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. 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 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 one has to analyse the action of these groups on
.
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),](/images/math/3/9/2/3924ac1fc2fb12e726f72ae614524f24.png)
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
![\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).](/images/math/0/8/3/0837a3016fe4671c5d6f87337ba90512.png)
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
![\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.](/images/math/6/e/4/6e4c1c95832c7e254833cf0fa6282c82.png)
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 a homomophism
![\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.](/images/math/8/3/1/8314a35756bc568945624f1cf79e13eb.png)
Its kernel 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
.
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
![\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.](/images/math/4/a/2/4a2ec3ae9dfd040f89ff12611c7b6471.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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
![\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.](/images/math/8/f/8/8f82becf6578414ab635c8d45dbb32c4.png)
- groups
defined by the following presentation
![\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,](/images/math/4/6/9/4693c9214169b6317cbe8e23f69067a0.png)
- 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.
![\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,](/images/math/4/a/1/4a1b4b9076a480fb20fa82f8d3702c7b.png)
where are relatively prime integers and
![\displaystyle r \equiv -1 \pmod{k},](/images/math/b/b/3/bb330f98cfce36a13f7e754cff5e79b5.png)
![\displaystyle r \equiv 1 \pmod{l}.](/images/math/4/9/5/495812f582658e8fa09e6fabd8abcc53.png)
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
![3](/images/math/0/9/9/0994084a0f5f0cf820f737bbeca8a4b0.png)
![SO(4)](/images/math/0/f/4/0f479d19ba432a39dbb076e41511c02a.png)
![S^3](/images/math/0/e/b/0ebf327477419ed9318cd89c957c1f9c.png)
![SO(4)](/images/math/0/f/4/0f479d19ba432a39dbb076e41511c02a.png)
![Q(8n,k,l)](/images/math/0/8/7/08752954d005a712d84815f2b3f0bd2b.png)
![S^3](/images/math/0/e/b/0ebf327477419ed9318cd89c957c1f9c.png)
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 one has to analyse the action of these groups on
.
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),](/images/math/3/9/2/3924ac1fc2fb12e726f72ae614524f24.png)
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
![\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).](/images/math/0/8/3/0837a3016fe4671c5d6f87337ba90512.png)
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
![\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.](/images/math/6/e/4/6e4c1c95832c7e254833cf0fa6282c82.png)
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 a homomophism
![\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.](/images/math/8/3/1/8314a35756bc568945624f1cf79e13eb.png)
Its kernel 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
.
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
![\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.](/images/math/4/a/2/4a2ec3ae9dfd040f89ff12611c7b6471.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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
![\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.](/images/math/8/f/8/8f82becf6578414ab635c8d45dbb32c4.png)
- groups
defined by the following presentation
![\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,](/images/math/4/6/9/4693c9214169b6317cbe8e23f69067a0.png)
- 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.
![\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,](/images/math/4/a/1/4a1b4b9076a480fb20fa82f8d3702c7b.png)
where are relatively prime integers and
![\displaystyle r \equiv -1 \pmod{k},](/images/math/b/b/3/bb330f98cfce36a13f7e754cff5e79b5.png)
![\displaystyle r \equiv 1 \pmod{l}.](/images/math/4/9/5/495812f582658e8fa09e6fabd8abcc53.png)
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
![3](/images/math/0/9/9/0994084a0f5f0cf820f737bbeca8a4b0.png)
![SO(4)](/images/math/0/f/4/0f479d19ba432a39dbb076e41511c02a.png)
![S^3](/images/math/0/e/b/0ebf327477419ed9318cd89c957c1f9c.png)
![SO(4)](/images/math/0/f/4/0f479d19ba432a39dbb076e41511c02a.png)
![Q(8n,k,l)](/images/math/0/8/7/08752954d005a712d84815f2b3f0bd2b.png)
![S^3](/images/math/0/e/b/0ebf327477419ed9318cd89c957c1f9c.png)
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 one has to analyse the action of these groups on
.
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),](/images/math/3/9/2/3924ac1fc2fb12e726f72ae614524f24.png)
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
![\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).](/images/math/0/8/3/0837a3016fe4671c5d6f87337ba90512.png)
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
![\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.](/images/math/6/e/4/6e4c1c95832c7e254833cf0fa6282c82.png)
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 a homomophism
![\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.](/images/math/8/3/1/8314a35756bc568945624f1cf79e13eb.png)
Its kernel 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
.
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
![\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.](/images/math/4/a/2/4a2ec3ae9dfd040f89ff12611c7b6471.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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
![\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.](/images/math/8/f/8/8f82becf6578414ab635c8d45dbb32c4.png)
- groups
defined by the following presentation
![\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,](/images/math/4/6/9/4693c9214169b6317cbe8e23f69067a0.png)
- 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.
![\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,](/images/math/4/a/1/4a1b4b9076a480fb20fa82f8d3702c7b.png)
where are relatively prime integers and
![\displaystyle r \equiv -1 \pmod{k},](/images/math/b/b/3/bb330f98cfce36a13f7e754cff5e79b5.png)
![\displaystyle r \equiv 1 \pmod{l}.](/images/math/4/9/5/495812f582658e8fa09e6fabd8abcc53.png)
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
![3](/images/math/0/9/9/0994084a0f5f0cf820f737bbeca8a4b0.png)
![SO(4)](/images/math/0/f/4/0f479d19ba432a39dbb076e41511c02a.png)
![S^3](/images/math/0/e/b/0ebf327477419ed9318cd89c957c1f9c.png)
![SO(4)](/images/math/0/f/4/0f479d19ba432a39dbb076e41511c02a.png)
![Q(8n,k,l)](/images/math/0/8/7/08752954d005a712d84815f2b3f0bd2b.png)
![S^3](/images/math/0/e/b/0ebf327477419ed9318cd89c957c1f9c.png)
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 one has to analyse the action of these groups on
.
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),](/images/math/3/9/2/3924ac1fc2fb12e726f72ae614524f24.png)
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
![\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).](/images/math/0/8/3/0837a3016fe4671c5d6f87337ba90512.png)
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
![\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.](/images/math/6/e/4/6e4c1c95832c7e254833cf0fa6282c82.png)
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 a homomophism
![\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.](/images/math/8/3/1/8314a35756bc568945624f1cf79e13eb.png)
Its kernel 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
.
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
![\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.](/images/math/4/a/2/4a2ec3ae9dfd040f89ff12611c7b6471.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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
![\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.](/images/math/8/f/8/8f82becf6578414ab635c8d45dbb32c4.png)
- groups
defined by the following presentation
![\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,](/images/math/4/6/9/4693c9214169b6317cbe8e23f69067a0.png)
- 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.
![\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,](/images/math/4/a/1/4a1b4b9076a480fb20fa82f8d3702c7b.png)
where are relatively prime integers and
![\displaystyle r \equiv -1 \pmod{k},](/images/math/b/b/3/bb330f98cfce36a13f7e754cff5e79b5.png)
![\displaystyle r \equiv 1 \pmod{l}.](/images/math/4/9/5/495812f582658e8fa09e6fabd8abcc53.png)
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