The signature and the elliptic genus of \(\pi_{2}\)-finite manifolds with circle actions.
\textit{M. Atiyah} and \textit{F. Hirzebruch} [Essays Topol. Relat. Top., Mem. dediés a Georges de Rham 18--28 (1970; Zbl 0193.52401)] proved that if \(M\) is a closed connected spin manifold which admits a non-trivial smooth circle action, then the Â-genus of \(M\) vanishes. \textit{P. Landweber} and \textit{R. Stong} [Topology 27, No. 2, 145--161 (1988; Zbl 0647.57013)] proved that if \(M\) is a closed spin manifold with a smooth circle action, and the action is of odd type, then the signature of \(M\) vanishes. The authors prove these and similar results for oriented connected closed even-dimensional smooth manifolds with a non-trivial smooth circle action, which are not required to be spin manifolds, but, instead, the second homotopy group is supposed to be finite.
scientific article; zbMATH DE number 2037257
scientific article