Difference between free G-action and effective G-action?

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SUMMARY

The discussion clarifies the distinction between free G-actions and effective G-actions on manifolds. A free G-action occurs when the group action has no fixed points, meaning every group element acts without leaving any point unchanged. In contrast, an effective G-action requires that the only group element that acts as the identity is the identity element itself. The definitions referenced are sourced from Isham's work on the topic.

PREREQUISITES
  • Understanding of group actions in topology
  • Familiarity with manifold theory
  • Knowledge of the concepts of fixed points and identity elements in group theory
  • Basic comprehension of mathematical notation and definitions
NEXT STEPS
  • Study the definitions of group actions in "Geometry, Topology and Physics" by Isham
  • Explore the implications of free and effective actions in algebraic topology
  • Research examples of manifolds exhibiting free G-actions
  • Investigate the role of symmetry in effective G-actions on manifolds
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Mathematicians, particularly those specializing in topology and group theory, as well as students seeking to deepen their understanding of manifold theory and group actions.

pellman
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What is the difference between an effective G-action on a manifold and a free G-action on a manifold? Including a screenshot of the page from Isham where they are defined. Seem to be the same thing to me. Maybe I'm just not seeing the difference?
 

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Oops. Never mind. I got it. Don't see an option for deleting the thread.
 

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