Fundamental groups of 3-dimensional spherical space forms

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Every finite subgroup of $SO(3)$ is either
Every finite subgroup of $SO(3)$ is either
* a [[Wikipedia:Cyclic_group|cyclic group]],
* a [[Wikipedia:Cyclic_group|cyclic group]],
−
* a [[Wikipedia:Dihedral_group|dihedral] group], i.e., $D_{2n} = \langle a,x \mid a^n = 1, x^2 = 1, x^{-1} a x = a^{-1} \rangle$, $n \geq 1$,
+
* a [[Wikipedia:Dihedral_group|dihedral group]], i.e., $D_{2n} = \langle a,x \mid a^n = 1, x^2 = 1, x^{-1} a x = a^{-1} \rangle$, $n \geq 1$,
* a [[Wikipedia:Tetrahedral_group|tetrahedral group]] $T$, i.e., the symmetry group of the regular tetrahedron,
* a [[Wikipedia:Tetrahedral_group|tetrahedral group]] $T$, i.e., the symmetry group of the regular tetrahedron,
* a [[Wikipedia:Octahedral_group|octahedral group]] $O$, i.e., the symmetry group of the regular octahedron,
* a [[Wikipedia:Octahedral_group|octahedral group]] $O$, i.e., the symmetry group of the regular octahedron,

Revision as of 13:11, 11 August 2022

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Contents

1 Introduction

The purpose of this article is to describe fundamental groups of 3-dimensional spherical space forms. For the historical context refer to this article. Today the list of groups which arise as fundamental groups of 3-dimensional spherical space forms is known. These are exactly the groups which admit a fixed-point free representation in SO(4). In 1950's Milnor in [Milnor1957] compiled a list of all finite groups which could possibly act freely but not necessarily linearly on S^3. Apart from the groups admitting fixed-point free representations in SO(4), Milnor's list also included a family of finite groups denoted by Q(8n,k,l). Question whether these groups can act on S^3 remained unsolved until the proof of the Geometrization Conjecture was finished by Perelman. The exposition in this article is based on Chapter 7.5 of [Wolf2011], which surveys results of Hopf [Hopf1926] and Seifert-Threlfall [Threlfall&Seifert1931], [Threlfall&Seifert1933].

2 Finite subgroups of SO(4)

The list of finite subgroups of SO(4) can be determined in three steps:

  • determine finite subgroups of SO(3),
  • use the covering map S^3 \to SO(3) to determine finite subgroups of S^3,
  • use the fact that SO(4) is doubly covered by 
    Tex syntax error
     to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of SO(3) one has to analyse the action of these groups on S^2. From Riemann-Hurwitz formula we obtain the following equation

\displaystyle 2 \left(1 - \frac{1}{N} \right) = \sum_{i = 1}^q \left(1 - \frac{1}{n_i} \right),

where N denotes the order of the group, q denotes number of orbits with non-trivial isotropy groups, and n_i denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of SO(3).

Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of SO(3) is either

These groups are called polyhedral groups.

2.2 Finite subgroups of S^3

Let \mathbb{H} denote the algebra of quaternions and treat S^3 as a subset of \mathbb{H} of quaternions of norm 1. Consider the action of S^3 on \mathbb{H} by conjugation

\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).

This action preserves 1, so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation \pi \colon S^3 \to SO(3) with kernel equal to \{\pm 1\}. Hence, \pi is a 2-fold covering map.

If G is a finite subgroup of S^3, let F = \pi(G). If F = G, then, since -1 is the only element of S^3 of order 2, F and G must be both of odd order. Therefore comparing this with the list of finite subgroups of SO(3) yields that F and G are both cyclic of odd order. On the other hand, if F \neq G, then G is an extension of the form

\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.

These considerations yields the following theorem.

Theorem 2.2 [Wolf2011]. Every finite subgroup of S^3 is either

These groups are called binary polyhedral groups.

2.3 Finite subgroups of SO(4)

To perform the final step, consider a homomophism

\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.

Its kernel is equal to \{(1,1), (-1,-1)\}.

