Semifree \(S^ 3\)-actions and elliptic genera
type: Item
instance of: scholarly article
title: Semifree \(S^ 3\)-actions and elliptic genera
zbMATH Open document ID: 0855.57032
DOI: 10.1016/0166-8641(96)00010-7
published in: Topology and its Applications
publication date: 1997-02-03T00:00:00Z
review text: 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 \(\widehat {A} (M) =0\).
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 58J26
zbMATH DE Number: 896624
zbMATH Keywords: \(\widehat {A}\)-genus
zbMATH Keywords: spin-manifold
zbMATH Keywords: \(S^ 1\)-action
zbMATH Keywords: characteristic numbers
zbMATH Keywords: \(S^ 3\)-actions
zbMATH Keywords: Pontryagin numbers
zbMATH Keywords: elliptic genera
zbMATH Keywords: fixed point set
author: K. H. Mayer
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
full work available at URL: 0166-8641(96)00010-7
OpenAlex ID: W2019719686
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cites work: \(S^ 1\)-actions and elliptic genera
cites work: Cobordism groups of semi-free \(S^ 1\)- and \(S^ 3\)-actions
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description: scientific article; zbMATH DE number 896624