HW2 MA5311: Due February 3

Exercise 1. Let $g : \mathbf{R}^2 \to \mathbf{R}^4$ , $(u,v) \to (\cos u , \sin u, \cos v, \sin v)$ . Show that $g( \mathbf{R}^2)$ is a 2-dimensional smooth manifold homeomorphic to the torus $\mathbf{S}^1 \times \mathbf{S}^1$ .

Exercise 2. Let $h_+: \mathbf{S}^2-N \to \mathbf{C}$ be the stereographic projection from the north pole $N$ , and $h_-$ be the stereographic projection from the south pole $S$ .

Show that for $z \neq 0$ , $h_+ h_-^{-1} =\frac{1}{\bar z}$ .
Show that if $P$ is a non constant polynomial, the map $f=h_+^{-1} P h_+$ , $f(N)=N$ is smooth.
More generally, if $Q: \mathbf{C} \to \mathbf{C}$ is smooth, find a condition on $Q$ so that $f=h_+^{-1} Q h_+$ , $f(N)=N$ is smooth.
By following Milnor’s argument in the proof of the fundamental theorem of algebra, find sufficient conditions so that a smooth map $Q: \mathbf{C} \to \mathbf{C}$ is onto.