Finite subgroups of 
Tex syntax error
 can be determined by Goursat's lemma. This lemma says, that every finite subgroup of 
Tex syntax error
 is isomorphic to the fibre product G \times_{Q} H, where G and H are finite subgroups of S^3 and Q is a common quotient of G and H.

2.4 Finite fixed-point free subgroups of SO(4)

Not every finite subgroup of SO(4) act freely on S^3. Following lemma gives necessary and sufficient condition for F(q_1,q_2) to be fixed point free for q_1, q_2 \in S^3.

Lemma 2.3 [Wolf2011]. Let q_1, q_2 be unit quaternions, then F(q_1,q_2) has a fixed point on S^3 if, and only if, q_1 is conjgate to q_2 in S^3.

Proof. This is a simple observation

\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.
\square

Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of SO(4) belongs to the following list

\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.
  • groups P_{8 \cdot 3^{k}}' defined by the following presentation
\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,
  • direct product of any of the above group with a cyclic group of relatively prime order.

3 Milnor's contribution

In [Milnor1957] author proves the following theorem.

Theorem 3.1. If G is a finite group which admits a fixed-point free action on a sphere S^{2n+1}, then for every prime p every subgroup of G of order 2p is cyclic.

This theorem allowed him to choose all possible candidates, from the list compiled by Zassenhaus and Suzuki of groups with periodic cohomology, which could possibly act on S^3. Apart from groups which admit a fixed point free representation in SO(4) he obtained the following family of groups.

\displaystyle Q(8n,k,l) = \langle x,y,z \mid x^2 = (xy)^2=y^{2n}, z^{kl}=1, xzx^{-1} = z^{r}, yzy^{-1} = z^{-1} \rangle,

where 8n,k,l are relatively prime integers and

\displaystyle r \equiv -1 \pmod{k},
\displaystyle r \equiv 1 \pmod{l}.

Groups Q(8n,k,l) were excluded from the list of fundamental groups of 3-manifolds only after resolution of the Geometrization Conjecture.

4 References

$. Consider the action of $S^3$ on $\mathbb{H}$ by conjugation $$q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).$$ This action preserves 3-dimensional spherical space forms. For the historical context refer to this article. Today the list of groups which arise as fundamental groups of 3-dimensional spherical space forms is known. These are exactly the groups which admit a fixed-point free representation in SO(4). In 1950's Milnor in [Milnor1957] compiled a list of all finite groups which could possibly act freely but not necessarily linearly on S^3. Apart from the groups admitting fixed-point free representations in SO(4), Milnor's list also included a family of finite groups denoted by Q(8n,k,l). Question whether these groups can act on S^3 remained unsolved until the proof of the Geometrization Conjecture was finished by Perelman. The exposition in this article is based on Chapter 7.5 of [Wolf2011], which surveys results of Hopf [Hopf1926] and Seifert-Threlfall [Threlfall&Seifert1931], [Threlfall&Seifert1933].

2 Finite subgroups of SO(4)

The list of finite subgroups of SO(4) can be determined in three steps:

  • determine finite subgroups of SO(3),
  • use the covering map S^3 \to SO(3) to determine finite subgroups of S^3,
  • use the fact that SO(4) is doubly covered by 
    Tex syntax error
     to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of SO(3) one has to analyse the action of these groups on S^2. From Riemann-Hurwitz formula we obtain the following equation

\displaystyle 2 \left(1 - \frac{1}{N} \right) = \sum_{i = 1}^q \left(1 - \frac{1}{n_i} \right),

where N denotes the order of the group, q denotes number of orbits with non-trivial isotropy groups, and n_i denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of SO(3).

Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of SO(3) is either

These groups are called polyhedral groups.

2.2 Finite subgroups of S^3

Let \mathbb{H} denote the algebra of quaternions and treat S^3 as a subset of \mathbb{H} of quaternions of norm 1. Consider the action of S^3 on \mathbb{H} by conjugation

\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).

This action preserves 1, so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation \pi \colon S^3 \to SO(3) with kernel equal to \{\pm 1\}. Hence, \pi is a 2-fold covering map.

If G is a finite subgroup of S^3, let F = \pi(G). If F = G, then, since -1 is the only element of S^3 of order 2, F and G must be both of odd order. Therefore comparing this with the list of finite subgroups of SO(3) yields that F and G are both cyclic of odd order. On the other hand, if F \neq G, then G is an extension of the form

\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.

