nLab
group action on an sphere

Contents

Idea

The possible actions of well-behaved topological groups (such as compact Lie groups) on topological n-spheres display various interesting patterns in their classification.

This entry is meant to eventually list and discuss some of these. For the moment it mainly just collects some references.

Properties

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Proposition

Given an continuous action of the circle group on the topological 4-sphere, its fixed point space is of one of two types:

  1. either it is the 0-sphere S 0S 4S^0 \hookrightarrow S^4

  2. or it has the rational homotopy type of an even-dimensional sphere.

(Félix-Oprea-Tanré 08, Example 7.39)

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References

Discussion of free group actions on spheres by finite groups includes

Discussion of circle group-actions on spheres includes

The subgroups of SO(8) which act freely on S 7S^7 have been classified in

and lifted to actions of Spin(8) in

Further discussion of these actions of Spin(8)Spin(8) on the 7-sphere is in

where they are related to the black M2-brane BPS-solutions of 11-dimensional supergravity at ADE-singularities.