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Right-angled Artin groups, polyhedral products and the ${{\sf {TC}}}$-generating function

Part of: Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  11 June 2021

Jorge Aguilar-Guzmán ,
Jesús González Open the ORCID record for Jesús González [Opens in a new window]  and
John Oprea
Jorge Aguilar-Guzmán
Affiliation:
Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del I.P.N., México City 07000, Mexico ([email protected]; [email protected])
Jesús González
Affiliation:
Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del I.P.N., México City 07000, Mexico ([email protected]; [email protected])
John Oprea
Affiliation:
Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115, USA ([email protected])
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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

Axial mapgenerating functionLusternik–Schnirelmann categorypolyhedral productright-angled Artin grouptopological complexity

MSC classification

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 649 - 673
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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