These considerations yields the following theorem.

Theorem 2.2 [Wolf2011]. Every finite subgroup of S^3 is either

These groups are called binary polyhedral groups.

2.3 Finite subgroups of SO(4)

To perform the final step, consider a homomophism

\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.

Its kernel is equal to \{(1,1), (-1,-1)\}.

Finite subgroups of 
Tex syntax error
 can be determined by Goursat's lemma. This lemma says, that every finite subgroup of 
Tex syntax error
 is isomorphic to the fibre product G \times_{Q} H, where G and H are finite subgroups of S^3 and Q is a common quotient of G and H.

2.4 Finite fixed-point free subgroups of SO(4)

Not every finite subgroup of SO(4) act freely on S^3. Following lemma gives necessary and sufficient condition for F(q_1,q_2) to be fixed point free for q_1, q_2 \in S^3.

Lemma 2.3 [Wolf2011]. Let q_1, q_2 be unit quaternions, then F(q_1,q_2) has a fixed point on S^3 if, and only if, q_1 is conjgate to q_2 in S^3.

Proof. This is a simple observation

\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.
\square

Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of SO(4) belongs to the following list

\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.
  • groups P_{8 \cdot 3^{k}}' defined by the following presentation
\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,
  • direct product of any of the above group with a cyclic group of relatively prime order.

3 Milnor's contribution

In [Milnor1957] author proves the following theorem.

Theorem 3.1. If G is a finite group which admits a fixed-point free action on a sphere S^{2n+1}, then for every prime p every subgroup of G of order 2p is cyclic.

This theorem allowed him to choose all possible candidates, from the list compiled by Zassenhaus and Suzuki of groups with periodic cohomology, which could possibly act on S^3. Apart from groups which admit a fixed point free representation in SO(4) he obtained the following family of groups.

\displaystyle Q(8n,k,l) = \langle x,y,z \mid x^2 = (xy)^2=y^{2n}, z^{kl}=1, xzx^{-1} = z^{r}, yzy^{-1} = z^{-1} \rangle,

where 8n,k,l are relatively prime integers and

\displaystyle r \equiv -1 \pmod{k},
\displaystyle r \equiv 1 \pmod{l}.

Groups Q(8n,k,l) were excluded from the list of fundamental groups of 3-manifolds only after resolution of the Geometrization Conjecture.

4 References

$, so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation $\pi \colon S^3 \to SO(3)$ with kernel equal to $\{\pm 1\}$. Hence, $\pi$ is a 2-fold covering map. If $G$ is a finite subgroup of $S^3$, let $F = \pi(G)$. If $F = G$, then, since $-1$ is the only element of $S^3$ of order $, $F$ and $G$ must be both of odd order. Therefore comparing this with the list of finite subgroups of $SO(3)$ yields that $F$ and $G$ are both cyclic of odd order. On the other hand, if $F \neq G$, then $G$ is an extension of the form $3-dimensional spherical space forms. For the historical context refer to this article. Today the list of groups which arise as fundamental groups of 3-dimensional spherical space forms is known. These are exactly the groups which admit a fixed-point free representation in SO(4). In 1950's Milnor in [Milnor1957] compiled a list of all finite groups which could possibly act freely but not necessarily linearly on S^3. Apart from the groups admitting fixed-point free representations in SO(4), Milnor's list also included a family of finite groups denoted by Q(8n,k,l). Question whether these groups can act on S^3 remained unsolved until the proof of the Geometrization Conjecture was finished by Perelman. The exposition in this article is based on Chapter 7.5 of [Wolf2011], which surveys results of Hopf [Hopf1926] and Seifert-Threlfall [Threlfall&Seifert1931], [Threlfall&Seifert1933].

