nLab group actions on spheres

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 $n$-spheres.

Example

(action of $\mathbb{Z}/2$ by antipodal involution)
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.

Remark

While an analogous argument as in Ex. 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).

Free involutions

The above examples of free actions on spheres are all on odd-dimensional spheres, except when the group is $\mathbb{Z}/2$, 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 $n$-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 $n$-spheres (Ex. below).

Proposition

(only $\mathbb{Z}/2$ has free actions on even-dimensional spheres)
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

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

Proposition

Every free involution

• on an even-dimensional sphere is orientation-reversing;

• on an odd-dimensional sphere is orientation-preserving.

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 $\sigma \,\colon\, S^{n} \to S^n$ is, as an element in the homotopy groups of spheres,

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

  $$[\sigma] \,=\, - 1 \,\in\, \mathbb{Z} \,\simeq\, \pi_{2k}(S^{2k})$$

• on an odd-dimensional $n$-sphere, $n = 2k+1$, equal to

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

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

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

Example

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

• a single free involution on the 3-sphere $S^3$ (hence the antipodal Ex. )

  (Livesay 1960)

• at least 2 distinct free involutions on the 4-sphere $S^4$

  (Fintushel & Stern 1981)

• 4 distinct free involutions on the 5-sphere $S^5$

• 4 distinct free involutions on the 6-sphere $S^6$

• $card(\mathbb{Z}/2 \oplus \mathbb{Z}/28 \oplus \mathbb{Z})$ distinct free involutions on the 7-sphere $S^7$

  (López de Medrano, Sec. V.6.1)

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

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 $n$-sphere, and hence is actually homeomorphic to an $n$-sphere.

Therefore, topological involutions on $n$-spheres should have the same classification up to homeomorphism as homeomorphism types of topological manifolds which are homotopy equivalent to $\mathbb{R}P^n$s, 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 $n$-sphere: One obstruction is Smith’s “$p^2$-condition” (Prop. below), another is Milnor’s “$2 p$-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 $p^2$-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))

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 $p^2$ are cyclic, i.e. isomorphic to $\mathbb{Z}/p^2$.

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

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

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

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

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

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

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

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

4. Every abelian subgroup of $G$ is cyclic.

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

Proposition

(Milnor’s $2 p$-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. $G$ satisfies the $2p$-condition (Def. ) for all $p$.

(Milnor 1957, Cor 1 & p. 627)

Proposition

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

Moreover, if $G$ is solvable, then the converse holds: Every solvable group satisfying all $p q$-conditions has a free orthogonal action on some $n$-sphere.

(Zassenhaus 1935, Vincent 1947)

Proposition

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

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

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

for all prime numbers $p$ (Def. ).

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 $S^{2 k \cdot e(G) - 1}$, for all $k \in \mathbb{N}$, where $e(G) \,\in\, \mathbb{N}_+$ is the Artin-Lam induction exponent of $G$, 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)

Classification results on groups satisfying these conditions are discussed in Wall 2013.

Example

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

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

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

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

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

On the other hand, the plain dihedral groups $D_{2(r+3)} \,\subset\, SO(3)$ have order $2 \cdot (r + 3)$ and hence finite subgroups of SO(3) may violate Milnor’s $2 p$-condition (Prop. ). Certainly their canonical actions on $S^2$ 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 $S^0 \hookrightarrow S^4$

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

(Félix-Oprea-Tanré 08, Example 7.39)

(…)

Related concepts

• lens space

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 $p^2$-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 $2 p$-condition for continuous actions:

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

See also:

• Richard G. Swan, Periodic Resolutions for Finite Groups, Annals of Mathematics Second Series 72 2 (1960) 267-291 [doi:10.2307/1970135]

Existence results:

• C. T. C. Wall, Free actions of finite groups on spheres, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 (pdf)

• Ib Madsen, Charles B. Thomas, C. T. C. Wall, The topological spherical space form problem II: existence of free actions, Topology Volume 15, Issue 4, 1976, Pages 375-382 (doi:10.1016/0040-9383(76)90031-8)

• Ib Madsen, Charles B. Thomas, C. T. C. Wall, Topological spherical space form problem III: Dimensional bounds and smoothing, Pacific J. Math. 106(1): 135-143 (1983) (pjm:1102721110)

Classification results:

• C. T. C. Wall, On the Structure of Finite Groups with Periodic Cohomology, 381-413 in: Lie Groups: Structure, Actions, and Representations, Progress in Mathematics 306 Birkhäuser 2013 (doi:10.1007/978-1-4614-7193-6_16)

following

• Joseph Wolf, Part III of: Spaces of constant curvature, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 (doi:10.1090/chel/372)

Discussion of free actions on products of spheres:

• Alejandro Adem, Constructing and deconstructing group actions, in Paul Goerss, Stewart Priddy, Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Contemporary Mathematics 346 AMS 2004 (arXiv:0212280, doi:10.1090/conm/346)

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:

• George R. Livesay: Fixed Point Free Involutions on the 3-Sphere, Annals of Mathematics Second Series, Vol. 72, No. 3 (Nov., 1960), pp. 603-611 (jstor:1970232)

• Santiago López de Medrano, Involutions on Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 59, Springer 1971 (doi:10.1007/978-3-642-65012-3)

• Ronald Fintushel, Ronald J. Stern, An Exotic Free Involution on $S^4$, Annals of Mathematics Second Series, Vol. 113, No. 2 (Mar., 1981), pp. 357-365 (jstor:2006987)

Classification of finite free actions by isometries

The original article identifying the $p q$-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:

• Joseph Wolf, Spaces of constant curvature, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 (doi:10.1090/chel/372)

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, ISBN:9781571463012)

streamlined re-proof:

• Daniel Allcock, Spherical space forms revisited, Trans. Amer. Math. Soc. 370 (2018), 5561-5582 (arXiv:1509.00906, doi:10.1090/tran/7167)

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

• Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)

• Paul de Medeiros, José Figueroa-O'Farrill, Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)

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:

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