Smooth actions of finite Oliver groups on spheres
type: Item
instance of: scholarly article
title: Smooth actions of finite Oliver groups on spheres
zbMATH Open document ID: 1019.57020
DOI: 10.1016/S0040-9383(02)00008-3
author: Masaharu Morimoto
author: Krzysztof Pawałowski
published in: Topology
publication date: 2003-03-11T00:00:00Z
review text: The paper is concerned with the characterization of the smooth (closed) manifolds \(F\) which occur as the fixed point sets of smooth actions of a given finite group \(G\) on spheres \(S\) and of the real \(G\)-vector bundles \(\nu\) over \(F\) which occur as the equivariant normal bundle. It extends the results of \textit{B.Oliver} [Topology 35, 583-615 (1996; Zbl 0861.57047)]. For a finite Oliver group \(G\) admitting a smooth one fixed point \(G\)-action on a sphere or on a disk and satisfying a gap condition, the answer is given by the Oliver obstruction of \(\nu \) in terms of real \(K\)-theory. Specific criteria are derived for finite perfect groups and for Oliver groups with a \(pqr\)-cyclic quotient. In the latter case, one can realise any sequence of Chern (resp.~Pontryagin numbers) of a smooth manifold as those of a \(G\)-fixed point set. The author's main contributions rely on equivariant thickening and surgery, an equivariant bundle extension method and an equivariant bundle subtraction property.
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57R67
zbMATH DE Number: 1880020
zbMATH Keywords: Oliver group
zbMATH Keywords: gap group
zbMATH Keywords: fixed point set
zbMATH Keywords: Oliver obstruction
zbMATH Keywords: equivariant surgery
zbMATH Keywords: equivariant thickening
reviewed by: Martin Raussen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1880020