Fake Lie groups with maximal tori. IV
type: Item
instance of: scholarly article
title: Fake Lie groups with maximal tori. IV
zbMATH Open document ID: 0761.55013
DOI: 10.1007/BF01934316
author: Dietrich Notbohm
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164992
review text: This paper is a continuation of a series of papers on fake Lie groups of \textit{L. Smith} and the author [ibid. 288, 637-661, 663-673 (1990; Zbl 0697.55006 and Zbl 0697.55007); ibid. 290, 629-642 (1991; Zbl 0741.55004)]. A fake Lie group of type \(G\), \(G\) a compact connected Lie group, is a finite loop space \(X\) such that the classifying space \(BX\) has the same genus as \(BG\); i.e., both spaces are \(p\)-local equivalent for any prime \(p\). For such spaces there exists a notion of a maximal torus. A map \(BT_ X\to BX\), \(T_ X\) a torus, is a maximal torus of \(X\) if the homotopy fiber is equivalent to a finite CW-complex and if \(\text{rank}(T_ X)=\text{rank}(X)\). The main result classifies the fake Lie groups with maximal torus. A fake Lie group \(X\) of type \(G\) has a maximal torus iff there exists a compact connected Lie group \(H\) such that \(BX\) and \(BH\) are homotopy-equivalent. If \(G\) is simply connected or if \(G\) is simple, then \(H=G\), i.e., \(BX\) and \(BG\) are equivalent.
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55R15
zbMATH DE Number: 126359
zbMATH Keywords: fake Lie groups
zbMATH Keywords: finite loop space
zbMATH Keywords: classifying space
zbMATH Keywords: genus
zbMATH Keywords: maximal torus
Wikidata QID: Q115393045
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 126359