Fundamental groups of 3-dimensional spherical space forms

Latest revision as of 23:45, 11 August 2022

1 Introduction

The purpose of this article is to describe fundamental groups of ${{Authors|Wpolitarczyk}}{{Stub}} == Introduction == <wikitex>; The purpose of this article is to describe fundamental groups of $-dimensional spherical space forms. For the historical context refer to [[Space forms: a history | this article]]. Today the list of groups which arise as fundamental groups of $-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 {{cite|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 $Q(8n,k,l)$ (see Theorem 3.1 below for their definition). The problem 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 {{cite|Wolf2011}}, which surveys results of Hopf {{cite|Hopf1926}} and Seifert-Threlfall {{cite|Threlfall&Seifert1931}}, {{cite|Threlfall&Seifert1933}}. </wikitex> == Finite subgroups of SO(4) == <wikitex>; 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 $S^3 \times S^3$ to determine its finite groups. </wikitex> === Finite subgroups of SO(3) === <wikitex>; To classify finite subgroups of $SO(3)$ one has to analyse the action of these groups on $S^2$. From [[Wikipedia:Riemann–Hurwitz_formula | Riemann-Hurwitz formula]] we obtain the following equation $ \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 $SO(3)$. {{beginthm|Theorem|{{cite|Wolf2011|thm 2.6.5.}}}} Every finite subgroup of $SO(3)$ is either * 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: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:Icosahedral_group|icosahedral group]] $I$, i.e., the symmetry group of the regular icosahedron. These groups are called polyhedral groups. {{endthm}} </wikitex> === Finite subgroups of S^3 === <wikitex>; Let $\mathbb{H}$ denote the algebra of quaternions and treat $S^3$ as a subset of $\mathbb{H}$ of quaternions of norm 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 $Q(8n,k,l)$ (see Theorem 3.1 below for their definition). The problem 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

  $S^3 \times S^3$ to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of $SO(3)$, we have to analyze 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 $SO(3)$.

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.

2.3 Finite subgroups of SO(4)

To perform the final step, consider the homomophism

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

where $q \in \mathbb{H} = \mathbb{R}^4$. The kernel of $F$ is equal to $\{(1,1), (-1,-1)\}$.

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Consequently, any finite subgroup of $SO(4)$ can be presented as a quotient $K / (K \cap \{(1,1),(-1,-1)\})$, where $K = G \times_Q H$ is the fiber product of two finite subgroups $G$ and $H$ of $S^3$.

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

Not every finite subgroup of $SO(4)$ can act freely on $S^3$. The Following lemma gives a necessary and sufficient condition for the map $F(q_1,q_2)$, defined in the previous section, to be fixed-point free for $q_1, q_2 \in S^3$.

Proof. This is a simple observation

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

$\square$

Using the above lemma and the classification of finite subgroups of $SO(4)$ described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of $SO(4)$.

Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of $SO(4)$ belongs to the following list:

• finite cyclic groups,
• generalised quaternion groups $Q_{8k}$, $k \geq 1$, i.e., a binary dihedral group (see Theorem 2.2 above) , where $n=2k$ is even,
• the binary octahedral group $O^{\ast}$,
• the binary tetrahedral group $T^{\ast}$,
• the binary icosahedral group $I^{\ast}$,
• groups $D_{2^k(2n+1)}$, for $k \geq 2$ and $n \geq 1$, with presentation

$$\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$$

• groups $P_{8 \cdot 3^{k}}'$ defined by the following presentation

$$\displaystyle \langle x,y,z \mid x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,$$

• direct product of any of the above groups with a cyclic group of relatively prime order.

3 Milnor's contribution

Milnor in [Milnor1957] proved the following theorem.

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

• [Hopf1926] H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, (1926), 313-339. Zbl 51.0439.05
• [Milnor1957] J. Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
• [Threlfall&Seifert1931] W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, Math. Ann. 104 (1931), no.1, 1–70. MR1512649 Zbl 0006.03403
• [Threlfall&Seifert1933] W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes (Schluß), Math. Ann. 107 (1933), no.1, 543–586. MR1512817 Zbl 58.1203.01
• [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219

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

  $S^3 \times S^3$ to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of $SO(3)$, we have to analyze 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 $SO(3)$.

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.

2.3 Finite subgroups of SO(4)

To perform the final step, consider the homomophism

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

where $q \in \mathbb{H} = \mathbb{R}^4$. The kernel of $F$ is equal to $\{(1,1), (-1,-1)\}$.

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Consequently, any finite subgroup of $SO(4)$ can be presented as a quotient $K / (K \cap \{(1,1),(-1,-1)\})$, where $K = G \times_Q H$ is the fiber product of two finite subgroups $G$ and $H$ of $S^3$.

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

Not every finite subgroup of $SO(4)$ can act freely on $S^3$. The Following lemma gives a necessary and sufficient condition for the map $F(q_1,q_2)$, defined in the previous section, to be fixed-point free for $q_1, q_2 \in S^3$.

Proof. This is a simple observation

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

$\square$

Using the above lemma and the classification of finite subgroups of $SO(4)$ described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of $SO(4)$.

Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of $SO(4)$ belongs to the following list:

• finite cyclic groups,
• generalised quaternion groups $Q_{8k}$, $k \geq 1$, i.e., a binary dihedral group (see Theorem 2.2 above) , where $n=2k$ is even,
• the binary octahedral group $O^{\ast}$,
• the binary tetrahedral group $T^{\ast}$,
• the binary icosahedral group $I^{\ast}$,
• groups $D_{2^k(2n+1)}$, for $k \geq 2$ and $n \geq 1$, with presentation

$$\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$$

• groups $P_{8 \cdot 3^{k}}'$ defined by the following presentation

$$\displaystyle \langle x,y,z \mid x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,$$

• direct product of any of the above groups with a cyclic group of relatively prime order.

3 Milnor's contribution

Milnor in [Milnor1957] proved the following theorem.

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

• [Hopf1926] H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, (1926), 313-339. Zbl 51.0439.05
• [Milnor1957] J. Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
• [Threlfall&Seifert1931] W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, Math. Ann. 104 (1931), no.1, 1–70. MR1512649 Zbl 0006.03403
• [Threlfall&Seifert1933] W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes (Schluß), Math. Ann. 107 (1933), no.1, 543–586. MR1512817 Zbl 58.1203.01
• [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219

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

  $S^3 \times S^3$ to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of $SO(3)$, we have to analyze 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 $SO(3)$.

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.

2.3 Finite subgroups of SO(4)

To perform the final step, consider the homomophism

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

where $q \in \mathbb{H} = \mathbb{R}^4$. The kernel of $F$ is equal to $\{(1,1), (-1,-1)\}$.

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Consequently, any finite subgroup of $SO(4)$ can be presented as a quotient $K / (K \cap \{(1,1),(-1,-1)\})$, where $K = G \times_Q H$ is the fiber product of two finite subgroups $G$ and $H$ of $S^3$.

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

Not every finite subgroup of $SO(4)$ can act freely on $S^3$. The Following lemma gives a necessary and sufficient condition for the map $F(q_1,q_2)$, defined in the previous section, to be fixed-point free for $q_1, q_2 \in S^3$.

Proof. This is a simple observation

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

$\square$

Using the above lemma and the classification of finite subgroups of $SO(4)$ described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of $SO(4)$.

Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of $SO(4)$ belongs to the following list:

• finite cyclic groups,
• generalised quaternion groups $Q_{8k}$, $k \geq 1$, i.e., a binary dihedral group (see Theorem 2.2 above) , where $n=2k$ is even,
• the binary octahedral group $O^{\ast}$,
• the binary tetrahedral group $T^{\ast}$,
• the binary icosahedral group $I^{\ast}$,
• groups $D_{2^k(2n+1)}$, for $k \geq 2$ and $n \geq 1$, with presentation

$$\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$$

• groups $P_{8 \cdot 3^{k}}'$ defined by the following presentation

$$\displaystyle \langle x,y,z \mid x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,$$

• direct product of any of the above groups with a cyclic group of relatively prime order.

3 Milnor's contribution

Milnor in [Milnor1957] proved the following theorem.

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

• [Hopf1926] H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, (1926), 313-339. Zbl 51.0439.05
• [Milnor1957] J. Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
• [Threlfall&Seifert1931] W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, Math. Ann. 104 (1931), no.1, 1–70. MR1512649 Zbl 0006.03403
• [Threlfall&Seifert1933] W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes (Schluß), Math. Ann. 107 (1933), no.1, 543–586. MR1512817 Zbl 58.1203.01
• [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219

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

  $S^3 \times S^3$ to determine its finite groups.

2.1 Finite subgroups of SO(3)

To classify finite subgroups of $SO(3)$, we have to analyze 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 $SO(3)$.

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.

2.3 Finite subgroups of SO(4)

To perform the final step, consider the homomophism

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

where $q \in \mathbb{H} = \mathbb{R}^4$. The kernel of $F$ is equal to $\{(1,1), (-1,-1)\}$.

Tex syntax error

Tex syntax error

Consequently, any finite subgroup of $SO(4)$ can be presented as a quotient $K / (K \cap \{(1,1),(-1,-1)\})$, where $K = G \times_Q H$ is the fiber product of two finite subgroups $G$ and $H$ of $S^3$.

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

Not every finite subgroup of $SO(4)$ can act freely on $S^3$. The Following lemma gives a necessary and sufficient condition for the map $F(q_1,q_2)$, defined in the previous section, to be fixed-point free for $q_1, q_2 \in S^3$.

Proof. This is a simple observation

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

$\square$

Using the above lemma and the classification of finite subgroups of $SO(4)$ described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of $SO(4)$.

Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of $SO(4)$ belongs to the following list:

• finite cyclic groups,
• generalised quaternion groups $Q_{8k}$, $k \geq 1$, i.e., a binary dihedral group (see Theorem 2.2 above) , where $n=2k$ is even,
• the binary octahedral group $O^{\ast}$,
• the binary tetrahedral group $T^{\ast}$,
• the binary icosahedral group $I^{\ast}$,
• groups $D_{2^k(2n+1)}$, for $k \geq 2$ and $n \geq 1$, with presentation

$$\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$$

• groups $P_{8 \cdot 3^{k}}'$ defined by the following presentation

$$\displaystyle \langle x,y,z \mid x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,$$

• direct product of any of the above groups with a cyclic group of relatively prime order.

3 Milnor's contribution

Milnor in [Milnor1957] proved the following theorem.

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

• [Hopf1926] H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, (1926), 313-339. Zbl 51.0439.05
• [Milnor1957] J. Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
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