nLab
group actions on spheres

Contents

Contents

Idea

The possible actions of well-behaved topological groups (such as compact Lie groups) on topological or smooth 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.

Spherical space forms

Let GG be a discrete group and ρ\rho an action of GG on the n-sphere by isometries, which is free and properly discontinuous.

The induced quotient spaces S n/GS^n/G in this case are also called spherical space forms.

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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

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

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

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

and lifted to actions of Spin(8) in

  • Sunil Gadhia, Supersymmetric quotients of M-theory and supergravity backgrounds, PhD thesis, School of Mathematics, University of Edinburgh, 2007 (spire:1393845)

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 April 24, 2018 at 10:52:38. See the history of this page for a list of all contributions to it.