Article contents
Right-angled Artin groups, polyhedral products and the
${{\sf {TC}}}$-generating function
Published online by Cambridge University Press: 11 June 2021
Abstract
For a graph $\Gamma$, let
$K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group
$H_{\Gamma }$ defined by
$\Gamma$. We use the relationship between the combinatorics of
$\Gamma$ and the topological complexity of
$K(H_{\Gamma },\,1)$ 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
$\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree
$n$. Additionally, motivated by the fact that
$K(H_{\Gamma },\,1)$ 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.
Keywords
MSC classification
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 649 - 673
- Copyright
- Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
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