## 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 $S$.

1. Show that for $z \neq 0$, $h_+ h_-^{-1} =\frac{1}{\bar z}$.
2. Show that if $P$ is a non constant polynomial, the map $f=h_+^{-1} P h_+$, $f(N)=N$ is smooth.
3. 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.
4. 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.

Exercise 3.  By using Sard’s theorem, prove that the set of regular values of a smooth map $f : M \to N$ is dense in $N$.

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