## Higher limits via subgroup complexes.(English)Zbl 1004.55008

Let $$G$$ be a finite group. A collection $${\mathcal C}$$ is a set of subgroups in $$G$$, closed under conjugation in $$G$$. Denote by $$|{\mathcal C}|$$ the nerve of the poset $${\mathcal C}$$ and by $${\mathbf O}_{\mathcal C}$$ the full subcategory of the orbit category of $$G$$ with objects the transitive $$G$$-sets with isotropy groups in $${\mathcal C}$$. The purpose of this paper is to give a new finite model for the higher derived functors $$\lim_{{\mathbf O}_{\mathcal C}}^*F= \{\lim_{{\mathbf O}_{\mathcal C}}^i \}_{i\geq 1}F$$ of the inverse limit when $$F$$ is a contravariant functor with domain $${\mathbf O}_{\mathcal C}$$ and codomain the category of $$\mathbb{Z}_{(p)}$$-modules. Such functors arise in the homotopy representation theory of compact Lie groups.
Theorem 1: (a) If $${\mathcal C}$$ is closed under passage to $$p$$-radical over-subgroups then $$\lim_{{\mathbf O}_{\mathcal C}}^* F=H_G^*(|{\mathcal C}|;{\mathcal F})$$ where $${\mathcal F}$$ is a $$G$$-local coefficient system on $$|{\mathcal C}|$$ naturally associated to $$F$$ and $${\mathcal C}$$. (Here $$H_G^*(-;{\mathcal F})$$ denotes the Bredon cohomology.) (b) If $$F$$ is a functor concentrated on conjugates of a single subgroup $$P$$ then $$\lim_{{\mathbf O}_{\mathcal C}}^iF= H_G^{i-1} (\operatorname{Hom}_{NP/P} \text{St}_* (NP/P);F(P))$$ where $$\text{St}_*(W)$$ denotes the Steinberg complex of $$W$$.
Theorem 2: (a) Let $${\mathcal C}$$ be as in Theorem 1. If for all $$P\in{\mathcal C}$$, the centralizer $$CP$$ acts trivially on $$F(P)$$ then, for any collection $${\mathcal C}'\subset{\mathcal C}$$ which contains all $$p$$-centric subgroups of $${\mathcal C}$$, $$\lim_{{\mathbf O}_{\mathcal C}}^*F= \lim_{{\mathbf O}_{{\mathcal C}'}}^*F= H_G^*(|{\mathcal C}'|;{\mathcal F})$$ where $${\mathcal F}$$ is a $$G$$-local coefficient system on $$|{\mathcal C}|$$ naturally associated to $$F$$ and $${\mathcal C}'$$. (b) If $$F$$ is a functor concentrated on conjugates of a single $$p$$-centric subgroup $$P$$ then $$\lim_{{\mathbf O}_{\mathcal C}}^iF= H_G^{i-1} (\operatorname{Hom}_{NP/PCP} \text{St}_* (NP/PCP); F(P))$$ where $$\text{St}_*(W)$$ denotes the Steinberg complex of $$W$$.
This last result is in a certain sense equivalent to a version of Quillen’s conjecture on the contractibility of the nerve of the collection of nontrivial $$p$$-subgroups. The author continues with the construction of a spectral sequence $$E_1^{i,j} \Rightarrow \lim_{{\mathbf O}_{\mathcal C}}^{i+j}F$$ where the term $$E_1^{i,j}$$ is explicitly defined by Theorems 1 and 2 applied to “atomic functors”. The paper ends with applications to group cohomology and other Mackey functors and an application to $$\operatorname {mod}p$$ decomposition of $$BG$$.

### MSC:

 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 18G10 Resolutions; derived functors (category-theoretic aspects) 55R37 Maps between classifying spaces in algebraic topology 20J06 Cohomology of groups 55N91 Equivariant homology and cohomology in algebraic topology

### Keywords:

orbit category; Mackey functor; Steinberg complex
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