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Differentiable pseudo-free circle actions.

D Montgomery1, C T Yang

  • 1Institute for Advanced Study, Princeton, N.J. 08540.

Proceedings of the National Academy of Sciences of the United States of America
|May 1, 1971
PubMed
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This study classifies pseudofree differentiable actions on homotopy seven spheres. Unlike linear actions, these actions can have any finite number of exceptional orbits with cyclic isotropy groups.

Area of Science:

  • Topology
  • Differential Geometry
  • Group Actions

Background:

  • Pseudofree group actions are defined by specific orbit and isotropy group properties.
  • Homotopy seven spheres are manifolds with rich topological structures.
  • Understanding group actions is crucial in various fields of mathematics.

Purpose of the Study:

  • To classify all pseudofree differentiable actions of the circle group (G) on homotopy seven spheres.
  • To analyze the structure of exceptional orbits and their isotropy groups.

Main Methods:

  • Utilizing concepts from differential geometry and algebraic topology.
  • Investigating the properties of orbits and isotropy subgroups under pseudofree actions.
  • Classifying actions based on the number and nature of exceptional orbits.

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Main Results:

  • All pseudofree differentiable actions of the circle group on homotopy seven spheres are classified.
  • It is shown that there can be any finite number of exceptional orbits.
  • The isotropy group on exceptional orbits is a finite cyclic group Z(k) where k > 1.

Conclusions:

  • The classification provides a comprehensive understanding of pseudofree actions on these specific manifolds.
  • The findings contrast with the linear case, highlighting unique aspects of differentiable actions.
  • This work contributes to the study of transformation groups and their geometric manifestations.