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Right-angled Artin groups, polyhedral products and the -generating function
Proceedings of the Royal Society of Edinburgh Section A: Mathematics ( IF 1.3 ) Pub Date : 2021-06-11 , DOI: 10.1017/prm.2021.26
Jorge Aguilar-Guzmán , Jesús González , John Oprea

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.



中文翻译:

直角 Artin 群、多面体积和生成函数

对于图$\Gamma$,令$K(H_{\Gamma },\,1)$表示与直角 Artin (RAA) 群$H_{\Gamma }$相关的 Eilenberg–Mac Lane 空间,定义为$\伽玛$。我们使用$\Gamma$的组合和$K(H_{\Gamma },\,1)$的拓扑复杂性之间的关系来解释并推广到更高的 TC 领域,Dranishnikov 的观察覆盖空间可以大于基础空间。在此过程中,对于任何正整数$n$,我们构造一个图$\mathcal {O}_n$ ,其 TC 生成函数具有次数为$n$的多项式分子. 此外,由于$K(H_{\Gamma },\,1)$可以实现为多面体积,我们研究了更一般的多面体积空间的 LS 类别和拓扑复杂性。特别是,我们使用强轴映射的概念,以便在许多情况下对多面体乘积的拓扑复杂性进行估计,该多面体乘积的因子是实射影空间。我们的估计显示了 RAA 组中不存在的混合 cat-TC 现象。

更新日期:2021-06-11
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