O(2) actions on the 5-sphere
type: Item
instance of: scholarly article
title: O(2) actions on the 5-sphere
zbMATH Open document ID: 0613.57018
DOI: 10.1007/BF01389237
author: Ronald Fintushel
author: Ronald Stern
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143430
review text: A smooth \(S^ 1\)-action is called pseudofree, if it is free except for finitely many exceptional orbits. The authors study the following question, which they call Montgomery-Yang problem: Must a pseudofree \(S^ 1\)-action on \(S^ 5\) have \(\leq 3\) exceptional orbits ? The answer is ''yes'' under the added hypothesis that the pseudofree \(S^ 1\)-action extends to an action of O(2). For a proof assume that the pseudofree \(S^ 1\)-action on \(S^ 5\) has n exceptional orbits. In a former paper [Ann. Math., II. Ser. 122, 335-364 (1985; Zbl 0602.57013)] the authors had associated an integral invariant R(X) to the orbit space \(X=S^ 5/S^ 1\) and proved that \(R(X)<0\). The proof used techniques of gauge theory. Again using techniques of gauge theory now it is proved that \(2n-9-R(X)<0\). Because R(X) is negative, this is only possible for \(n\leq 3\). Finally the authors construct pseudofree \(S^ 1\)-actions on rational homology 5-spheres from given orbit spaces.
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 81T08
zbMATH DE Number: 3992458
zbMATH Keywords: smooth \(S^ 1\)-action
zbMATH Keywords: pseudofree \(S^ 1\)-action on \(S^ 5\)
zbMATH Keywords: exceptional orbits
zbMATH Keywords: gauge theory
zbMATH Keywords: pseudofree \(S^ 1\)-actions on rational homology 5-spheres
reviewed by: K. H. Mayer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3992458