group action on an sphere




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.




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)



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

  • J. A. Wolf, Spaces of constant curvature, Publish or Perish, Boston, Third ed., 1974

and lifted to actions of Spin(8) in

  • S. Gadhia, Supersymmetric quotients of M-theory and supergravity backgrounds, PhD thesis, School of Mathematics, University of Edinburgh, 2007

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.

Last revised on December 6, 2019 at 10:05:59. See the history of this page for a list of all contributions to it.