# Isometry classification of spherical space forms

A spherical space form is a compact Riemannian manifold of constant sectional curvature $1$, or equivalently, the quotient of the unit sphere by a finite group of orthogonal transformations that have no fixed points.

It is known that diffeomorphic spherical space forms are isometric. This was proved by de Rham in 1950. His paper is not readily available to me, so I ask

1. Is there a modern treatment of this result in the literature?

2. Is there a simple proof for 3-dimensional lens spaces? (The proof based on Hamilton's theorem that Ricci flow preserves positive curvature in dimension 3 does not count as simple).

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Doesn't it suffice to prove that Diff(M) is path connected? I think Smale gave a simple proof of something along these lines, but I could well be mixing things up. – Matthew Stover Apr 19 '10 at 16:21
$Diff(M)$ for a spherical 3-manifold $M$ is rarely path-connected. – Ryan Budney Apr 19 '10 at 16:33
Here is Darryl McCullough's paper where he computes the isometry group of lens spaces: front.math.ucdavis.edu/0010.5077 – Ryan Budney Apr 19 '10 at 16:36
Is there an easy example that isn't? – Matthew Stover Apr 19 '10 at 17:13
Ryan, how is computing isometry group of lens spaces relevant to my question? – Igor Belegradek Apr 19 '10 at 17:16

## 2 Answers

That the diffeomorphism and isometry problem is the same for spherical 3-manifolds (i.e. spherical space forms or they're also called elliptic manifolds) goes back to people like Reidemeister and Heinz Hopf if I understand correctly. But I admit I've heard a rather confusing array of names associated to this problem and I'm not certain who did what when. I've also heard Paul Olum's name associated with this. I don't think I've heard De Rham's name though.

There are a few steps: 1) Getting the list of spherical 3-manifolds, this is Heinz Hopf's work, I'm reasonably certain. 2) computing the diffeomorphism relationship among spherical space forms. This is the most subtle for lens spaces where one uses either Reidemeister or Whitehead torsion. I like the approach in:

Przytycki; Yasuhara (2003), "Symmetry of Links and Classification of Lens Spaces", Geom. Ded. 98 (1)

where they use the Alexander polynomial. Francis Bonahon also has a nice approach using Heegaard splittings, it is written up in Hatcher's 3-manifolds notes.

But the diffeomorphism classification for the spherical manifolds that are not lens spaces I think this isn't so hard. Other than the dihedral/prism manifolds, the fundamental groups are products of binary tetrahedral, binary octahedral or binary icosahedral groups with cyclic groups. But the classification of dihedral/prism fundamental groups isn't so complicated, these groups are central extensions of dihedral groups, and this central extension gives you the isomorphism type of the this class of groups.

I'm curious what others have to say.

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Milnor in his survey on Whitehead torsion attributed the result to de Rham [Complexes à automorphismes et homéomorphie différentiable Ann. Inst. Fourier Grenoble 2 (1950)], and to Franz (1935) in the lens space case. He does not restrict to dimension 3; perhaps in 3d care the references go even earlier. – Igor Belegradek Apr 19 '10 at 17:25
As far as I know, on the classification of spherical 3-manifolds, P.Scott's "The geometries of 3-manifolds" on Bull. London Math. Soc. and P.Orlik's book "Seifert manifolds" are the best literature. – X.M. Du Apr 20 '10 at 14:19

The classification of spherical space forms in any dimension is covered in Wolf's Spaces of constant curvature, which I think qualifies as "modern". Having said that, there are discrepancies between different editions, especially for the eight-dimensional space forms.

The three-dimensional case is much simpler, of course, and does not use the full machinery in Wolf's book. As stated in the question, it boils down to classifying the finite subgroups of $\mathrm{SO}(4)$ which act freely on $S^3$. As a preliminary step, one classifies the finite subgroups of the double cover $$\mathrm{Spin}(4) \cong \mathrm{Sp}(1) \times \mathrm{Sp}(1),$$ where $\mathrm{Sp}(1)$ is the group of unit-norm quaternions.

The action of $(a,b) \in \mathrm{Sp}(1) \times \mathrm{Sp}(1)$ on a unit-norm quaternion $u \in S^3 \subset \mathbb{H}$ is given by $a u \overline{b}$, whence the action has kernel the order-2 subgroup generated by $(-1,-1)$.

The classification of subgroups of $\mathrm{Sp}(1) \times \mathrm{Sp}(1)$ is an application of Goursat's lemma. The details are in Conway and Smith's On quaternions and octonions: their geometry, arithmetic, and symmetry, Section III.4.

Edit: As pointed out by Ryan below, I've answered the wrong question. That topology and geometry determine each other for three-dimensional space forms is well-known in cosmology as I tried to expain in my answer to this question. The paper Topological lensing in spherical spaces, coauthored by Jeffrey Weeks, has a nice treatment and points to two papers by Threlfall and Seifert of the early 1930s Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, which supposedly contain this result. This, of course, may not be "modern".

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The question was about the difference between the isometry and diffeomorphism classifications, not just the isometry classification. – Ryan Budney Apr 19 '10 at 17:41
"Modern" is not the point. I just want to know the simplest (mathematical) proof available. – Igor Belegradek Apr 19 '10 at 23:56