Semifree \(S^ 3\)-actions and elliptic genera
The Atiyah-Hirzebruch theorem on vanishing of the \(\widehat {A}\)-genus [\textit{M. F. Atiyah} and \textit{F. Hirzebruch}, in: Essays Topol. Relat. Top., 18-28 (1970; Zbl 0193.52401)] asserts that if a compact connected smooth spin-manifold \(M\) admits a nontrivial smooth \(S^1\)-action, then \(\widehat{A} (M)=0\). There are generalizations of the Atiyah-Hirzebruch theorem to the effect that the results relate the characteristic numbers of a compact oriented smooth manifold to the existence of a smooth nontrivial \(S^1\)-action. In the paper under review, the author deals with a similar problem for semifree smooth \(S^3\)-actions. More precisely, he shows that the Pontryagin numbers of a compact oriented smooth manifold \(M\), which admits a semifree smooth \(S^3\)-action, can be computed from the Pontryagin numbers of certain quaternionic projective bundles over some components of the fixed point set \(F\). The proof makes use of the fact that the normal bundle of each connected component of \(F\) in \(M\) has the structure of a quaternionic vector bundle provided the action is semifree. Using the multiplicativity property for the elliptic genera, the author describes a relation between the elliptic genera of a manifold with a semifree \(S^3\)-action and the elliptic genera of the fixed point set. In particular, he obtains the vanishing theorem: if a compact connected smooth manifold \(M\) admits a nontrivial smooth semifree \(S^3\)-action, then \(\widehat {A} (M) =0\).
scientific article; zbMATH DE number 896624
scientific article