# 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.

## Properties

### Free actions by finite groups and spherical space forms

We discuss some aspects of (continuous) free actions of finite groups on n-spheres.

If the action is by isometries then the topological quotient spaces of such actions are known as spherical space forms.

#### Basic examples

###### Example

(action of $\mathbb{Z}/2$ by antipodal inversion)
For all $n \in \mathbb{N}$, the group $\mathbb{Z}/2$ has a continuous free action on the n-sphere, given by antipodal reflection (hence reversing the orientation).

The topological quotient space of this action is the real projective space $S^{n+1}/(\mathbb{Z}/2) \,\simeq\, \mathbb{R}P^n$.

###### Example

(Finite cyclic groups act freely on odd-dimensional spheres)
For every non-trivial finite cyclic group $\mathbb{Z}/n \,\subset\, U(1)$, its left-multiplication action on the $2n+1$-sphere, regarded as the unit sphere in the complex $n$-space

$S^{2n+1} \,\simeq\, S\big(\mathbb{R}^{2n+2}\big) \,\simeq\, S\big(\mathbb{C}^{n+1}\big) \,,$

is (a continuous group action and) free.

This follows verbatim as the (slightly more interesting) Ex. below, interchanging the complex numbers with the quaternions.

###### Example

(Finite ADE-groups act freely on $S^{4n+3}$-spheres)
For every non-trivial finite subgroup of SU(2) $G \,\subset\,$ SU(2) $\simeq$ Sp(1) its quaternionic left-multiplication action on the $4n+3$-sphere, regarded as the unit sphere in the quaternionic $n$-space

$S^{4n+3} \,\simeq\, S\big(\mathbb{R}^{4n+4}\big) \,\simeq\, S\big(\mathbb{H}^{n+1}\big) \,,$

is (a continuous group action and) free.

###### Proof

Here are the elementary and straightforward details:

Under the given identification, points in $S^{4n+3}$ correspond to $(n+1)$-tuples of quaternions $\vec v \,=\, (v_1, \cdots, v_{n+1})$, $v_i \,\in\, \mathbb{H}$ (i.e. quaternionic vectors) such that

(1)\begin{aligned} \left\vert \vec v \right\vert^2 & \;\coloneqq\; \vec v^\dagger \cdot \vec v \\ & \;\coloneqq\; \bar v_1 \cdot v_1 + \cdots + \vec v_{n+1} \cdot v_{n+1} \\ & \;\overset{!}{=}\; 1 \end{aligned} \,,

where $\overline{(-)}$ denotes quaternionic conjugation.

Since Sp(1) is the subgroup of the quaternionic group of units on the unit-norm elements

$Sp(1) \;\coloneqq\; \big\{ q \,\in\, \mathbb{H}^\times \,\vert\, \bar q \cdot q \,=\, 1 \big\} \;\subset\; \mathbb{H}^\times$

we have

\begin{aligned} \left\vert q \cdot \vec v \right\vert & \;=\; (q \cdot \vec v)^\dagger \cdot (q \cdot \vec v) \\ & \;=\; \vec v^\dagger \cdot \underset{ = 1}{\underbrace{q^\dagger \cdot q}} \cdot \vec v \\ & \;=\; \vec v^\dagger \cdot \vec v \\ & \;=\; \left\vert \vec v\right\vert^2 \,, \end{aligned}

showing that the left multiplication action of $Sp(1)$ on $\mathbb{H}^{n+1}$ does restrict to an action on its unit sphere. Moreover, since quaternion-multiplication is clearly continuous with respect to the Euclidean topology on $\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4$, this yields a continuous action on $S^{4n+3}$.

Finally, since $\mathbb{H} \,\ni\, q \,\mapsto\, \bar q \cdot q \,\in\, \mathbb{R}$ is positive definite ($\bar q \cdot q = 0 \,\;\Leftrightarrow\;\, q = 0$ ), at least one of the components $v_i$ of $\vec v$ needs to be non-zero in order for (1) to hold. But on this component the left action $v_i \,\mapsto\, q \cdot v_i$ is left-multiplication in the group of units $\mathbb{H}^{\times} \,=\, \mathbb{H} \setminus \{0\}$ and hence is free, as the multiplication action of any group on itself is free.