2 Finite subgroups of SO(4)

The list of finite subgroups of SO(4) can be determined in three steps:

  • determine finite subgroups of SO(3),
  • use the covering map S^3 \to SO(3) to determine finite subgroups of S^3,
  • use the fact that SO(4) is doubly covered by 
    Tex syntax error
     to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of SO(3) one has to analyse the action of these groups on S^2. From Riemann-Hurwitz formula we obtain the following equation

\displaystyle 2 \left(1 - \frac{1}{N} \right) = \sum_{i = 1}^q \left(1 - \frac{1}{n_i} \right),

where N denotes the order of the group, q denotes number of orbits with non-trivial isotropy groups, and n_i denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of SO(3).

Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of SO(3) is either

These groups are called polyhedral groups.

2.2 Finite subgroups of S^3

Let \mathbb{H} denote the algebra of quaternions and treat S^3 as a subset of \mathbb{H} of quaternions of norm 1. Consider the action of S^3 on \mathbb{H} by conjugation

\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).

This action preserves 1, so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation \pi \colon S^3 \to SO(3) with kernel equal to \{\pm 1\}. Hence, \pi is a 2-fold covering map.

If G is a finite subgroup of S^3, let F = \pi(G). If F = G, then, since -1 is the only element of S^3 of order 2, F and G must be both of odd order. Therefore comparing this with the list of finite subgroups of SO(3) yields that F and G are both cyclic of odd order. On the other hand, if F \neq G, then G is an extension of the form

\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.

These considerations yields the following theorem.

Theorem 2.2 [Wolf2011]. Every finite subgroup of S^3 is either

These groups are called binary polyhedral groups.

2.3 Finite subgroups of SO(4)

To perform the final step, consider a homomophism

\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.

Its kernel is equal to \{(1,1), (-1,-1)\}.

Finite subgroups of 
Tex syntax error
 can be determined by Goursat's lemma. This lemma says, that every finite subgroup of 
Tex syntax error
 is isomorphic to the fibre product G \times_{Q} H, where G and H are finite subgroups of S^3 and Q is a common quotient of G and H.

2.4 Finite fixed-point free subgroups of SO(4)

Not every finite subgroup of SO(4) act freely on S^3. Following lemma gives necessary and sufficient condition for F(q_1,q_2) to be fixed point free for q_1, q_2 \in S^3.

Lemma 2.3 [Wolf2011]. Let q_1, q_2 be unit quaternions, then F(q_1,q_2) has a fixed point on S^3 if, and only if, q_1 is conjgate to q_2 in S^3.

Proof. This is a simple observation

\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.
\square

Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of SO(4) belongs to the following list

\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.
  • groups P_{8 \cdot 3^{k}}' defined by the following presentation
\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,
  • direct product of any of the above group with a cyclic group of relatively prime order.

3 Milnor's contribution

In [Milnor1957] author proves the following theorem.

Theorem 3.1. If G is a finite group which admits a fixed-point free action on a sphere S^{2n+1}, then for every prime p every subgroup of G of order 2p is cyclic.

This theorem allowed him to choose all possible candidates, from the list compiled by Zassenhaus and Suzuki of groups with periodic cohomology, which could possibly act on S^3. Apart from groups which admit a fixed point free representation in SO(4) he obtained the following family of groups.

\displaystyle Q(8n,k,l) = \langle x,y,z \mid x^2 = (xy)^2=y^{2n}, z^{kl}=1, xzx^{-1} = z^{r}, yzy^{-1} = z^{-1} \rangle,

where 8n,k,l are relatively prime integers and

\displaystyle r \equiv -1 \pmod{k},
\displaystyle r \equiv 1 \pmod{l}.

Groups Q(8n,k,l) were excluded from the list of fundamental groups of 3-manifolds only after resolution of the Geometrization Conjecture.

