\(KO\)-obstructions of non-abelian group action on spin manifolds (Q1371946)

If a compact Lie group \(G\) of positive dimension acts non-trivially on a closed spin manifold \(M\), then by a theorem of \textit{M. Atiyah} and \textit{F. Hirzebruch} [Essays Topol. Relat. Top., 18-28 (1970; Zbl 0193.52401)] the \(\widehat A\)-genus of \(M\) vanishes. This can be viewed as a \(K\)-theoretical obstruction for the existence of \(G\)-actions. Now, in the case that \(G\) is non-abelian, the article proves \(KO\)-theoretical obstructions for the existence of effective \(G\)-actions. There are interesting relations to the existence of metrics of positive scalar curvature.