Orbit spaces, Quillen’s Theorem A and Minami’s formula for compact Lie groups

Let $G$ be a compact Lie group. We present a criterion for the orbit spaces of two $G$-spaces to be homotopy equivalent and use it to obtain a quick proof of Webb's conjecture for compact Lie groups. We establish two Minami type formulae which present the $p$-localised spectrum
$\Sigma^\infty BG_+$ as an alternating sum of $p$-localised spectra $\Sigma^\infty BH_+$ for subgroup $H$ of $G$. The subgroups $H$ are calculated from the collections of the non-trivial elementary abelian $p$-subgroups of $G$ and the non-trivial $p$-radical subgroups of $G$. We also show that the Bousfield-Kan spectral sequences of the normaliser decompositions associated to these collections and to any $p$-local cohomology theory $h^*$, collapse at their $E_2$-pages to their vertical axes, and converge to $h^*(BG)$. An important tool is a topological version of Quillen's Theorem A which we prove.