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

Assaf Libman¹

¹ Department of Mathematical Sciences King's College University of Aberdeen Aberdeen AB24 3UE, Scotland, U.K.

Fundamenta Mathematicae, Tome 213 (2011) no. 2, pp. 115-167

Résumé

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 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 $E_2$-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.

Détail

DOI : 10.4064/fm213-2-2

Keywords: compact lie group present criterion orbit spaces g spaces homotopy equivalent obtain quick proof webbs conjecture compact lie groups establish minami type formulae which present p localised spectrum sigma infty alternating sum p localised spectra sigma infty subgroups subgroups calculated collections non trivial elementary abelian p subgroups non trivial p radical subgroups bousfield kan spectral sequences normaliser decompositions associated these collections p local cohomology theory * collapse their pages their vertical axes converge * important tool topological version quillens theorem which prove