Fundamental groups of 3-dimensional spherical space forms
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | The purpose of this article is to describe fundamental groups of $3$-dimensional spherical space forms. For the historical context refer to [[Space forms: a history | this article]]. | + | The purpose of this article is to describe fundamental groups of $3$-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 $3$-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 {{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> | </wikitex> | ||
− | == Finite subgroups of | + | == Finite subgroups of SO(4) == |
<wikitex>; | <wikitex>; | ||
− | + | The list of finite subgroups of $SO(4)$ can be determined in three steps: | |
* determine finite subgroups of $SO(3)$, | * determine finite subgroups of $SO(3)$, | ||
* use the covering map $S^3 \to SO(3)$ to determine finite subgroups of $S^3$, | * use the covering map $S^3 \to SO(3)$ to determine finite subgroups of $S^3$, | ||
− | * use the fact that $SO(4)$ doubly | + | * use the fact that $SO(4)$ is doubly covered by $S^3 \times S^3$ to determine its finite groups. |
</wikitex> | </wikitex> | ||
− | === Finite subgroups of | + | === Finite subgroups of SO(3) === |
<wikitex>; | <wikitex>; | ||
− | + | To classify finite subgroups of $SO(3)$, we have to analyze the action of these groups on $S^2$. | |
+ | From [[Wikipedia:Riemann–Hurwitz_formula | Riemann-Hurwitz formula]] we obtain the following equation | ||
$$2 \left(1 - \frac{1}{N} \right) = \sum_{i = 1}^q \left(1 - \frac{1}{n_i} \right),$$ | $$2 \left(1 - \frac{1}{N} \right) = \sum_{i = 1}^q \left(1 - \frac{1}{n_i} \right),$$ | ||
− | where $N$ denotes the order of the group, $q$ denotes number of orbits with non-trivial isotropy and $n_i$ denotes the order of the respective isotropy | + | 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.}}}} | {{beginthm|Theorem|{{cite|Wolf2011|thm 2.6.5.}}}} | ||
− | Every finite subgroup of $SO(3)$ is either [[Wikipedia:Cyclic_group|cyclic]], [[Wikipedia:Dihedral_group|dihedral]], [[Wikipedia:Tetrahedral_group|tetrahedral]], [[Wikipedia:Octahedral_group|octahedral]] | + | 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}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
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=== Finite subgroups of $S^3$ === | === Finite subgroups of $S^3$ === | ||
<wikitex>; | <wikitex>; | ||
− | Let $\mathbb{H}$ denote the algebra of quaternions and treat $S^3$ as a subset of $\mathbb{H}$ of quaternions of norm $1$. Consider | + | 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 |
$$q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).$$ | $$q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).$$ | ||
− | This action preserves $1$, so it | + | 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 $S^3$, | + | 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 |
$$1 \to \{\pm 1\} \to G \to F \to 1.$$ | $$1 \to \{\pm 1\} \to G \to F \to 1.$$ | ||
+ | These considerations yields the following theorem. | ||
{{beginthm|Theorem|{{cite|Wolf2011}}}} | {{beginthm|Theorem|{{cite|Wolf2011}}}} | ||
− | Every finite subgroup of $S^3$ is either [[Wikipedia:Cyclic_group|cyclic group]], [[Wikipedia:Dicyclic_group|binary dihedral group]], [[Wikipedia:Binary_tetrahedral_group|binary tetrahedral group]], [[Wikipedia:Binary_octahedral_group|binary octahedral group]] | + | Every finite subgroup of $S^3$ is either |
+ | * a [[Wikipedia:Cyclic_group|cyclic group]], | ||
+ | * a [[Wikipedia:Dicyclic_group|binary dihedral group]], $Q_{4n} = \langle a, x \mid a^{2n}=1, x^2 = a^n, x^{-1} a x = a^{-1} \rangle$, $n \geq 1$, | ||
+ | * a [[Wikipedia:Binary_tetrahedral_group|binary tetrahedral group]], $T^{\ast} = \langle s,t \mid (st)^{2} = s^{3} = t^{3}\rangle$, | ||
