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Artin relation for smooth representations.

K H Dovermann1, T Petrie

  • 1Department of Mathematics, University of Chicago, Chicago, Illinois 60637.

Proceedings of the National Academy of Sciences of the United States of America
|October 1, 1980
PubMed
Summary

Smooth representations of finite groups on spheres are studied. The dimension of fixed sets is a universal function of subgroup fixed sets only for noncyclic p-groups, not cyclic ones.

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Area of Science:

  • Mathematics
  • Algebraic Topology
  • Group Theory

Background:

  • Smooth group actions on spheres are fundamental in topology.
  • Understanding fixed set dimensions provides insights into group actions.

Purpose of the Study:

  • To determine conditions under which the dimension of fixed sets in smooth group representations is a universal function.
  • To identify specific types of finite groups (G) that exhibit this property.

Main Methods:

  • Investigating smooth representations of finite groups (G) acting on closed homotopy spheres (S).
  • Analyzing the relationship between the dimension of the fixed set S(G) and the dimensions of fixed sets S(H) for proper subgroups (H).

Main Results:

  • A function h(G) is established such that dimension S(G) = h(G){dimension S(H) for H proper subgroup of G} if and only if G has prime power order and is noncyclic.
  • This universality holds exclusively for noncyclic p-groups.

Conclusions:

  • The study characterizes finite groups whose smooth representations on spheres have universal fixed set dimensions.
  • Results are contrasted with existing theorems concerning orthogonal representations, like Artin's theorem.

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