Notice that, while an analogous argument shows that with $G \subset Sp(1)$ also the direct product group $G^{n+1}$ canonically acts on $S^{4n+3}$ by componentwise left multiplication, for $n \geq 2$ this action is no longer free, as the $k$th factor subgroup now fixes the elements whose $k$th component vanishes. In fact, higher direct product powers of cyclic groups in general have no free action on spheres of any dimension, see Smith’s $p^2$-condition (Prop. below).

The above examples of free actions are all on odd-dimensional spheres, except when the group is $\mathbb{Z}/2$. Indeed, this must be so in general:

###### Proposition

The only finite group with any free continuous action on a sphere $S^{2n}$ of positive even dimension is $\mathbb{Z}_2$.

###### Proof

Given a fixed-point free-action, the quotient space coprojection $S^{2n} \xrightarrow{q} S^{2n/G}$ is a covering space with degree $deg(q) = ord(G)$ equal to the order of $G$. Passing to Euler characteristics $\chi(-)$ and using that

1. $\chi(-)$ is always an integer (by definition);

2. $\chi(S^{2n}) = 2$ (by this Prop.);

3. $\chi(-)$ of any finite covering coprojection is multiplication by the degree (this Prop.)

we obtain the equation

$2 \;=\; \chi(S^{2n}) \;=\; ord(G) \cdot \chi(S^{2n}/G) \,.$

The only solutions for this algebraic equation (over the integers) have $ord(G) \,\in\, \{1, 2\}$. But $ord(G) = 1$ implies that $G$ is the trivial group, whose action certainly has fixed points. Hence the only admissible solution is $ord(G) = 2$ and the only group of that order is $\mathbb{Z}_2$. That this does have at least one fixed-point free action is Ex. .

###### Remark

The nature of the fixed loci of finite group actions on even-dimensional spheres is discussed in Craciun 2013.

###### Remark

Moreover, the Lefschetz fixed point theorem implies (see this example) that the only free action on an even dimensional sphere, necessarily by $\mathbb{Z}/2$ according to Prop. , is orientation reversing, as in Ex. .

#### Obstructions and existence

###### Proposition

(Smith’s $p^2$-condition)
For $p$ a prime number, the direct product group $\mathbb{Z}/p \times \mathbb{Z}/p$ and more generally the higher powers $(\mathbb{Z}/p)^{\geq 2}$ of the prime cyclic group do not have any continuous free action on any n-sphere.

(Smith 1944, p. 107 (4 of 5))

###### Definition

($p q$ condition) For $p, q$ a pair of prime numbers, not necessarily distinct, a finite group $G$ is said to satisfy the $p q$-condition if all subgroups of order $p \cdot q$ are cyclic groups:

$H \,\subset\, G \; \text{with} \; ord(H) \,=\, p q \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; H \,\simeq\, \mathbb{Z}/(p q) \,.$

###### Proposition

A finite group $G$ has a continuous free action on some n-sphere if and only if it satisfies the $p^2$-condition and the $2 p$-condition (Def. ) for all prime numbers $p$.

In this case there exists also a smooth structure on some n-sphere (possibly an exotic smooth structure) such that $G$ has a smooth free action on it.

Specifically, such free smooth actions exist in particular on all $S^n$ for which $n+1$ is any multiple of the Artin-Lam induction exponent, hence exist on spheres of arbitrarily large dimension.

### Fixed loci of the circle group acting on spheres

###### 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.

(…)

The non-existence of free actions of $(\mathbb{Z}/p)^{\geq 2}$ on any n-sphere:

• P. A. Smith, Permutable Periodic Transformations, Proceedings of the National Academy of Sciences of the United States of America Vol. 30, No. 5 (May 15, 1944), pp. 105-108 (jstor:87918)

Discussion of free group actions on spheres by finite groups:

Discussion of free actions on products of spheres:

Discussion of the fixed point-sets of finite group actions on even-dimensional spheres:

• Gheorghe Craciun, Most homeomorphisms with a fixed point have a Cantor set of fixed points, Archiv der Mathematik volume 100, pages 95–99 (2013) (doi:10.1007/s00013-012-0466-z)

Classification of free finite group actions by isometries, hence with quotient spaces being spherical space forms:

review:

• Ian Hambleton, Topological spherical space forms, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston (arXiv:1412.8187)

streamlined re-proof:

Discussion of circle group-actions on spheres:

• 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 Wolf 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.