+ | * a [[Wikipedia:Binary_octahedral_group|binary octahedral group]], $O^{\ast} = \langle s,t \mid (st)^{2} = s^{3} = t^{4} \rangle$, | ||
+ | * a [[Wikipedia:Binary_icosahedral_group|binary icosahedralhedral group]], $I^{\ast} = \langle s,t \mid (st)^{2} = s^{3} = t^{5} \rangle$. | ||
+ | These groups are called binary polyhedral groups. | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
− | === Finite subgroups of | + | === Finite subgroups of SO(4) === |
<wikitex>; | <wikitex>; | ||
− | To perform the final step, consider | + | To perform the final step, consider the homomophism |
− | $$F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1} | + | $$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}, $$ |
− | + | where $q \in \mathbb{H} = \mathbb{R}^4$. | |
+ | The kernel of $F$ is equal to $\{(1,1), (-1,-1)\}$. | ||
Finite subgroups of $S^3 \times S^3$ can be determined by [[Wikipedia:Goursat%27s_lemma|Goursat's lemma]]. This lemma says, that every finite subgroup of $S^3 \times S^3$ is isomorphic to the fibre product $G \times_{Q} H$, where $G$ and $H$ are finite subgroups of $S^3$ and $Q$ is a common quotient of $G$ and $H$. | Finite subgroups of $S^3 \times S^3$ can be determined by [[Wikipedia:Goursat%27s_lemma|Goursat's lemma]]. This lemma says, that every finite subgroup of $S^3 \times S^3$ is isomorphic to the fibre product $G \times_{Q} H$, where $G$ and $H$ are finite subgroups of $S^3$ and $Q$ is a common quotient of $G$ and $H$. | ||
+ | Consequently, any finite subgroup of $SO(4)$ can be presented as a quotient $K / (K \cap \{(1,1),(-1,-1)\})$, where $K = G \times_Q H$ is the fiber product of two finite subgroups $G$ and $H$ of $S^3$. | ||
</wikitex> | </wikitex> | ||
− | === Finite fixed-point free subgroups of | + | === Finite fixed-point free subgroups of SO(4) === |
<wikitex>; | <wikitex>; | ||
− | Not every finite subgroup of $SO(4)$ act freely on $S^3$. Following lemma gives necessary and sufficient condition for $F(q_1,q_2)$ to be fixed point free for $q_1, q_2 \in S^3$. | + | Not every finite subgroup of $SO(4)$ can act freely on $S^3$. The Following lemma gives a necessary and sufficient condition for the map $F(q_1,q_2)$, defined in the previous section, to be fixed-point free for $q_1, q_2 \in S^3$. |
+ | |||
{{beginthm|Lemma|{{cite|Wolf2011}}}} | {{beginthm|Lemma|{{cite|Wolf2011}}}} | ||
Let $q_1, q_2$ be unit quaternions, then $F(q_1,q_2)$ has a fixed point on $S^3$ if, and only if, $q_1$ is conjgate to $q_2$ in $S^3$. | Let $q_1, q_2$ be unit quaternions, then $F(q_1,q_2)$ has a fixed point on $S^3$ if, and only if, $q_1$ is conjgate to $q_2$ in $S^3$. | ||
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$$q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.$$ | $$q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.$$ | ||
{{endproof}} | {{endproof}} | ||
+ | |||
+ | Using the above lemma and the classification of finite subgroups of $SO(4)$ described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of $SO(4)$. | ||
{{beginthm|Theorem|{{cite|Wolf2011}}}} | {{beginthm|Theorem|{{cite|Wolf2011}}}} | ||
− | + | Any finite, fixed-point free subgroup of $SO(4)$ belongs to the following list: | |
− | * [[Wikipedia:Cyclic_group|cyclic | + | * [[Wikipedia:Cyclic_group|finite cyclic groups]], |
− | * [[Wikipedia:Quaternion_group|generalised quaternion | + | * [[Wikipedia:Quaternion_group|generalised quaternion groups]] $Q_{8k}$, $k \geq 1$, i.e., a binary dihedral group (see Theorem 2.2 above) , where $n=2k$ is even, |
− | * [[Wikipedia:Binary_tetrahedral_group|binary tetrahedral group]] $T^{\ast}$, | + | * the [[Wikipedia:Binary_octahedral_group|binary octahedral group]] $O^{\ast}$, |
− | * [[Wikipedia:Binary_icosahedral_group|binary icosahedral group]] $I^{\ast}$, | + | * the [[Wikipedia:Binary_tetrahedral_group|binary tetrahedral group]] $T^{\ast}$, |
