nLab group action on an sphere

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 $S^0 \hookrightarrow S^4$

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

• C. T. C. Wall, Free actions of finite groups on spheres, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 (pdf)

• Alejandro Adem, Constructing and deconstructing group actions (arXiv:0212280)

Discussion of circle group-actions on spheres includes

• Yves Félix, John Oprea, Daniel Tanré, Algebraic Models in Geometry, Oxford University Press 2008

The subgroups of SO(8) which act freely on $S^7$ 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 $Spin(8)$ on the 7-sphere is in

• Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)

• Paul de Medeiros, José Figueroa-O'Farrill, Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)

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