by the fact that K(HΓ, 1) can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

On the LS-category and topological complexity of projective product spaces

“…The upper bound in Theorem 1.3 is obtained through the construction of an explicit motion planner which uses the characterization of TC(P n 1 ) in terms of non-singular maps due to M. Farber, S. Yuzvinsky and S. Tabachnikov [8]. We note that, using a strong version of non-singular maps, an explicit motion planner for polyhedral products of real projective spaces has recently been constructed in [1].…”