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Abstract

Let $p$ be a prime number. We prove that if $G$ is a compact Lie group with a non-trivial $p$-subgroup, then the orbit space $(B{\cal A}_p(G))/G$ of the classifying space of the category associated to the $G$-poset ${\cal A}_p(G)$ of all non-trivial elementary abelian $p$-subgroups of $G$ is contractible. This gives, for every $G$-CW-complex $X$ each of whose isotropy groups contains a non-trivial $p$-subgroup, a decomposition of $X/G$ as a homotopy colimit of the functor $X^{E_n}/(NE_0\cap \mathinner {\ldotp \ldotp \ldotp }\cap NE_n)$ defined over the poset $(\mathop {\rm sd}\nolimits {\cal A}_p(G))/G$, where $\mathop {\rm sd}\nolimits $ is the barycentric subdivision. We also investigate some other equivariant homotopy and homology decompositions of $X$ and prove that if $G$ is a compact Lie group with a non-trivial $p$-subgroup, then the map $EG\times _G B{\cal A}_p(G)\to BG$ induced by the $G$-map $B{\cal A}_p(G)\to *$ is a mod $p$ homology isomorphism.