Exercise 10, § 3

a) Prove that, for every integer $n\geqslant 2$, the group SU($n,\mathbf{C}$) is simply connected. (The space SU(2$,\mathbf{C}$) is homeomorphic to $\mathbf{S}_3$. Have SU($n,\mathbf{C}$) act on the unit sphere of $\mathbf{C}^n$, identify the stabilizer of the point $(0, . . . ,0,1)$ with the group SU($n-1,\mathbf{C}$) and use Exercise 9.)

b) Prove that for every integer $n\geqslant 2$, the group $\mathbf{S}\mathbf{L}(n,\mathbf{C})$ is simply connected. (Let $B_+$ be the subgroup of $\mathbf{S}\mathbf{L}(n,\mathbf{C})$ consisting of upper triangular matrices whose diagonal coefficients are real numbers $>0$. Prove that the mapping from SU($n,\mathbf{C}$)$\times B_+$ into $\mathbf{S}\mathbf{L}(n,\mathbf{C})$ given by $(g, u)\mapsto g\cdot u$ is a homeomorphism.)