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.

Hence f is a diffeomorphism.

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