Fake Lie groups with maximal tori. IV. (English) Zbl 0761.55013
This paper is a continuation of a series of papers on fake Lie groups of 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.
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.
Reviewer: D.Notbohm
MSC:
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 55P15 | Classification of homotopy type |
| 55R15 | Classification of fiber spaces or bundles in algebraic topology |
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