Icosahedral group actions on \(\mathbb{R}{}^ 3\)
type: Item
instance of: scholarly article
title: Icosahedral group actions on \(\mathbb{R}{}^ 3\)
zbMATH Open document ID: 0771.57014
DOI: 10.1007/BF02100611
author: Reinhard Schultz
author: Slawomir Kwasik
published in: Inventiones Mathematicae
publication date: 1993-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/144000
review text: The main result of this paper is Theorem B: Every smooth action of a compact Lie group \(G\) on \(\mathbb{R}^ 3\) or \(D^ 3\) is differentiably equivalent to a linear action. A similar conclusion holds for smooth actions on \(S^ 3\) with nonempty fixed point sets. The theorem had been well-known for actions of positive-dimensional compact Lie groups and was proved earlier for orientation-preserving actions of finite groups non-isomorphic to \(A_ 5\) in [\textit{W. H. Meeks III} and \textit{S.-T. Yau}, Chapt. IX in ``The Smith conjecture'', Pure Appl. Math. 112, 167-180 (1984; Zbl 0599.57007)]. Hence, the authors have to prove the main result for the group \(A_ 5\) and to extend the techniques to the non-orientation preserving case. Using a stabilization with copies of the product cobordism \(T^ 2\times [0,1]\) on an induced action on \(S^ 2\times [0,1]\), the authors show that a smooth \(A_ 5\)-action on \(S^ 3\) with a nonempty fixed point set has an equivariant Heegaard splitting of genus \(60=| A_ 5|\). If one fixed point is removed from \(S^ 3\) the induced action is then shown to be equivariantly diffeomorphic to an invariant open neighborhood of the origin in \(\mathbb{R}^ 3\) with an orthogonal action. Standard cancellation techniques then show the topological equivalence to an orthogonal action. Essentially by proving the uniqueness of equivariant smoothings of 3-manifolds, smooth equivalence to an orthogonal action can be concluded, too.
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 149679
zbMATH Keywords: smooth group action
zbMATH Keywords: icosahedral group
zbMATH Keywords: infinite process
zbMATH Keywords: equivariant smoothing
reviewed by: Martin Raussen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 149679