Compact differentiable transformation groups on exotic spheres
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 \(\Sigma^ n \notin bP_{n + 1}\).
compact Lie subgroups of \(\text{Diff}(M)\)
scientific article; zbMATH DE number 683149
scientific article