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Artin relation for smooth representations.
1Department of Mathematics, University of Chicago, Chicago, Illinois 60637.
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.
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.