$G$-functors, $G$-posets and homotopy decompositions of $G$-spaces
Volume 169 / 2001
Abstract
We describe a unifying approach to a variety of homotopy decompositions of classifying spaces, mainly of finite groups. For a group $G$ acting on a poset ${\bf W}$ and an isotropy presheaf $d:{\bf W}\rightarrow {\cal S}(G)$ we construct a natural $G$-map $ \mathop {\rm hocolim}\nolimits _{{\cal W}_d}G/d(-)\rightarrow |{\bf W}|$ which is a (non-equivariant) homotopy equivalence, hence $ \mathop {\rm hocolim}\nolimits _{{\cal W}_d}EG\times _GF_d \rightarrow EG\times _G|{\bf W}|$ is a homotopy equivalence. Different choices of $G$-posets and isotropy presheaves on them lead to homotopy decompositions of classifying spaces. We analyze higher limits over the categories associated to isotropy presheaves ${\cal W}_d$; in some important cases they vanish in dimensions greater than the length of ${\bf W}$ and can be explicitly calculated in low dimensions. We prove a cofinality theorem for functors $F:{\cal C}\rightarrow {\cal O}(G)$ into the category of $G$-orbits which guarantees that the associated map $\alpha _F:\mathop {\rm hocolim}\nolimits _{{\cal C}}EG\times _G F(-)\rightarrow BG$ is a mod-$p$-homology decomposition.
