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

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)115-167
Number of pages53
JournalFundamenta Mathematicae
Volume213
Issue number2
DOIs
Publication statusPublished - 2011

Fingerprint

Orbit Space
Compact Lie Group
Subgroup
Theorem
Local Cohomology
G-space
Spectral Sequence
Homotopy
Vertical
Converge
Decompose

Cite this

Orbit spaces, Quillen’s Theorem A and Minami’s formula for compact Lie groups. / Libman, Assaf.

In: Fundamenta Mathematicae, Vol. 213, No. 2, 2011, p. 115-167.

Research output: Contribution to journalArticle

@article{448f6d4366ba401a8ce9a7cc374f6396,
title = "Orbit spaces, Quillen’s Theorem A and Minami’s formula for compact Lie groups",
abstract = "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.",
author = "Assaf Libman",
year = "2011",
doi = "10.4064/fm213-2-2",
language = "English",
volume = "213",
pages = "115--167",
journal = "Fundamenta Mathematicae",
issn = "0016-2736",
publisher = "Instytut Matematyczny",
number = "2",

}

TY - JOUR

T1 - Orbit spaces, Quillen’s Theorem A and Minami’s formula for compact Lie groups

AU - Libman, Assaf

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

U2 - 10.4064/fm213-2-2

DO - 10.4064/fm213-2-2

M3 - Article

VL - 213

SP - 115

EP - 167

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 2

ER -