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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