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 (with respect to the round metric) then the topological quotient spaces of such free actions are traditionally known as spherical space forms (going back to Killing 1891). Later (maybe starting with Madsen, Thomas & Wall 1976) authors also speak of topological spherical space forms for quotients by actions which are continuous but not necessarily by isometries.

Basic examples

We make explicit some elementary but important examples of free continuous finite group actions on nn-spheres.

Example

(action of /2\mathbb{Z}/2 by antipodal involution)
For all nn \in \mathbb{N}, the group / 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/(/2)P 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 /nU(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+1S( 2n+2)S( 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) GG \,\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+3S( 4n+4)S( 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 v=(v 1,,v n+1)\vec v \,=\, (v_1, \cdots, v_{n+1}), v iv_i \,\in\, \mathbb{H} (i.e. quaternionic vectors) such that

(1)|v| 2 v v v¯ 1v 1++v n+1v 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 ()¯\overline{(-)} denotes quaternionic conjugation.

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

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

we have

|qv| =(qv) (qv) =v q q=1v =v v =|v| 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 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 4\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4, this yields a continuous action on S 4n+3S^{4n+3}.

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

Remark

While an analogous argument as in Ex. shows that with GSp(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 n2n \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).

Free involutions

The above examples of free actions on spheres are all on odd-dimensional spheres, except when the group is / 2 \mathbb{Z}/2 , hence when the action is by involutions (Rem. below), where the antipodal free action (Ex. ) exists in all dimensions. Indeed, it holds in general that the only free actions of finite groups on even-dimensional spheres are involutions (Prop. below), and, as in the canonical example, all free involutions on even-dimensional spheres are orientation-reversing, and those on odd-dimensional spheres are orientation-preserving (Prop. below).

While, therefore, all free involutions on a given nn-sphere are homotopic to the antipodal involution (Rem. below), at least in the categories of piecewise-linear and of smooth actions there are families of distinct exotic involutions on nn-spheres (Ex. below).

Proposition

(only /2\mathbb{Z}/2 has free actions on even-dimensional spheres)
The only finite group with any free continuous action on a sphere S 2nS^{2n} of positive even dimension is 2 \mathbb{Z}_2 .

Proof

Given a fixed-point free-action, the quotient space coprojection S 2nqS 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 χ()\chi(-) and using that

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

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

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

we obtain the equation

2=χ(S 2n)=ord(G)χ(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 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

( / 2 \mathbb{Z}/2 -actions are equivalently involutions)
Any group action ρ:/2×S nS n\rho \,\colon\, \mathbb{Z}/2 \times S^n \to S^n of the / 2 \mathbb{Z}/2 is determined by its value ρ(σ):S nS n\rho(\sigma) \,\colon\, S^n \to S^n on the single non-trivial element σ/2\sigma \,\in\, \mathbb{Z}/2, which is an involution, and every involution corresponds to a unique /2\mathbb{Z}/2-action this way.

Proposition

Every free involution

Proof

By the Lefschetz fixed point theorem, see this example for details.

Remark

Prop. implies that the homotopy class of any fixed-point free involution σ:S nS n\sigma \,\colon\, S^{n} \to S^n is, as an element in the homotopy groups of spheres,

  • on an even-dimensional nn-sphere, n=2kn = 2k, equal to

    [σ]=1π 2k(S 2k) [\sigma] \,=\, - 1 \,\in\, \mathbb{Z} \,\simeq\, \pi_{2k}(S^{2k})
  • on an odd-dimensional nn-sphere, n=2k+1n = 2k+1, equal to

    [σ]=+1π 2k+1(S 2k+1). [\sigma] \,=\, + 1 \,\in\, \mathbb{Z} \,\simeq\, \pi_{2k+1}(S^{2k+1}) \,.

which means that every free involution on any nn-sphere is homotopic to the antipodal involution (Ex.).

Nevertheless, there are in general several involutions on any given nn-sphere which are not isomorphic to the antipodal involution as /2\mathbb{Z}/2-actions, certainly if regarded in the category of smooth (or just piecewise-linear) actions:

Example

(non-standard free smooth involutions on nn-spheres)
There are, in the category smooth /2\mathbb{Z}/2-actions, up to isomorphism:

Here all spheres are equipped with their standard smooth structure (?), it is just the extra involutions which are non-standard.

Similarly, there are non-standard involutions on nn-spheres in the category of piecewise-linear structures (López de Medrano, p. 2 (11 of 114)) following a similar pattern.

Example

(non-standard free continuous involutions on nn-spheres) There are also plenty of non-standard involutions on nn-spheres in the topological category (both for even and for odd nn):

By the analog of the Poincaré conjecture-theorem in higher dimensions (e.g. Newman 1966, Thm. 7), every topological manifold of the homotopy type of a real projective space has a double covering space which is a manifold of the homotopy type of an nn-sphere, and hence is actually homeomorphic to an nn-sphere.

Therefore, topological involutions on nn-spheres should have the same classification up to homeomorphism as homoemorphism types of topological manifolds which are homotopy equivalent to P n\mathbb{R}P^ns, and large classes of non-trivial examples of these are known (Belegradek, MO:407390).

General obstructions and existence

Not all finite groups have any free continuous action on any nn-sphere: One obstruction is Smith’s “p 2p^2-condition” (Prop. below), another is Milnor’s “2p2 p-condition” (Prop. below). But these are the only two obstructions, and a finite group that evades these is guaranteed to act not just continuously on a sphere of a single dimension, but smoothly on spheres of any dimension which is a multiple of its Artin-Lam induction exponent (Prop. below).