+ | * the [[Wikipedia:Binary_icosahedral_group|binary icosahedral group]] $I^{\ast}$, | ||
* groups $D_{2^k(2n+1)}$, for $k \geq 2$ and $n \geq 1$, with presentation | * groups $D_{2^k(2n+1)}$, for $k \geq 2$ and $n \geq 1$, with presentation | ||
$$\langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$$ | $$\langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$$ | ||
* groups $P_{8 \cdot 3^{k}}'$ defined by the following presentation | * groups $P_{8 \cdot 3^{k}}'$ defined by the following presentation | ||
− | $$\langle x,y,z | + | $$\langle x,y,z \mid x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,$$ |
− | * direct product of any of the above | + | * direct product of any of the above groups with a cyclic group of relatively prime order. |
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
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== Milnor's contribution == | == Milnor's contribution == | ||
<wikitex>; | <wikitex>; | ||
+ | Milnor in {{cite|Milnor1957}} proved 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 $2p$ 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 $8n,k,l$ 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 $3$-manifolds only after resolution of the Geometrization Conjecture. | ||
</wikitex> | </wikitex> | ||
Latest revision as of 23:45, 11 August 2022
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Contents |
1 Introduction
The purpose of this article is to describe fundamental groups of -dimensional spherical space forms. For the historical context refer to 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 . In 1950's Milnor in [Milnor1957] compiled a list of all finite groups which could possibly act freely but not necessarily linearly on . Apart from the groups admitting fixed-point free representations in , Milnor's list also included a family of finite groups denoted by (see Theorem 3.1 below for their definition). The problem whether these groups can act on 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 can be determined in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that is doubly covered by
Tex syntax error
to determine its finite groups.
2.1 Finite subgroups of SO(3)
To classify finite subgroups of , we have to analyze the action of these groups on . From Riemann-Hurwitz formula we obtain the following equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy groups, and denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of .
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either
- a cyclic group,
- a dihedral group, i.e., , ,
- a tetrahedral group , i.e., the symmetry group of the regular tetrahedron,
- a octahedral group , i.e., the symmetry group of the regular octahedron,
- a icosahedral group , i.e., the symmetry group of the regular icosahedron.
These groups are called polyhedral groups.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider the action of on by conjugation
This action preserves , so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation with kernel equal to . Hence, is a 2-fold covering map.
If is a finite subgroup of , let . If , then, since is the only element of of order , and must be both of odd order. Therefore comparing this with the list of finite subgroups of yields that and are both cyclic of odd order. On the other hand, if , then is an extension of the form
These considerations yields the following theorem.
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either
- a cyclic group,
- a binary dihedral group, , ,
- a binary tetrahedral group, ,
- a binary octahedral group, ,
- a binary icosahedralhedral group, .
These groups are called binary polyhedral groups.
2.3 Finite subgroups of SO(4)
To perform the final step, consider the homomophism
where . The kernel of is equal to .
