Fundamental groups of 3-dimensional spherical space forms
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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. Fundamental groups of -dimensional spherical space forms can be divided into two families. First family consists of groups which admit a linear fixed-point free representation into thus yielding a -manifold admitting riemannian metric of constant curvature. Second family consists of groups which do not admit a representation into , yet they can act without fixed points on . The exposition is based on [Wolf2011].
2 Finite subgroups of $SO(4)$
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers
Tex syntax error
to determine its finite groups.
2.1 Finite subgroups of $SO(3)$
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and denotes the order of the respective isotropy subgroup. Solutions to this equation yield the desired list.
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of $SO(4)$
To perform the final step, consider a homomophism
Its kernel 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 .
2.4 Finite fixed-point free subgroups of $SO(4)$
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for 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
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion 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
4 References
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219
2 Finite subgroups of $SO(4)$
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers
Tex syntax error
to determine its finite groups.
2.1 Finite subgroups of $SO(3)$
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and denotes the order of the respective isotropy subgroup. Solutions to this equation yield the desired list.
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of $SO(4)$
To perform the final step, consider a homomophism
Its kernel 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 .
2.4 Finite fixed-point free subgroups of $SO(4)$
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for 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
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion 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
4 References
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219
2 Finite subgroups of $SO(4)$
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers
Tex syntax error
to determine its finite groups.
2.1 Finite subgroups of $SO(3)$
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and denotes the order of the respective isotropy subgroup. Solutions to this equation yield the desired list.
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of $SO(4)$
To perform the final step, consider a homomophism
Its kernel 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 .
2.4 Finite fixed-point free subgroups of $SO(4)$
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for 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
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion 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
4 References
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219
2 Finite subgroups of $SO(4)$
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers
Tex syntax error
to determine its finite groups.
2.1 Finite subgroups of $SO(3)$
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and denotes the order of the respective isotropy subgroup. Solutions to this equation yield the desired list.
Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of $S^3$
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of $SO(4)$
To perform the final step, consider a homomophism
Its kernel 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 .
2.4 Finite fixed-point free subgroups of $SO(4)$
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for 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
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion 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
4 References
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219