Can anyone give me an explicit isomorphism between $SU(2)$ and the three sphere?


What about for higher spheres? This question [link text](https://mathoverflow.net/questions/11628/complex-projective-space-as-a-u1-quotient) seems to indicate that there exists a homeomorphism from $SU(n)/SU(n-1)$ to the $(2n-1)$-sphere.



$SU(2)$ and the three sphere
Elements of $SU(2)$ look like this:
$$ x = \begin{pmatrix} a & - \overline{b} \\ b & \overline{a} \end{pmatrix},$$
where $|a|^2 + |b|^2 = 1$. This follows easily from $x^\* = x^{-1}$. So you map that matrix to the point $(a,b)$ in $\mathbb{C}^2$, and this is your diffeomorphism.


