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 as an alternating sum of p-localised spectra for subgroups 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 E₂-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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-2-2,
author = {Assaf Libman},
title = {Orbit spaces, Quillen's Theorem A and Minami's formula for compact Lie groups},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {115-167},
zbl = {1244.55011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-2-2}
}
Assaf Libman. Orbit spaces, Quillen's Theorem A and Minami's formula for compact Lie groups. Fundamenta Mathematicae, Tome 215 (2011) pp. 115-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-2-2/