Fake Lie groups and maximal tori. I
type: Item
instance of: scholarly article
title: Fake Lie groups and maximal tori. I
zbMATH Open document ID: 0697.55006
DOI: 10.1007/BF01444556
author: Dietrich Notbohm
author: Larry Smith
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164758
review text: [Part II is reviewed below (see Zbl 0697.55007).] A finite loop space X is called a fake Lie group of type G, where G is a compact connected Lie group, if the classifying spaces BX and BG have the same genus. (Recall that two nilpotent spaces of finite type \(Y'\), \(Y''\) are said to have the same genus if their p-localizations are homotopy equivalent for all primes.) Using Rector's idea of a maximal torus for a finite loop space we prove that a fake Lie group is genuine iff it has a maximal torus. For each maximal torus for X there is a Weyl group. We prove that these all coincide in a strong sense with the classical Weyl group of G. Moreover we show that the homotopy fibre of a maximal torus f: BT\(\to BX\) is homotopy equivalent to G/T. Finally we show that a fake Lie group X has a maximal torus iff \(\tilde K(BX)\cong \tilde K(BG)\) as \(\lambda\)-rings.
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 4143031
zbMATH Keywords: finite loop space
zbMATH Keywords: fake Lie group
zbMATH Keywords: compact connected Lie group
zbMATH Keywords: classifying spaces
zbMATH Keywords: nilpotent spaces
zbMATH Keywords: genus
zbMATH Keywords: p-localizations
zbMATH Keywords: maximal torus for a finite loop space
zbMATH Keywords: Weyl group
zbMATH Keywords: lambda-rings
Wikidata QID: Q115393539
MaRDI profile type: Publication
cites work: Characteristic Classes and Homogeneous Spaces, I
cites work: Homotopy limits, completions and localizations
cites work: Evaluation of the transfer and the universal surgery classes
cites work: The realization of polynomial algebras as cohomology rings
cites work: Locally finite approximation of Lie groups. I
cites work: On the genus of generalized flag manifolds
cites work: PULLBACK THEOREMS AND PHANTOM MAPS
cites work: THE HOMOTOPY CLASSIFICATION OF SELF-MAPS OF INFINITE QUATERNIONIC PROJECTIVE SPACE
cites work: Maps between classifying spaces
cites work: Fake Lie groups with maximal tori. IV
cites work: Fake Lie groups and maximal tori. II
cites work: Subgroups of finite dimensional topological groups
cites work: Q4084973
cites work: Genetics of homotopy theory and the Adams conjecture
cites work: Lambda-rings, binomial domains, and vector bundles over cp(∞)
description: scientific article; zbMATH DE number 4143031