Finite subgroups ofTex syntax errorcan be determined by Goursat's lemma. This lemma says, that every finite subgroup of
Tex syntax erroris isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
Consequently, any finite subgroup of can be presented as a quotient , where is the fiber product of two finite subgroups and of .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of can act freely on . The Following lemma gives a necessary and sufficient condition for the map , defined in the previous section, to be fixed-point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Using the above lemma and the classification of finite subgroups of described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of .
Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of belongs to the following list:
- finite cyclic groups,
- generalised quaternion groups , , i.e., a binary dihedral group (see Theorem 2.2 above) , where is even,
- the binary octahedral group ,
- the binary tetrahedral group ,
- the binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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.
Theorem 3.1. If is a finite group which admits a fixed-point free action on a sphere , then for every prime every subgroup of of order 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 . Apart from groups which admit a fixed point free representation in he obtained the following family of groups.
where are relatively prime integers and
Groups were excluded from the list of fundamental groups of -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 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 can be determined in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that is doubly covered by
Tex syntax error
to determine its finite groups.
2.1 Finite subgroups of SO(3)
To classify finite subgroups of , we have to analyze the action of these groups on . From Riemann-Hurwitz formula we obtain the following equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy groups, and denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of .
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either
- a cyclic group,
- a dihedral group, i.e., , ,
- a tetrahedral group , i.e., the symmetry group of the regular tetrahedron,
- a octahedral group , i.e., the symmetry group of the regular octahedron,
- a icosahedral group , i.e., the symmetry group of the regular icosahedron.
These groups are called polyhedral groups.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider the action of on by conjugation
This action preserves , so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation with kernel equal to . Hence, is a 2-fold covering map.
If is a finite subgroup of , let . If , then, since is the only element of of order , and must be both of odd order. Therefore comparing this with the list of finite subgroups of yields that and are both cyclic of odd order. On the other hand, if , then is an extension of the form
These considerations yields the following theorem.
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either
- a cyclic group,
- a binary dihedral group, , ,
- a binary tetrahedral group, ,
- a binary octahedral group, ,
- a binary icosahedralhedral group, .
These groups are called binary polyhedral groups.
2.3 Finite subgroups of SO(4)
To perform the final step, consider the homomophism
where . The kernel of is equal to .
Finite subgroups ofTex syntax errorcan be determined by Goursat's lemma. This lemma says, that every finite subgroup of
Tex syntax erroris isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
Consequently, any finite subgroup of can be presented as a quotient , where is the fiber product of two finite subgroups and of .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of can act freely on . The Following lemma gives a necessary and sufficient condition for the map , defined in the previous section, to be fixed-point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Using the above lemma and the classification of finite subgroups of described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of .
Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of belongs to the following list:
- finite cyclic groups,
- generalised quaternion groups , , i.e., a binary dihedral group (see Theorem 2.2 above) , where is even,
- the binary octahedral group ,
- the binary tetrahedral group ,
- the binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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.
Theorem 3.1. If is a finite group which admits a fixed-point free action on a sphere , then for every prime every subgroup of of order 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 . Apart from groups which admit a fixed point free representation in he obtained the following family of groups.
where are relatively prime integers and
Groups were excluded from the list of fundamental groups of -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 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 can be determined in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that is doubly covered by
Tex syntax error
to determine its finite groups.
2.1 Finite subgroups of SO(3)
To classify finite subgroups of , we have to analyze the action of these groups on . From Riemann-Hurwitz formula we obtain the following equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy groups, and denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of .
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either
- a cyclic group,
- a dihedral group, i.e., , ,
- a tetrahedral group , i.e., the symmetry group of the regular tetrahedron,
- a octahedral group , i.e., the symmetry group of the regular octahedron,
- a icosahedral group , i.e., the symmetry group of the regular icosahedron.
These groups are called polyhedral groups.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider the action of on by conjugation
This action preserves , so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation with kernel equal to . Hence, is a 2-fold covering map.
If is a finite subgroup of , let . If , then, since is the only element of of order , and must be both of odd order. Therefore comparing this with the list of finite subgroups of yields that and are both cyclic of odd order. On the other hand, if , then is an extension of the form
These considerations yields the following theorem.
