The signature and the elliptic genus of \(\pi_{2}\)-finite manifolds with circle actions.
type: Item
instance of: scholarly article
title: The signature and the elliptic genus of \(\pi_{2}\)-finite manifolds with circle actions.
zbMATH Open document ID: 1039.57018
DOI: 10.1016/S0166-8641(03)00260-8
author: Haydeé Herrera
author: Rafael Herrera
published in: Topology and its Applications
publication date: 2004-02-03T00:00:00Z
review text: \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.
reviewed by: Július Korbaš
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 58J26
zbMATH DE Number: 2037257
zbMATH Keywords: elliptic genus
zbMATH Keywords: signature
zbMATH Keywords: Â-genus
zbMATH Keywords: Pontrjagin characteristic numbers
zbMATH Keywords: circle action
zbMATH Keywords: \(\pi_ 2\)-finite manifold
zbMATH Keywords: non-spin manifold
zbMATH Keywords: Dirac operator
zbMATH Keywords: homotopy group
zbMATH Keywords: fixed points
zbMATH Keywords: representation
zbMATH Keywords: rigidity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2037257