Fundamental groups of 3-dimensional spherical space forms
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The purpose of this article is to describe fundamental groups of -dimensional spherical space forms. For the historical context refer to this article. Today we know the whole list of groups which arise as fundamental groups of -dimensional spherical space forms. These are exactly groups which admit a fixed-point free representation in . In 1950's Milnor in [Milnor1957] provided a list of all finite groups which could possibly act freely but not necessarilly linearly on . All groups mentioned earlier belong to this list however there was also included a family of finite groups denoted by . Question whether these groups act on remained unsolved until the proof of the Geometrization Conjecture was finished by Perelman. The exposition of the first part 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 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 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 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
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 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 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
- cyclic group,
- binary dihedral group,
- binary tetrahedral group,
- binary octahedral group or
- 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
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
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
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.