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Homotopy nilpotent Lie groups have no torsion in homology. (English) Zbl 0880.22005

An \(H\)-space \(X\) is homotopy nilpotent if and only if the groups \([-,X]\) are always nilpotent. The author shows that if \(G\) is a compact connected Lie group that has \(p \)-torsion in homology, then \(G\) localized at \(p\) is not homotopy nilpotent. This implies that a connected Lie group is homotopy nilpotent if and only if it has no torsion in homology. Also proven is the fact that the localization \(G_{(p)}\) is homotopy nilpotent if and only if \(H(G; \mathbb{Z}_{(p)})\) is torsion free. The if-part is due to M. Hopkins.

MSC:

22E41 Continuous cohomology of Lie groups
55Q05 Homotopy groups, general; sets of homotopy classes
55P60 Localization and completion in homotopy theory
57T15 Homology and cohomology of homogeneous spaces of Lie groups
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