# nLab group actions on spheres

Contents

### Context

#### Spheres

n-sphere

low dimensional n-spheres

#### Representation theory

representation theory

geometric representation theory

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

The induced quotient spaces $S^n/G$ 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 $S^0 \hookrightarrow S^4$

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

(…)

## 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 $S^7$ 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 $Spin(8)$ 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 1, 2019 at 14:14:47. See the history of this page for a list of all contributions to it.