Fundamental groups of 3-dimensional spherical space forms
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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 $S^3$. | + | 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> | ||
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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 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 of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is 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]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary octahedral group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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 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 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 of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is 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]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary octahedral group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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 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 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 of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is 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]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary octahedral group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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 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 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 of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is 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]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary octahedral group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- 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 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