Proposition

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

Remark

By the fundamental theorem of finitely generated abelian groups, Prop. immediately implies that if a finite group has any continuous free action on any n-sphere, then it must satisfy the condition that all its subgroups of order p 2p^2 are cyclic, i.e. isomorphic to /p 2\mathbb{Z}/p^2.

This condition has come to be called the “p 2p^2-condition” (Def. below).

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 pqp \cdot q are cyclic groups:

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

Proposition

(equivalent statements of the p 2p^2-condition)
For a finite group GG, the following are equivalent:

  1. For all prime numbers pp, GG satisfies the p 2p^2-condition (Def. ).

  2. For all primes pp, the Sylow p-subgroup of GG is either cyclic or (if p=2p = 2) generalized quaternion.

  3. GG has “periodic cohomology” in that there exists kk \in \mathbb{N} such that its integral group cohomology group in degree kk is cyclic:

    kN kH grp k(G;)/n k \underset{k \in \mathbb{N}}{\exists} \; \underset{N_k \in \mathbb{N}}{\exists} \;\; H^k_{grp}(G;\, \mathbb{Z}) \,\simeq\, \mathbb{Z}/{n_k}

    In this case the possible “periods” kk form a subgroup of \mathbb{Z} (Cartan & Eilenberg 1956, p. 261 (283 of 421), with H k(G;)H k(G;)H^{-k}(G;\, \mathbb{Z}) \coloneqq H_k(G;\, \mathbb{Z}) by p. 232 (254 of 421))

  4. Every abelian subgroup of GG is cyclic.

(Cartan & Eilenberg 1956, Thm. IV 11.6, p. 262 (284 of 421))

Proposition

(Milnor’s 2p2 p-condition)
If a finite group has any continuous free action on any n-sphere, then

  1. every element of order 2 is in its center

  2. GG satisfies the 2p2p-condition (Def.) for all pp.

(Milnor 1957, Cor 1 & p. 627)

Proposition

(Zassenhaus’s pqp q-condition) If a finite group has an orthogonal free action on some n-sphere, namely a free action through a group homomorphism GO(n+1)G \xrightarrow O(n+1) to the orthogonal group regarded with its canonical action on the unit sphere S nS( n+1)S^n \,\simeq\, S(\mathbb{R}^{n+1}), then GG satisfies all pqp q-conditions (Def. ) for all pairs of primes p,qp,q.

Moreover, if GG is solvable, then the converse holds: Every solvable group satisfying all pqp q-conditions has a free orthogonal action on some nn-sphere.

(Zassenhaus 1935, Vincent 1947)

Proposition

(Madsen-Thomas-Wall theorem)
A finite group GG has a continuous free action on some n-sphere if and only if it satisfies

  1. the p 2p^2-condition (see Prop. )

  2. the 2p2 p-condition (see Prop. )

for all prime numbers pp (Def. ).

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 2ke(G)1S^{2 k \cdot e(G) - 1}, for all kk \in \mathbb{N}, where e(G) +e(G) \,\in\, \mathbb{N}_+ is the Artin-Lam induction exponent of GG, hence exist on spheres of arbitrarily large dimension.

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

Example

(Finite subgroups of SU(2)SU(2) satisfy the p 2p^2 and 2p2p conditions)
Every non-trivial finite subgroup of SU(2) GSU(2)Sp(1)G \,\subset\, SU(2) \,\simeq\, Sp(1) has a free smooth action on S 4k+3S^{4k+3}, for all kk \in \mathbb{N}, by Ex. . By Prop. this implies that all subgroups of GG of order p 2p^2 or 2p2 p must be cyclic.

Indeed, by the ADE-classification of finite subgroups of SU(2), the only possible non-cyclic subgroups are:

  1. the binary dihedral groups 2D 2(r+3)2 D_{2(r+3)} of order 2 2(r+3)2^2 \cdot (r + 3),

  2. the binary tetrahedral group of order 24=2 3324 \,=\, 2^3 \cdot 3,

  3. the binary octahedral group of order 48=2 4348 \,=\, 2^4 \cdot 3,

  4. the binary icosahedral group of order 120=2 335120 \,=\, 2^3 \cdot 3 \cdot 5.

None of these orders is of the form p 2p^2 or 2p2 p, in accord with Prop. .

On the other hand, the plain dihedral groups D 2(r+3)SO(3)D_{2(r+3)} \,\subset\, SO(3) have order 2(r+3)2 \cdot (r + 3) and hence finite subgroups of SO(3) may violate Milnor’s 2p2 p-condition (Prop. ). Certainly their canonical actions on S 2 S^2 are not free (as each element must act by a rotation which necessarily fixes an antipodal pair of points).


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 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)

(…)

Literature

See also the references at spherical space form.

Characterization of finite free actions by homeomorphisms

Discussion of free actions by finite groups:

The original article identifying the p 2p^2-condition for continuous actions:

  • 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)

The original article identifying the 2p2 p-condition for continuous actions:

  • John Milnor, Groups Which Act on S nS^n Without Fixed Point, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 (jstor:2372566)

Existence results:

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 involutions:

Classification of finite free actions by isometries

The original article identifying the pqp q-conditions for orthogonal actions:

  • Hans Zassenhaus, Über endliche Fastkörper, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg. Vol. 11. No. 1. Springer-Verlag, 1935 (pdf)

  • Georges Vincent, Les groupes linéaires finis sans points fixes, Commentarii Mathematici Helvetici 20.1 (1947): 117-171 (pdf)

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

review:

streamlined re-proof:

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.

Discussion of circle-actions

Discussion of circle group-actions on spheres:

Last revised on November 17, 2021 at 11:19:14. See the history of this page for a list of all contributions to it.