4 References

\to \{\pm 1\} \to G \to F \to 1.$$ These considerations yields the following theorem. {{beginthm|Theorem|{{cite|Wolf2011}}}} Every finite subgroup of $S^3$ is either * [[Wikipedia:Cyclic_group|cyclic group], * [[Wikipedia:Dicyclic_group|binary dihedral group]], $Q_{4n} = \langle a, x \mid a^{2n}=1, x^2 = a^n, x^{-1} a x = a^{-1} \rangle$, $n \geq 1$, * [[Wikipedia:Binary_tetrahedral_group|binary tetrahedral group]], T = \langle s,t \mid (st)^{2} = s^{3} = t^{3}\rangle$, * [[Wikipedia:Binary_octahedral_group|binary octahedral group]], O = \langle s,t \mid (st)^{2} = s^{3} = t^{4} \rangle$, * [[Wikipedia:Binary_icosahedral_group|binary icosahedralhedral group]], I = \langle s,t \mid (st)^{2} = s^{3} = t^{5} \rangle$. These groups are called binary polyhedral groups. {{endthm}} === Finite subgroups of SO(4) === ; To perform the final step, consider a homomophism $$F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}. $$ Its kernel is equal to $\{(1,1), (-1,-1)\}$. Finite subgroups of $S^3 \times S^3$ can be determined by [[Wikipedia:Goursat%27s_lemma|Goursat's lemma]]. This lemma says, that every finite subgroup of $S^3 \times S^3$ is isomorphic to the fibre product $G \times_{Q} H$, where $G$ and $H$ are finite subgroups of $S^3$ and $Q$ is a common quotient of $G$ and $H$. === Finite fixed-point free subgroups of SO(4) === ; Not every finite subgroup of $SO(4)$ act freely on $S^3$. Following lemma gives necessary and sufficient condition for $F(q_1,q_2)$ to be fixed point free for $q_1, q_2 \in S^3$. {{beginthm|Lemma|{{cite|Wolf2011}}}} Let $q_1, q_2$ be unit quaternions, then $F(q_1,q_2)$ has a fixed point on $S^3$ if, and only if, $q_1$ is conjgate to $q_2$ in $S^3$. {{endthm}} {{beginproof}} This is a simple observation $$q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.$$ {{endproof}} {{beginthm|Theorem|{{cite|Wolf2011}}}} Finite fixed-point free subgroup of $SO(4)$ belongs to the following list * [[Wikipedia:Cyclic_group|cyclic group]], * [[Wikipedia:Quaternion_group|generalised quaternion group]] $Q_{8k}$, * [[Wikipedia:Binary_octahedral_group|binary octahedral group]] $O^{\ast}$, * [[Wikipedia:Binary_tetrahedral_group|binary tetrahedral group]] $T^{\ast}$, * [[Wikipedia:Binary_icosahedral_group|binary icosahedral group]] $I^{\ast}$, * groups $D_{2^k(2n+1)}$, for $k \geq 2$ and $n \geq 1$, with presentation $$\langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$$ * groups $P_{8 \cdot 3^{k}}'$ defined by the following presentation $$\langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,$$ * direct product of any of the above group with a cyclic group of relatively prime order. {{endthm}} == Milnor's contribution == ; In {{cite|Milnor1957}} author proves the following theorem. {{beginthm|Theorem|}} If $G$ is a finite group which admits a fixed-point free action on a sphere $S^{2n+1}$, then for every prime $p$ every subgroup of $G$ of order p$ is cyclic. {{endthm}} This theorem allowed him to choose all possible candidates, from the list compiled by Zassenhaus and Suzuki of groups with periodic cohomology, which could possibly act on $S^3$. Apart from groups which admit a fixed point free representation in $SO(4)$ he obtained the following family of groups. $$Q(8n,k,l) = \langle x,y,z \mid x^2 = (xy)^2=y^{2n}, z^{kl}=1, xzx^{-1} = z^{r}, yzy^{-1} = z^{-1} \rangle,$$ where n,k,l$ are relatively prime integers and $$r \equiv -1 \pmod{k},$$ $$r \equiv 1 \pmod{l}.$$ Groups $Q(8n,k,l)$ were excluded from the list of fundamental groups of $-manifolds only after resolution of the Geometrization Conjecture. == References == {{#RefList:}} [[Category:Theory]]3-dimensional spherical space forms. For the historical context refer to this article. Today the list of groups which arise as fundamental groups of 3-dimensional spherical space forms is known. These are exactly the groups which admit a fixed-point free representation in SO(4). In 1950's Milnor in [Milnor1957] compiled a list of all finite groups which could possibly act freely but not necessarily linearly on S^3. Apart from the groups admitting fixed-point free representations in SO(4), Milnor's list also included a family of finite groups denoted by Q(8n,k,l). Question whether these groups can act on S^3 remained unsolved until the proof of the Geometrization Conjecture was finished by Perelman. The exposition in this article is based on Chapter 7.5 of [Wolf2011], which surveys results of Hopf [Hopf1926] and Seifert-Threlfall [Threlfall&Seifert1931], [Threlfall&Seifert1933].

