nLab
group actions on spheres

Contents

Context

Spheres

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 β„€/2\mathbb{Z}/2 by antipodal inversion)
For all nβˆˆβ„•n \in \mathbb{N}, the group β„€ / 2 \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/(β„€/2)≃ℝP nS^{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 β„€/nβŠ‚U(1)\mathbb{Z}/n \,\subset\, U(1), its left-multiplication action on the 2 n + 1 2n+1 -sphere, regarded as the unit sphere in the complex nn-space

S 2n+1≃S(ℝ 2n+2)≃S(β„‚ n+1), 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+3S^{4n+3}-spheres)
For every non-trivial finite subgroup of SU(2) GβŠ‚G \,\subset\, SU(2) ≃\simeq Sp(1) its quaternionic left-multiplication action on the 4 n + 3 4n+3 -sphere, regarded as the unit sphere in the quaternionic nn-space

S 4n+3≃S(ℝ 4n+4)≃S(ℍ n+1), 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+3S^{4n+3} correspond to (n+1)(n+1)-tuples of quaternions vβ†’=(v 1,β‹―,v n+1)\vec v \,=\, (v_1, \cdots, v_{n+1}), v iβˆˆβ„v_i \,\in\, \mathbb{H} (i.e. quaternionic vectors) such that

(1)|vβ†’| 2 ≔vβ†’ †⋅vβ†’ ≔vΒ― 1β‹…v 1+β‹―+vβ†’ n+1β‹…v n+1 =!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 (βˆ’)Β―\overline{(-)} denotes quaternionic conjugation.

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

Sp(1)≔{qβˆˆβ„ Γ—|qΒ―β‹…q=1}βŠ‚β„ Γ— Sp(1) \;\coloneqq\; \big\{ q \,\in\, \mathbb{H}^\times \,\vert\, \bar q \cdot q \,=\, 1 \big\} \;\subset\; \mathbb{H}^\times

we have

|qβ‹…vβ†’| =(qβ‹…vβ†’) †⋅(qβ‹…vβ†’) =vβ†’ †⋅q †⋅q⏟=1β‹…vβ†’ =vβ†’ †⋅vβ†’ =|vβ†’| 2, \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)Sp(1) on ℍ n+1\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 ℍ≃ ℝℝ 4\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4, this yields a continuous action on S 4n+3S^{4n+3}.

Finally, since β„βˆ‹q↦qΒ―β‹…qβˆˆβ„\mathbb{H} \,\ni\, q \,\mapsto\, \bar q \cdot q \,\in\, \mathbb{R} is positive definite (qΒ―β‹…q=0⇔q=0\bar q \cdot q = 0 \,\;\Leftrightarrow\;\, q = 0 ), at least one of the components v iv_i of vβ†’\vec v needs to be non-zero in order for (1) to hold. But on this component the left action v i↦qβ‹…v iv_i \,\mapsto\, q \cdot v_i is left-multiplication in the group of units ℍ Γ—=β„βˆ–{0}\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 GβŠ‚Sp(1)G \subset Sp(1) also the direct product group G n+1G^{n+1} canonically acts on S 4n+3S^{4n+3} by componentwise left multiplication, for nβ‰₯2n \geq 2 this action is no longer free, as the kkth factor subgroup now fixes the elements whose kkth 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 2p^2-condition (Prop. below).

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

Proposition

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

Proof

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

  1. Ο‡(βˆ’)\chi(-) is always an integer (by definition);

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

  3. Ο‡(βˆ’)\chi(-) of any finite covering coprojection is multiplication by the degree (this Prop.)

we obtain the equation

2=Ο‡(S 2n)=ord(G)β‹…Ο‡(S 2n/G). 2 \;=\; \chi(S^{2n}) \;=\; ord(G) \cdot \chi(S^{2n}/G) \,.

The only solutions for this algebraic equation (over the integers) have ord(G)∈{1,2}ord(G) \,\in\, \{1, 2\}. But ord(G)=1ord(G) = 1 implies that GG is the trivial group, whose action certainly has fixed points. Hence the only admissible solution is ord(G)=2ord(G) = 2 and the only group of that order is β„€ 2 \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 β„€/2\mathbb{Z}/2 according to Prop. , is orientation reversing, as in Ex. .

Obstructions and existence

Proposition

(Smith’s p 2p^2-condition)
For pp a prime number, the direct product group β„€/pΓ—β„€/p\mathbb{Z}/p \times \mathbb{Z}/p and more generally the higher powers (β„€/p) β‰₯2(\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

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

HβŠ‚Gwithord(H)=pqβ‡’H≃℀/(pq). H \,\subset\, G \; \text{with} \; ord(H) \,=\, p q \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; H \,\simeq\, \mathbb{Z}/(p q) \,.

Proposition

(Madsen-Thomas-Wall theorem)
A finite group GG has a continuous free action on some n-sphere if and only if it satisfies the p 2p^2-condition and the 2p2 p-condition (Def. ) for all prime numbers pp.

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

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

(Madsen, Thomas and Wall 1976, Thm. 0.5-0.6, 1983, Thm. 5, reviewed as Hambleton 2014, Thm. 6.1)

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 0β†ͺS 4S^0 \hookrightarrow S^4

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

(FΓ©lix-Oprea-TanrΓ© 08, Example 7.39)

(…)

References

The non-existence of free actions of (β„€/p) β‰₯2(\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 FΓ©lix, John Oprea, Daniel TanrΓ©, Algebraic Models in Geometry, Oxford University Press 2008

The subgroups of SO(8) which act freely on S 7S^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)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 October 27, 2021 at 11:58:38. See the history of this page for a list of all contributions to it.