Compact differentiable transformation groups on exotic spheres
type: Item
instance of: scholarly article
title: Compact differentiable transformation groups on exotic spheres
zbMATH Open document ID: 0808.57026
DOI: 10.1007/BF01459789
published in: Mathematische Annalen
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165214
review text: Exotic spheres constitute a natural family of fundamental testing spaces in the study of differentiable transformation groups. A systematic study of the symmetry properties of exotic spheres in \(\Sigma^ n\) is certainly also a natural way to improve our understanding of the rather mysterious structure of this family. We shall focus attention on a comparison study between large compact subgroups \(G\) of \(\text{Diff} (S^ n)\) and \(\text{Diff} (\Sigma^ n)\), as a natural testing ground for developing techniques in transformation groups which are truly sensitive to the change of differentiable structures. The notion of geometric weight system, introduced by Wu-yi Hsiang in the 1970's, enables us to analyze the possible imbeddings \(G \subset \text{Diff}(\Sigma^ n)\) by combinatorial means, in a way parallel to the classical representation theory. A simple measure of symmetry of a manifold \(M\) is the maximum dimension \(N(M)\) of compact Lie subgroups of \(\text{Diff}(M)\). The paper combines the technique of geometric weight system with the concordance classification of regular actions of classical groups to calculate the numbers \(N(\Sigma^ n)\) for all exotic spheres of Kervaire or Milnor type, and moreover, to prove that \(N(\Sigma^ n) < (3/2) (n+1)\) in the remaining cases \(\Sigma^ n \notin bP_{n + 1}\).
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57R60
zbMATH DE Number: 683149
zbMATH Keywords: exotic spheres
zbMATH Keywords: geometric weight system
zbMATH Keywords: compact Lie subgroups of \(\text{Diff}(M)\)
zbMATH Keywords: concordance
zbMATH Keywords: regular actions of classical groups
author: Eldar Straume
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 683149