13 March 1992

revised 26 November 2004

** Definition ** A * manifold * is a locally Euclidean Hausdorff space with
a countable basis of open sets.

** Definition ** A *smooth manifold* is a manifold together with a smooth
structure. (Roughly speaking, a smooth structure defines local coordinates on
the manifold that vary smoothly.)

In particular, **R**^{n} is a smooth manifold. And it can be shown that both

S^{3} = { (x_{1},x_{2},x_{3},x_{4}) ∈ **R**^{4} : x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1 }

and

SU(2) = { A ∈ Mat(2,**C**) : A^{}*A=**I** and det{A}=1 }

are smooth manifolds. Note: Mat(2,**C**) is the set of 2×2 matrices with complex-valued entries.

** Definition ** A * diffeomorphism * between two manifolds is a smooth
homeomorphism whose inverse is also smooth. We say two manifolds are * diffeomorphic *
when there is a diffeomorphism between them.

These definitions give precise meaning to one sense in which S^{3} and SU(2)
are the same, namely, they are diffeomorphic as smooth manifolds. To give
precise meaning to the second sense in which they are the same we require two
more definitions.

** Definition ** A *Lie group* is a smooth manifold that is also an
algebraic group whose operations (composition and inversion) are smooth maps.
(A Lie group is a special case of a topological group since smooth maps are a special case of continuous maps.)

** Definition ** A *Lie isomorphism* between Lie groups is a diffeomorphism that is also a group homomorphism (and hence group isomorphism). We say two Lie groups are
*(Lie) isomorphic* if there is a Lie isomorphism between them.

The group SU(2) is a Lie subgroup of the general linear group GL(2,**C**). The manifold S^{3}, on the other hand, has no inherent group structure. But we can give it
a group structure by identifying it with the unit quaternions

S**H** = { a+b**i**+c**j**+d**k** ∈ **H** : a^{2}+b^{2}+c^{2}+d^{2}=1 },

where the
group operation is quaternion multiplication (determined by
**i**^{2}=**j**^{2}=**k**^{2}=**ijk**=-1)
and the group inverse is quaternion conjugation (defined by
γ*=a-b**i**-c**j**-d**k** where γ=a+b**i**+c**j**+d**k** ∈ **H**) since γγ*=1 for all γ ∈ S**H**.
Since these operations are smooth, S**H** is a Lie group. In particular, it is a Lie subgroup of the quaternions **H**.

With this identification of S^{3} with S**H** we can now give precise meaning
to the second sense in which S^{3} and SU(2) are the same, namely, S**H**
and SU(2) are isomorphic as Lie groups.
To prove this, we first construct a map φ : **H** → Mat(2,**C**) as follows.

First note that there is an injection **C** → **H** given by a+ib → a+b**i** and a bijection **C**^{2} ↔ **H** via the correspondence

( |
| ) | ↔ a+bi+cj-dk
= (a+bi)+j(c+di), |

( |
| ) | ↔ z_{1}+jz_{2}, |

The above correspondence **C**^{2}↔**H** allows us to use quaternion multiplication in **H** to define a linear transformation of vectors in **C**^{2}. First, observe that an element γ=w_{1}+**j**w_{2} ∈ **H** defines a transformation γ : **H** → **H** given by
z_{1}+**j**z_{2} → γ⋅(z_{1}+**j**z_{2}). Expanding this product, using the fact that w**j**=**j**w*, one finds that
γ⋅(z_{1}+**j**z_{2})
= (w_{1}+**j**w_{2})⋅(z_{1}+**j**z_{2})
= w_{1}z_{1}+**j**w_{2}**j**z_{2}+**j**w_{2}z_{1}+w_{1}**j**z_{2}
= (w_{1}z_{1}-w_{2}*z_{2})+**j**(w_{2}z_{1}+w_{1}*z_{2}).
Thus, the quaternion multiplication γ : **H** → **H** corresponds to a linear transformation φ(γ) : **C**^{2} → **C**^{2} given by

( |
| ) | → | ( |
| ). |

φ(γ) | ( |
| ) | = | ( |
| ) | = | ( |
| ) | ( |
| ). |

φ(γ) = | ( |
| ). |

**Theorem**

* SU(2) is diffeomorphic to S ^{3} and Lie-isomorphic to SH. *

**Proof**

First assume γ ∈ S**H**. Then we can easily verify that det(φ(γ)) = w_{1}*w_{1}+w_{2}*w_{2} = 1 and φ(γ)^{}*φ(γ) = **I** .
So φ(γ) ∈ SU(2). Thus, φ : S**H** → SU(2).

Conversely, let A ∈ SU(2). Then the constraints det(A)=1 and A*A=**I** together imply that A must have the form

A = | ( |
| ), |

From the above discussion it is clear that φ and φ^{-1} are smooth maps since their components are linear functions of the coordinates. Moreover, since SU(2) and S**H** are submanifolds, the restrictions of φ and φ^{-1} to these submanifolds are also smooth. Therefore, φ is a diffeomorphism, and so S^{3}=S**H** is diffeomorphic to
SU(2).

Finally, we show that φ is also a Lie isomorphism between S**H** and SU(2). First, observe that φ is a group
homomorphism. Indeed,
φ(γ*) = φ((w_{1}+**j**w_{2})*) = φ(w_{1}*-**j**w_{2}) = φ(γ)* = φ(γ)^{-1},
and
(γ_{1}γ_{2})(z_{1}+**j**z_{2})
=
(γ_{1})(γ_{2})(z_{1}+**j**z_{2}) implies that
φ(γ_{1}γ_{2}) = φ(γ_{1})φ(γ_{2}). Therefore, φ is a Lie homomorphism. And since φ is a diffeomorphism, φ is a Lie isomorphism. This completes the proof.
♦