how do i define a bijective map form SU(2) to U(1) where SU(2) is the special unitary 2x2 matrix and U(1) is the unitary 1x1 matrix?
U(1) consists of complex numbers which are the inverse of their conjugate.
An element of SU(2) has the form $\displaystyle \left(\begin{array}{cc}\alpha&-\overline\beta\\ \beta&\overline\alpha\end{array}\right)$ such that the determinant is 1.
Can you finish from there?