Bredon cohomological dimensions for proper actions and Mackey functors.
type: Item
instance of: scholarly article
title: Bredon cohomological dimensions for proper actions and Mackey functors.
zbMATH Open document ID: 1337.20060
author: Dieter Degrijse
published in: Forum Mathematicum
publication date: 2015-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.1227
review text: The author compares cohomological dimension for a discrete group \(G\) for different types of cohomology groups built up using the family of finite subgroups of \(G\) and various types of coefficients. More concretely, let \(H^*_{\mathcal F}(G,-)\) denote the Bredon cohomology groups of \(G\) for the family \(\mathcal F\), of finite subgroups of \(G\). The author considers the following coefficients for these Bredon cohomology groups: contravariant functors from the orbit category \(\mathcal O_{\mathcal F}G\) to the category of abelian groups, Mod-\(\mathcal O_{\mathcal F}G\); the subcategory of the above consisting of Mackey functors, \(\mathcal M\); and the category of cohomological Mackey functors, \(co\mathcal M\). All of these give notions to the corresponding cohomological dimension for proper actions, \(\underline{cd}_{\mathcal N}(G)\), where \(\mathcal N\) is any of the above coefficients. On the other hand \textit{B. E. A. Nucinkis} defined a suitable cohomology theory, \(\mathcal FH^*(G;M)\) based on \(\mathcal F\) [in Topology Appl. 92, No. 2, 153-171 (1999; Zbl 0933.18010)]. The author proves the following relationship among the above cohomological dimensions: Theorem A. Let \(G\) be a discrete group of finite length \(\ell(G)\). Then \[ \mathcal Fcd(G)=\underline{cd}_{co\mathcal M}(G)=\underline{cd}_{\mathcal M}(G)\leq\underline{cd}(G)\leq\max_{H\in\mathcal F}\{\mathcal Fcd(W_G(H))+\ell(H)\}, \] where \(W_G(H)=N_G(H)/H\). In general the cohomological dimension does not behave well for short exact sequences of groups. However, the author has the following: Theorem B. Let \(1\to N\to G\to Q\to 1\) be a short exact sequence of groups such that every finite index overgroup of \(N\) in \(G\) has a uniform bound on the lengths of its finite subgroups that are not contained in \(N\). If \(\underline{cd}(G)<\infty\) or \(\underline{cd}_{co\mathcal M}(G)<\infty\), then \[ \underline{cd}_{co\mathcal M}(G)\leq\underline{cd}_{co\mathcal M}(N)+\underline{cd}_{co\mat…
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 55N25
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 18G20
zbMATH DE Number: 6505816
zbMATH Keywords: Bredon cohomological dimension
zbMATH Keywords: Bredon cohomology groups
zbMATH Keywords: Mackey functors
zbMATH Keywords: proper actions
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
OpenAlex ID: W2072894270
arXiv ID: 1305.1227
DOI: 10.1515/FORUM-2014-0003
Recommended article: On dimensions in Bredon homology.
Recommended article: The dominant dimension of cohomological Mackey functors
Recommended article: COHOMOLOGICAL DIMENSION OF MACKEY FUNCTORS FOR INFINITE GROUPS
Recommended article: A bound for the Bredon cohomological dimension
Recommended article: Dimension in Bredon cohomology
Recommended article: Bredon cohomology of finite dimensional \(C_p\)-spaces
Recommended article: Cohomological dimension and relative Cohen-Maculayness
Recommended article: Cohen-Macaulay homological dimensions
Recommended article: On equivalences for cohomological Mackey functors.
description: scientific article