Bredon cohomological dimensions for proper actions and Mackey functors.
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 \[ \underline{cd}_{co\mathcal M}(G)\leq\underline{cd}_{co\mathcal M}(N)+\underline{cd}_{co\mathcal M}(Q). \] The author also explores some closure properties for the class of groups with finite Bredon cohomological dimension.
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