SU(2) diffeomorphic to S^3 (3-sphere)

$SU(2)\cong S^3$ (diffeomorphic)

Proof:
We have that $\displaystyle SU(2)=\left\{\begin{pmatrix}\alpha &-\overline{\beta}\\ \beta &\overline{\alpha}\end{pmatrix}: \alpha, \beta\in\mathbb{C}, |\alpha|^2+|\beta|^2=1\right\}.$

Since $\mathbb{R}^4\cong\mathbb{C}^2$ , we may view $S^3$ as $\displaystyle S^3=\{(\alpha,\beta)\in\mathbb{C}^2: |\alpha|^2+|\beta|^2=1\}.$

Consider the map
$\begin{aligned}f: S^3&\to SU(2)\\ f(\alpha,\beta)&=\begin{pmatrix}\alpha &-\overline{\beta}\\ \beta &\overline{\alpha}\end{pmatrix}. \end{aligned}$

It is clear that $f$ is well-defined since if $(\alpha,\beta)\in S^3$ , then $f(\alpha,\beta)\in SU(2)$ .

If $f(\alpha_1,\beta_1)=f(\alpha_2,\beta_2)$ , it is clear that $(\alpha_1, \beta_1)=(\alpha_2,\beta_2)$ . So $f$ is injective. It is also clear that $f$ is surjective.

Note that $SU(2)\subseteq M(2,\mathbb{C})\cong\mathbb{R}^8$ , where $M(2,\mathbb{C})$ denotes the set of 2 by 2 complex matrices.

When $f$ is viewed as a function $\widetilde{f}: \mathbb{R}^4\to\mathbb{R}^8$ , it is clear that $\widetilde{f}$ and $\widetilde{f}^{-1}$ are smooth maps since their component functions are of class $C^\infty$ . Since $SU(2)$ and $S^3$ are submanifolds, the restrictions to these submanifolds (i.e.\ $f$ and $f^{-1}$ ) are also smooth.