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either
- a cyclic group,
- a binary dihedral group, , ,
- a binary tetrahedral group, ,
- a binary octahedral group, ,
- a binary icosahedralhedral group, .
These groups are called binary polyhedral groups.
2.3 Finite subgroups of SO(4)
To perform the final step, consider the homomophism
where . The kernel of is equal to .
Finite subgroups ofTex syntax errorcan be determined by Goursat's lemma. This lemma says, that every finite subgroup of
Tex syntax erroris isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
Consequently, any finite subgroup of can be presented as a quotient , where is the fiber product of two finite subgroups and of .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of can act freely on . The Following lemma gives a necessary and sufficient condition for the map , defined in the previous section, to be fixed-point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Using the above lemma and the classification of finite subgroups of described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of .
Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of belongs to the following list:
- finite cyclic groups,
- generalised quaternion groups , , i.e., a binary dihedral group (see Theorem 2.2 above) , where is even,
- the binary octahedral group ,
- the binary tetrahedral group ,
- the binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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.
Theorem 3.1. If is a finite group which admits a fixed-point free action on a sphere , then for every prime every subgroup of of order 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 . Apart from groups which admit a fixed point free representation in he obtained the following family of groups.
where are relatively prime integers and
Groups were excluded from the list of fundamental groups of -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 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 can be determined in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that is doubly covered by
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to determine its finite groups.
2.1 Finite subgroups of SO(3)
To classify finite subgroups of , we have to analyze the action of these groups on . From Riemann-Hurwitz formula we obtain the following equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy groups, and denotes the order of the respective isotropy group. Solutions to this equation yield the desired list of finite subgroups of .
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either
- a cyclic group,
- a dihedral group, i.e., , ,
- a tetrahedral group , i.e., the symmetry group of the regular tetrahedron,
- a octahedral group , i.e., the symmetry group of the regular octahedron,
- a icosahedral group , i.e., the symmetry group of the regular icosahedron.
These groups are called polyhedral groups.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider the action of on by conjugation
This action preserves , so it descends to a norm-preserving action on the set of imaginary quaternions. Therefore this action yields a surjective representation with kernel equal to . Hence, is a 2-fold covering map.
If is a finite subgroup of , let . If , then, since is the only element of of order , and must be both of odd order. Therefore comparing this with the list of finite subgroups of yields that and are both cyclic of odd order. On the other hand, if , then is an extension of the form
These considerations yields the following theorem.
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either
- a cyclic group,
- a binary dihedral group, , ,
- a binary tetrahedral group, ,
- a binary octahedral group, ,
- a binary icosahedralhedral group, .
These groups are called binary polyhedral groups.
2.3 Finite subgroups of SO(4)
To perform the final step, consider the homomophism
where . The kernel of is equal to .
Finite subgroups ofTex syntax errorcan be determined by Goursat's lemma. This lemma says, that every finite subgroup of
Tex syntax erroris isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
Consequently, any finite subgroup of can be presented as a quotient , where is the fiber product of two finite subgroups and of .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of can act freely on . The Following lemma gives a necessary and sufficient condition for the map , defined in the previous section, to be fixed-point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Using the above lemma and the classification of finite subgroups of described in the previous section, we can obtain a complete list of finite, fixed-point free subgroups of .
Theorem 2.4 [Wolf2011]. Any finite, fixed-point free subgroup of belongs to the following list:
- finite cyclic groups,
- generalised quaternion groups , , i.e., a binary dihedral group (see Theorem 2.2 above) , where is even,
- the binary octahedral group ,
- the binary tetrahedral group ,
- the binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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.
Theorem 3.1. If is a finite group which admits a fixed-point free action on a sphere , then for every prime every subgroup of of order 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 . Apart from groups which admit a fixed point free representation in he obtained the following family of groups.
where are relatively prime integers and
Groups were excluded from the list of fundamental groups of -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 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