Why is $SU(2)$ diffeomorphic to $S^3$? As the title,my question is why $SU(2)$ is diffeomorphic to $S^3$ ? How can I structure a diffeomorphic map between $SU(2)$ and $S^3$?

Please help me,and give me more details!

Thank you very much!
By definition, we have that  $\displaystyle SU(2)=\left\{\begin{pmatrix}a &amp;-\overline{b}\\  b &amp;\overline{a}\end{pmatrix}: a, b\in\mathbb{C}, |a|^2+|b|^2=1\right\}.$

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

Define a map from $S^3$to $SU(2)$ as:
\begin{aligned}F: S^3&amp;\to SU(2)\\  F(a,b)&amp;=\begin{pmatrix}a &amp;-\overline{b}\\  b &amp;\overline{a}\end{pmatrix}.  \end{aligned}

Can you show that $F$ is well defined, $F$ is injective, and $F$ is surjective (i.e. an isomorphism)? After that, you should show that $F$ and $F^{-1}$ are smooth (or $C^{\infty}$). (Hint: Show $\displaystyle SU(2)$ is a submanifold of of the set of 2 x 2 matrices with complex entries, or better yet $\mathbb{R}^8$. What can you say about $F$ and its inverse restricted to these submanifolds?)
Every SU2 matrix is a linear combination of Pauli matrices p1, p2, p3 + unity matrix I which is a base of the associated Lie algebra of the SU2 group.
So every SU2 matrix  X can be written a X=x1.p1 +x2.p2 +x3.p3 +x4.I and the condition that detX = 1 gives x1power2 + x2power2 +x3power2 +x4power2 = 1 which is the S3 sphere. and every point on that sphere represents an SU2 matrix