2 Finite subgroups of SO(4)

The list of finite subgroups of SO(4) can be determined in three steps:

  • determine finite subgroups of SO(3),
  • use the covering map S^3 \to SO(3) to determine finite subgroups of S^3,
  • use the fact that SO(4) is doubly covered by 
    Tex syntax error
     to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of SO(3) one has to analyse the action of these groups on S^2. From Riemann-Hurwitz formula we obtain the following equation

\displaystyle 2 \left(1 - \frac{1}{N} \right) = \sum_{i = 1}^q \left(1 - \frac{1}{n_i} \right),

where N denotes the order of the group, q denotes number of orbits with non-trivial isotropy groups, and n_i denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of SO(3).

Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of SO(3) is either

These groups are called polyhedral groups.

2.2 Finite subgroups of S^3

Let \mathbb{H} denote the algebra of quaternions and treat S^3 as a subset of \mathbb{H} of quaternions of norm 1. Consider the action of S^3 on \mathbb{H} by conjugation

\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).

This action preserves 1, so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation \pi \colon S^3 \to SO(3) with kernel equal to \{\pm 1\}. Hence, \pi is a 2-fold covering map.

If G is a finite subgroup of S^3, let F = \pi(G). If F = G, then, since -1 is the only element of S^3 of order 2, F and G must be both of odd order. Therefore comparing this with the list of finite subgroups of SO(3) yields that F and G are both cyclic of odd order. On the other hand, if F \neq G, then G is an extension of the form

\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.

These considerations yields the following theorem.

Theorem 2.2 [Wolf2011]. Every finite subgroup of S^3 is either

These groups are called binary polyhedral groups.

2.3 Finite subgroups of SO(4)

To perform the final step, consider a homomophism

\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.

Its kernel is equal to \{(1,1), (-1,-1)\}.

Finite subgroups of 
Tex syntax error
 can be determined by Goursat's lemma. This lemma says, that every finite subgroup of 
Tex syntax error
 is isomorphic to the fibre product G \times_{Q} H, where G and H are finite subgroups of S^3 and Q is a common quotient of G and H.

2.4 Finite fixed-point free subgroups of SO(4)

Not every finite subgroup of SO(4) act freely on S^3. Following lemma gives necessary and sufficient condition for F(q_1,q_2) to be fixed point free for q_1, q_2 \in S^3.

Lemma 2.3 [Wolf2011]. Let q_1, q_2 be unit quaternions, then F(q_1,q_2) has a fixed point on S^3 if, and only if, q_1 is conjgate to q_2 in S^3.

Proof. This is a simple observation

\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.
\square

Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of SO(4) belongs to the following list

\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.
  • groups P_{8 \cdot 3^{k}}' defined by the following presentation
\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,
  • direct product of any of the above group with a cyclic group of relatively prime order.

3 Milnor's contribution

In [Milnor1957] author proves the following theorem.

Theorem 3.1. If G is a finite group which admits a fixed-point free action on a sphere S^{2n+1}, then for every prime p every subgroup of G of order 2p is cyclic.

This theorem allowed him to choose all possible candidates, from the list compiled by Zassenhaus and Suzuki of groups with periodic cohomology, which could possibly act on S^3. Apart from groups which admit a fixed point free representation in SO(4) he obtained the following family of groups.

\displaystyle Q(8n,k,l) = \langle x,y,z \mid x^2 = (xy)^2=y^{2n}, z^{kl}=1, xzx^{-1} = z^{r}, yzy^{-1} = z^{-1} \rangle,

where 8n,k,l are relatively prime integers and

\displaystyle r \equiv -1 \pmod{k},
\displaystyle r \equiv 1 \pmod{l}.

Groups Q(8n,k,l) were excluded from the list of fundamental groups of 3-manifolds only after resolution of the Geometrization Conjecture.

4 References

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