Document Zbl 0796.57015 - zbMATH Open

From the author’s introduction: “If \(X\) is a finite homotopy associative \(H\)-space, then the functor \([-,X]\) takes values in the category of groups. Zabrodsky defined \(X\) to be homotopy nilpotent if \([-,X]\) takes values in nilpotent groups. By using Morava \(K\)-theories, Hopkins found cohomological criteria for a finite \(H\)-space to be homotopy nilpotent, and proved that \(H\)-spaces with no torsion in homology are homotopy nilpotent. Moreover, he conjectured the homotopy nilpotency of all homotopy associative finite \(H\)-spaces. However, V. K. Rao found that \(\text{Spin}(n)\), \(\text{SO}(n)\), \(n \geq 7\), and \(\text{SO}(3)\), \(\text{SO}(4)\) are not homotopy nilpotent. Using a result of Ravenel and Wilson and arguments and results of Rao, we find non-homotopy nilpotency for exceptional Lie groups with \(p\)-torsion in homology. Thus we obtain the following Theorem: Let \(G\) be a simply connected Lie group. Then for each prime \(p\), the \(p\)-localization \(G_{(p)}\) is homotopy nilpotent if and only if \(H^*(G)\) has no \(p\)-torsion.”.

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